structured/0704.1694.structured.json

{"paper_meta":{"paper_id":"arxiv:0704.1694","title":"0704.1694","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"arXiv:0704.1694v1 [cs.CC] 13 Apr 2007\nLocally Decodable Codes From Nice Subsets of Finite Fields\nand Prime Factors of Mersenne Numbers\nKiran S. Kedlaya\nMIT\nkedlaya@mit.edu\nSergey Yekhanin\nMIT\nyekhanin@mit.edu\nAbstract\nA k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that\none can probabilistically recover any bit xi of the message by querying only k bits of the codeword C(x), even\nafter some constant fraction of codeword bits has been corrupted. The major goal of LDC related research is to\nestablish the optimal trade-off between length and query complexity of such codes.\nRecently [34] introduced a novel technique for constructing locally decodable codes and vastly improved the\nupper bounds for code length. The technique is based on Mersenne primes. In this paper we extend the work\nof [34] and argue that further progress via these methods is tied to progress on an old number theory question\nregarding the size of the largest prime factors of Mersenne numbers.\nSpecifically, we show that every Mersenne number m = 2t −1 that has a prime factor p > mγ yields a family\nof k(γ)-query locally decodable codes of length exp\n n1/t \n. Conversely, if for some fixed k and all ǫ > 0 one can\nuse the technique of [34] to obtain a family of k-query LDCs of length exp (nǫ) ; then infinitely many Mersenne\nnumbers have prime factors larger than known currently.\n1\nIntroduction\nClassical error-correcting codes allow one to encode an n-bit string x into in N-bit codeword C(x), in such\na way that x can still be recovered even if C(x) gets corrupted in a number of coordinates. It is well-known\nthat codewords C(x) of length N = O(n) already suffice to correct errors in up to δN locations of C(x) for\nany constant δ < 1/4. The disadvantage of classical error-correction is that one needs to consider all or most\nof the (corrupted) codeword to recover anything about x. Now suppose that one is only interested in recovering\none or a few bits of x. In such case more efficient schemes are possible. Such schemes are known as locally\ndecodable codes (LDCs). Locally decodable codes allow reconstruction of an arbitrary bit xi, from looking only\nat k randomly chosen coordinates of C(x), where k can be as small as 2. Locally decodable codes have numerous\napplications in complexity theory [15, 29], cryptography [6, 11] and the theory of fault tolerant computation [24].\nBelow is a slightly informal definition of LDCs:\nA (k, δ, ǫ)-locally decodable code encodes n-bit strings to N-bit codewords C(x), such that for every i ∈[n],\nthe bit xi can be recovered with probability 1−ǫ, by a randomized decoding procedure that makes only k queries,\neven if the codeword C(x) is corrupted in up to δN locations.\nOne should think of δ > 0 and ǫ < 1/2 as constants. The main parameters of interest in LDCs are the length\nN and the query complexity k. Ideally we would like to have both of them as small as possible. The concept\nof locally decodable codes was explicitly discussed in various papers in the early 1990s [2, 28, 21]. Katz and\n\nTrevisan [15] were the first to provide a formal definition of LDCs. Further work on locally decodable codes\nincludes [3, 8, 20, 4, 16, 30, 34, 33, 14, 23].\nBelow is a brief summary of what was known regarding the length of LDCs prior to [34]. The length of optimal\n2-query LDCs was settled by Kerenidis and de Wolf in [16] and is exp(n).1 The best upper bound for the length\nof 3-query LDCs was exp\n n1/2 \ndue to Beimel et al. [3], and the best lower bound is ̃Ω(n2) [33]. For general\n(constant) k the best upper bound was exp\n nO(log log k/(k log k)) \ndue to Beimel et al. [4] and the best lower bound\nis ̃Ω\n n1+1/(⌈k/2⌉−1) \n[33].\nThe recent work [34] improved the upper bounds to the extent that it changed the common perception of what\nmay be achievable [12, 11]. [34] introduced a novel technique to construct codes from so-called nice subsets\nof finite fields and showed that every Mersenne prime p = 2t −1 yields a family of 3-query LDCs of length\nexp\n n1/t \n. Based on the largest known Mersenne prime [9], this translates to a length of less than exp\n \nn10−7 \n.\nCombined with the recursive construction from [4], this result yields vast improvements for all values of k > 2. It\nhas often been conjectured that the number of Mersenne primes is infinite. If indeed this conjecture holds, [34] gets\nthree query locally decodable codes of length N = exp\n \nnO\n“\n1\nlog log n\n” \nfor infinitely many n. Finally, assuming\nthat the conjecture of Lenstra, Pomerance and Wagstaff [31, 22, 32] regarding the density of Mersenne primes\nholds, [34] gets three query locally decodable codes of length N = exp\n \nn\nO\n“\n1\nlog1−ǫ log n\n” \nfor all n, for every ǫ >\n0.\n1.1\nOur results\nIn this paper we address two natural questions left open by [34]:\n1. Are Mersenne primes necessary for the constructions of [34]?\n2. Has the technique of [34] been pushed to its limits, or one can construct better codes through a more clever\nchoice of nice subsets of finite fields?\nWe extend the work of [34] and answer both of the questions above. In what follows let P(m) denote the\nlargest prime factor of m. We show that one does not necessarily need to use Mersenne primes. It suffices to have\nMersenne numbers with polynomially large prime factors. Specifically, every Mersenne number m = 2t −1 such\nthat P(m) ≥mγ yields a family of k(γ)-query locally decodable codes of length exp\n n1/t \n. A partial converse\nalso holds. Namely, if for some fixed k ≥3 and all ǫ > 0 one can use the technique of [34] to (unconditionally)\nobtain a family of k-query LDCs of length exp (nǫ) ; then for infinitely many t we have\nP(2t −1) ≥(t/2)1+1/(k−2).\n(1)\nThe bound (1) may seem quite weak in light of the widely accepted conjecture saying that the number of\nMersenne primes is infinite. However (for any k ≥3) this bound is substantially stronger than what is currently\nknown unconditionally. Lower bounds for P(2t −1) have received a considerable amount of attention in the\nnumber theory literature [25, 26, 10, 27, 19, 18]. The strongest result to date is due to Stewart [27]. It says that\nfor all integers t ignoring a set of asymptotic density zero, and for all functions ǫ(t) > 0 where ǫ(t) tends to zero\nmonotonically and arbitrarily slowly:\nP(2t −1) > ǫ(t)t (log t)2 / log log t.\n(2)\n1Throughout the paper we use the standard notation exp(x)\ndef\n= eO(x).\n\nThere are no better bounds known to hold for infinitely many values of t, unless one is willing to accept some\nnumber theoretic conjectures [19, 18]. We hope that our work will further stimulate the interest in proving lower\nbounds for P(2t −1) in the number theory community.\nIn summary, we show that one may be able to improve the unconditional bounds of [34] (say, by discovering a\nnew Mersenne number with a very large prime factor) using the same technique. However any attempts to reach\nthe exp (nǫ) length for some fixed query complexity and all ǫ > 0 require either progress on an old number theory\nproblem or some radically new ideas.\nIn this paper we deal only with binary codes for the sake of clarity of presentation. We remark however that\nour results as well as the results of [34] can be easily generalized to larger alphabets. Such generalization will be\ndiscussed in detail in [35].\n1.2\nOutline\nIn section 3 we introduce the key concepts of [34], namely that of combinatorial and algebraic niceness of\nsubsets of finite fields. We also briefly review the construction of locally decodable codes from nice subsets. In\nsection 4 we show how Mersenne numbers with large prime factors yield nice subsets of prime fields. In section 5\nwe prove a partial converse. Namely, we show that every finite field Fq containing a sufficiently nice subset, is an\nextension of a prime field Fp, where p is a large prime factor of a large Mersenne number. Our main results are\nsummarized in sections 4.3 and 5.4.\n2\nNotation\nWe use the following standard mathematical notation:\n• [s] = {1, . . . , s};\n• Zn denotes integers modulo n;\n• Fq is a finite field of q elements;\n• dH(x, y) denotes the Hamming distance between binary vectors x and y;\n• (u, v) stands for the dot product of vectors u and v;\n• For a linear space L ⊆Fm\n2 , L⊥denotes the dual space. That is, L⊥= {u ∈Fm\n2 | ∀v ∈L, (u, v) = 0};\n• For an odd prime p, ord2(p) denotes the smallest integer t such that p | 2t −1.\n3\nNice subsets of finite fields and locally decodable codes\nIn this section we introduce the key technical concepts of [34], namely that of combinatorial and algebraic\nniceness of subsets of finite fields. We briefly review the construction of locally decodable codes from nice\nsubsets. Our review is concise although self-contained. We refer the reader interested in a more detailed and\nintuitive treatment of the construction to the original paper [34]. We start by formally defining locally decodable\ncodes.\nDefinition 1 A binary code C : {0, 1}n →{0, 1}N is said to be (k, δ, ǫ)-locally decodable if there exists a\nrandomized decoding algorithm A such that\n\n1. For all x ∈{0, 1}n, i ∈[n] and y ∈{0, 1}N such that dH(C(x), y) ≤δN : Pr[Ay(i) = xi] ≥1−ǫ, where\nthe probability is taken over the random coin tosses of the algorithm A.\n2. A makes at most k queries to y.\nWe now introduce the concepts of combinatorial and algebraic niceness of subsets of finite fields. Our defini-\ntions are syntactically slightly different from the original definitions in [34]. We prefer these formulations since\nthey are more appropriate for the purposes of the current paper. In what follows let F∗\nq denote the multiplicative\ngroup of Fq.\nDefinition 2 A set S ⊆F∗\nq is called t combinatorially nice if for some constant c > 0 and every positive integer\nm there exist two n = ⌊cmt⌋-sized collections of vectors {u1, . . . , un} and {v1, . . . , vn} in Fm\nq , such that\n• For all i ∈[n], (ui, vi) = 0;\n• For all i, j ∈[n] such that i ̸= j, (uj, vi) ∈S.\nDefinition 3 A set S ⊆F∗\nq is called k algebraically nice if k is odd and there exists an odd k′ ≤k and two sets\nS0, S1 ⊆Fq such that\n• S0 is not empty;\n• |S1| = k′;\n• For all α ∈Fq and β ∈S : |S0 ∩(α + βS1)| ≡0 mod (2).\nThe following lemma shows that for an algebraically nice set S, the set S0 can always be chosen to be large. It\nis a straightforward generalization of [34, lemma 15].\nLemma 4 Let S ⊆F∗\nq be a k algebraically nice set. Let S0, S1 ⊆Fq be sets from the definition of algebraic\nniceness of S. One can always redefine the set S0 to satisfy |S0| ≥⌈q/2⌉.\nProof:\nLet L be the linear subspace of Fq\n2 spanned by the incidence vectors of the sets α + βS1, for α ∈Fq and\nβ ∈S. Observe that L is invariant under the actions of a 1-transitive permutation group (permuting the coordinates\nin accordance with addition in Fq). This implies that the space L⊥is also invariant under the actions of the same\ngroup. Note that L⊥has positive dimension since it contains the incidence vector of the set S0. The last two\nobservations imply that L⊥has full support, i.e., for every i ∈[q] there exists a vector v ∈L⊥such that vi ̸= 0. It\nis easy to verify that any linear subspace of Fq\n2 that has full support contains a vector of Hamming weight at least\n⌈q/2⌉. Let v ∈L⊥be such a vector. Redefining the set S0 to be the set of nonzero coordinates of v we conclude\nthe proof.\nWe now proceed to the core proposition of [34] that shows how sets exhibiting both combinatorial and algebraic\nniceness yield locally decodable codes.\nProposition 5 Suppose S ⊆F∗\nq is t combinatorially nice and k algebraically nice; then for every positive integer\nn there exists a code of length exp(n1/t) that is (k, δ, 2kδ) locally decodable for all δ > 0.\nProof:\nOur proof comes in three steps. We specify encoding and local decoding procedures for our codes and\nthen argue the lower bound for the probability of correct decoding. We use the notation from definitions 2 and 3.\nEncoding: We assume that our message has length n = ⌊cmt⌋for some value of m. (Otherwise we pad the\nmessage with zeros. It is easy to see that such padding does not not affect the asymptotic length of the code.) Our\n\ncode will be linear. Therefore it suffices to specify the encoding of unit vectors e1, . . . , en, where ej has length n\nand a unique non-zero coordinate j. We define the encoding of ej to be a qm long vector, whose coordinates are\nlabelled by elements of Fm\nq . For all w ∈Fm\nq we set:\nEnc(ej)w =\n 1,\nif (uj, w) ∈S0;\n0,\notherwise.\n(3)\nIt is straightforward to verify that we defined a code encoding n bits to exp(n1/t) bits.\nLocal decoding: Given a (possibly corrupted) codeword y and an index i ∈[n], the decoding algorithm A picks\nw ∈Fm\nq , such that (ui, w) ∈S0 uniformly at random, reads k′ ≤k coordinates of y, and outputs the sum:\nX\nλ∈S1\nyw+λvi.\n(4)\nProbability of correct decoding: First we argue that decoding is always correct if A picks w ∈Fm\nq such that\nall bits of y in locations {w + λvi}λ∈S1 are not corrupted. We need to show that for all i ∈[n], x ∈{0, 1}n and\nw ∈Fm\nq , such that (ui, w) ∈S0:\nX\nλ∈S1\n \n \nn\nX\nj=1\nxj Enc(ej)\n \n \nw+λvi\n= xi.\n(5)\nNote that\nX\nλ∈S1\n \n \nn\nX\nj=1\nxj Enc(ej)\n \n \nw+λvi\n=\nn\nX\nj=1\nxj\nX\nλ∈S1\nEnc(ej)w+λvi =\nn\nX\nj=1\nxj\nX\nλ∈S1\nI [(uj, w + λvi) ∈S0] ,\n(6)\nwhere I[γ ∈S0] = 1 if γ ∈S0 and zero otherwise. Now note that\nX\nλ∈S1\nI [(uj, w + λvi) ∈S0] =\nX\nλ∈S1\nI [(uj, w) + λ(uj, vi) ∈S0] =\n 1,\nif i = j,\n0,\notherwise.\n(7)\nThe last identity in (7) for i = j follows from: (ui, vi) = 0, (ui, w) ∈S0 and k′ = |S1| is odd. The last identity\nfor i ̸= j follows from (uj, vi) ∈S and the algebraic niceness of S. Combining identities (6) and (7) we get (5).\nNow assume that up to δ fraction of bits of y are corrupted. Let Ti denote the set of coordinates whose labels\nbelong to\n \nw ∈Fm\nq | (ui, w) ∈S0\n \n. Recall that by lemma 4, |Ti| ≥qm/2. Thus at most 2δ fraction of coor-\ndinates in Ti contain corrupted bits. Let Qi =\n \n{w + λvi}λ∈S1 | w : (ui, w) ∈S0\n \nbe the family of k′-tuples\nof coordinates that may be queried by A. (ui, vi) = 0 implies that elements of Qi uniformly cover the set Ti.\nCombining the last two observations we conclude that with probability at least 1 −2kδ A picks an uncorrupted\nk′-tuple and outputs the correct value of xi.\nAll locally decodable codes constructed in this paper are obtained by applying proposition 5 to certain nice\nsets. Thus all our codes have the same dependence of ǫ (the probability of the decoding error) on δ (the fraction\nof corrupted bits). In what follows we often ignore these parameters and consider only the length and query\ncomplexity of codes.\n\n4\nMersenne numbers with large prime factors yield nice subsets of prime fields\nIn what follows let ⟨2⟩⊆F∗\np denote the multiplicative subgroup of F∗\np generated by 2. In [34] it is shown\nthat for every Mersenne prime p = 2t −1 the set ⟨2⟩⊆F∗\np is simultaneously 3 algebraically nice and ord2(p)\ncombinatorially nice. In this section we prove the same conclusion for a substantially broader class of primes.\nLemma 6 Suppose p is an odd prime; then ⟨2⟩⊆F∗\np is ord2(p) combinatorially nice.\nProof: Let t = ord2(p). Clearly, t divides p −1. We need to specify a constant c > 0 such that for every positive\ninteger m there exist two n = ⌊cmt⌋-sized collections of m long vectors over Fp satisfying:\n• For all i ∈[n], (ui, vi) = 0;\n• For all i, j ∈[n] such that i ̸= j, (uj, vi) ∈⟨2⟩.\nFirst assume that m has the shape m =\n \nm′−1+(p−1)/t\n(p−1)/t\n \n, for some integer m′ ≥p −1. In this case [34, lemma\n13] gives us a collection of n =\n \nm′\np−1\n \nvectors with the right properties. Observe that n ≥cmt for a constant\nc that depends only on p and t. Now assume m does not have the right shape, and let m1 be the largest integer\nsmaller than m that does have it. In order to get vectors of length m we use vectors of length m1 coming from [34,\nlemma 13] padded with zeros. It is not hard to verify such a construction still gives us n ≥cmt large families of\nvectors for a suitably chosen constant c.\nWe use the standard notation F to denote the algebraic closure of the field F. Also let Cp ⊆F\n∗\n2 denote the\nmultiplicative subgroup of p-th roots of unity in F2. The next lemma generalizes [34, lemma 14].\nLemma 7 Let p be a prime and k be odd. Suppose there exist ζ1, . . . , ζk ∈Cp such that\nζ1 + . . . + ζk = 0;\n(8)\nthen ⟨2⟩⊆F∗\np is k algebraically nice.\nProof:\nIn what follows we define the set S1 ⊆Fp and prove the existence of a set S0 such that that together S0\nand S1 yield k algebraic niceness of ⟨2⟩. Identity 8 implies that there exists an odd integer k′ ≤k and k′ distinct\np-th roots of unity ζ′\n1, . . . , ζ′\nk ∈Cp such that\nζ′\n1 + . . . + ζ′\nk′ = 0.\n(9)\nLet t = ord2(p). Observe that Cp ⊆F2t. Let g be a generator of Cp. Identity (9) yields gγ1 + . . . + gγk′ = 0, for\nsome distinct values of {γi}i∈[k′]. Set S1 = {γ1, . . . , γk′}.\nConsider a natural one to one correspondence between subsets S′ of Fp and polynomials φS′(x) in the ring\nF2[x]/(xp −1) : φS′(x) = P\ns∈S′ xs. It is easy to see that for all sets S′ ⊆Fp and all α, β ∈Fp, such that β ̸= 0 :\nφα+βS′(x) = xαφS′(xβ).\nLet α be a variable ranging over Fp and β be a variable ranging over ⟨2⟩. We are going to argue the existence of a\nset S0 that has even intersections with all sets of the form α+βS1, by showing that all polynomials φα+βS1 belong\nto a certain linear space L ∈F2[x]/(xp −1) of dimension less than p. In this case any nonempty set T ⊆Fp such\nthat φT ∈L⊥can be used as the set S0. Let τ(x) = gcd(xp −1, φS1(x)). Note that τ(x) ̸= 1 since g is a common\nroot of xp −1 and φS1(x). Let L be the space of polynomials in F2[x]/(xp −1) that are multiples of τ(x). Clearly,\ndim L = p −deg τ. Fix some α ∈Fp and β ∈⟨2⟩. Let us prove that φα+βS1(x) is in L :\nφα+βS1(x) = xαφS1(xβ) = xα(φS1(x))β.\nThe last identity above follows from the fact that for any f ∈F2[x] and any integer i : f(x2i) = (f(x))2i.\n\nIn what follows we present sufficient conditions for the existence of k-tuples of p-th roots of unity in F2 that\nsum to zero. We treat the k = 3 case separately since in that case we can use a specialized argument to derive a\nmore explicit conclusion.\n4.1\nA sufficient condition for the existence of three p-th roots of unity summing to zero\nLemma 8 Let p be an odd prime. Suppose ord2(p) < (4/3) log2 p; then there exist three p-th roots of unity in F2\nthat sum to zero.\nProof: We start with a brief review of some basic concepts of projective algebraic geometry. Let F be a field, and\nf ∈F[x, y, z] be a homogeneous polynomial. A triple (x0, y0, z0) ∈F3 is called a zero of f if f(x0, y0, z0) = 0.\nA zero is called nontrivial if it is different from the origin. An equation f = 0 defines a projective plane curve χf.\nNontrivial zeros of f considered up to multiplication by a scalars are called F-rational points of χf. If F is a finite\nfield it makes sense to talk about the number of F-rational points on a curve.\nLet t = ord2(p). Note that Cp ⊆F2t. Consider a projective plane Fermat curve χ defined by\nx(2t−1)/p + y(2t−1)/p + z(2t−1)/p = 0.\n(10)\nLet us call a point a on χ trivial if one of the coordinates of a is zero. Cyclicity of F∗\n2t implies that χ contains\nexactly 3(2t −1)/p trivial F2t-rational points. Note that every nontrivial point of χ yields a triple of elements of\nCp that sum to zero. The classical Weil bound [17, p. 330] provides an estimate\n|Nq −(q + 1)| ≤(d −1)(d −2)√q\n(11)\nfor the number Nq of Fq-rational points on an arbitrary smooth projective plane curve of degree d. (11) implies\nthat in case\n2t + 1 >\n 2t −1\np\n−1\n 2t −1\np\n−2\n \n2t/2 + 32t −1\np\n(12)\nthere exists a nontrivial point on the curve (10). Note that (12) follows from\n2t + 1 >\n 2t\np\n 2t\np\n \n2t/2 −23t/2+1\np\n+ 3 ∗2t\np\n,\n(13)\nand (13) follows from\n2t > 22t+t/2/p2\nand\n2t/2+1 > 3.\nNow note that the first inequality above follows from t < (4/3) log2 p and the second follows from t > 1.\nNote that the constant 4/3 in lemma 8 cannot be improved to 2: there are no three elements of C13264529 that\nsum to zero, even though ord2(13264529) = 47 < 2 ∗log2 13264529 ≈47.3.\n4.2\nA sufficient condition for the existence of k p-th roots of unity summing to zero\nOur argument in this section comes in three steps. First we briefly review the notion of (additive) Fourier\ncoefficients of subsets of F2t. Next, we invoke a folklore argument to show that subsets of F2t with appropriately\nsmall nontrivial Fourier coefficients contain k-tuples of elements that sum to zero. Finally, we use a recent result of\nBourgain and Chang [5] (generalizing the classical estimate for Gauss sums) to argue that (under certain constraints\non p) all nontrivial Fourier coefficients of Cp are small.\nFor x ∈F2t let Tr(x) = x + x2 + . . . + x2t−1 denote the trace of x. It is not hard to verify that for all x,\nTr(x) ∈F2. Characters of F2t are homomorphisms from the additive group of F2t into the multiplicative group\n\n{±1}. There exist 2t characters. We denote characters by χa, where a ranges in F2t, and set χa(x) = (−1)Tr(ax).\nLet C(x) denote the incidence function of a set C ⊆F2t. For arbitrary a ∈Ft\n2 the Fourier coefficient χa(C) is\ndefined by χa(C) = P χa(x)C(x), where the sum is over all x ∈F2t. Fourier coefficient χ0(C) = |C| is called\ntrivial, and other Fourier coefficients are called nontrivial. In what follows P\nχ stands for summation over all 2t\ncharacters of F2t. We need the following two standard properties of characters and Fourier coefficients.\nX\nχ\nχ(x) =\n 2t,\nif x = 0;\n0,\notherwise.\n(14)\nX\nχ\nχ2(C) = 2t|C|.\n(15)\nThe following lemma is a folklore.\nLemma 9 Let C ⊆F2t and k ≥3 be a positive integer. Let F be the largest absolute value of a nontrivial Fourier\ncoefficient of C. Suppose\nF\n|C| <\n |C|\n2t\n 1/(k−2)\n(16)\nthen there exist k elements of C that sum to zero.\nProof:\nLet M(C) = # {ζ1, . . . , ζk ∈C | ζ1 + . . . + ζk = 0} . (14) yields\nM(C) = 1\n2t\nX\nx1,...,xk∈F2t\nC(x1) . . . C(xk)\nX\nχ\nχ(x1 + . . . + xk).\n(17)\nNote that χ(x1 + . . . + xk) = χ(x1) . . . χ(xk). Changing the order of summation in (17) we get\nM(C) = 1\n2t\nX\nχ\nX\nx1,...,xk∈F2t\nC(x1) . . . C(xk)χ(x1) . . . χ(xk) = 1\n2t\nX\nχ\nχk(C).\n(18)\nNote that\n1\n2t\nX\nχ\nχk(C) = |C|k\n2t\n+ 1\n2t\nX\nχ̸=χ0\nχk(C) ≥|C|k\n2t\n−F k−2 1\n2t\nX\nχ\nχ2(C) = |C|k\n2t\n−F k−2|C|,\n(19)\nwhere the last identity follows from (15). Combining (18) and (19) we conclude that (16) implies M(C) > 0.\nThe following lemma is a special case of [5, theorem 1].\nLemma 10 Assume that n | 2t −1 and satisfies the condition\ngcd\n \nn, 2t −1\n2t′ −1\n \n< 2t(1−ǫ)−t′,\nfor all 1 ≤t′ < t, t′ | t,\nwhere ǫ > 0 is arbitrary and fixed. Then for all a ∈F∗\n2t\n \nX\nx∈F2t\n(−1)Tr(axn)\n \n< c12t(1−δ),\n(20)\nwhere δ = δ(ǫ) > 0 and c1 = c1(ǫ) are absolute constants.\n\nBelow is the main result of this section. Recall that Cp denotes the set of p-th roots of unity in F2.\nLemma 11 For every c > 0 there exists an odd integer k = k(c) such that the following implication holds. If p is\nan odd prime and ord2(p) < c log2 p then some k elements of Cp sum to zero.\nProof:\nNote that if there exist k′ elements of a set C ⊆F2 that sum to zero, where k′ is odd; then there exist\nk elements of C that sum to zero for every odd k ≥k′. Also note that the sum of all p-th roots of unity is\nzero. Therefore given c it suffices to prove the existence of an odd k = k(c) that works for all sufficiently large\np. Let t = ord2(p). Observe that p > 2t/c. Assume p is sufficiently large so that t > 2c. Next we show that\nthe precondition of lemma 10 holds for n = (2t −1)/p and ǫ = 1/(2c). Let t′ | t and 1 ≤t′ < t. Clearly\ngcd(2t′ −1, p) = 1. Therefore\ngcd\n 2t −1\np\n, 2t −1\n2t′ −1\n \n=\n2t −1\np(2t′ −1) < 2t(1−1/c)\n2t′ −1 ,\n(21)\nwhere the inequality follows from p > 2t/c. Clearly, t > 2c yields 2t/(2c)/2 > 1. Multiplying the right hand side\nof (21) by 2t/(2c)/2 and using 2(2t′ −1) > 2t′ we get\ngcd\n 2t −1\np\n, 2t −1\n2t′ −1\n \n< 2t(1−1/(2c))−t′.\n(22)\nCombining (22) with lemma 10 we conclude that there exist δ > 0 and c1 such that for all a ∈F∗\n2t\n \nX\nx∈F2t\n(−1)Tr\n“\nax(2t−1)/p” \n< c12t(1−δ).\n(23)\nObserve that x(2t−1)/p takes every value in Cp exactly (2t −1)/p times when x ranges over F∗\n2t. Thus (23) implies\n(2t −1)(F/p) < c12t(1−δ),\n(24)\nwhere F denotes that largest nontrivial Fourier coefficient of Cp. (24) yields F/p < (2c1)2−δt. Pick k ≥3 to be\nthe smallest odd integer such that (1 −1/c)/(k −2) < δ. We now have\nF\np < 2−(1−1/c)t\n(k−2)\n(25)\nfor all sufficiently large values of p. Combining p > 2t/c with (25) we get\nF\n|Cp| <\n |Cp|\n2t\n 1/(k−2)\n,\nand the application of lemma 9 concludes the proof.\n4.3\nSummary\nIn this section we summarize our positive results and show that one does not necessarily need to use Mersenne\nprimes to construct locally decodable codes via the methods of [34]. It suffices to have Mersenne numbers with\npolynomially large prime factors. Recall that P(m) denotes the largest prime factor of an integer m. Our first\ntheorem gets 3-query LDCs from Mersenne numbers m with prime factors larger than m3/4.\n\nTheorem 12 Suppose P(2t −1) > 20.75t; then for every message length n there exists a three query locally\ndecodable code of length exp(n1/t).\nProof:\nLet P(2t −1) = p. Observe that p | 2t −1 and p > 20.75t yield ord2(p) < (4/3) log2 p. Combining\nlemmas 8,7 and 6 with proposition 5 we obtain the statement of the theorem.\nAs an example application of theorem 12 one can observe that P(223 −1) = 178481 > 2(3/4)∗23 ≈155872 yields\na family of three query locally decodable codes of length exp(n1/23). Theorem 12 immediately yields:\nTheorem 13 Suppose for infinitely many t we have P(2t −1) > 20.75t; then for every ǫ > 0 there exists a family\nof three query locally decodable codes of length exp(nǫ).\nThe next theorem gets constant query LDCs from Mersenne numbers m with prime factors larger than mγ for\nevery value of γ.\nTheorem 14 For every γ > 0 there exists an odd integer k = k(γ) such that the following implication holds.\nSuppose P(2t −1) > 2γt; then for every message length n there exists a k query locally decodable code of length\nexp(n1/t).\nProof:\nLet P(2t −1) = p. Observe that p | 2t −1 and p > 2γt yield ord2(p) < (1/γ) log2 p. Combining\nlemmas 22,7 and 6 with proposition 5 we obtain the statement of the theorem.\nAs an immediate corollary we get:\nTheorem 15 Suppose for some γ > 0 and infinitely many t we have P(2t −1) > 2γt; then there is a fixed k such\nthat for every ǫ > 0 there exists a family of k query locally decodable codes of length exp(nǫ).\n5\nNice subsets of finite fields yield Mersenne numbers with large prime factors\nDefinition 16 We say that a sequence\n \nSi ⊆F∗\nqi\n \ni≥1 of subsets of finite fields is k-nice if every Si is k alge-\nbraically nice and t(i) combinatorially nice, for some integer valued monotonically increasing function t.\nThe core proposition 5 asserts that a subset S ⊆F∗\nq that is k algebraically nice and t combinatorially nice yields\na family of k-query locally decodable codes of length exp(n1/t). Clearly, to get k-query LDCs of length exp(nǫ)\nfor some fixed k and every ǫ > 0 via this proposition, one needs to exhibit a k-nice sequence. In this section\nwe show how the existence of a k-nice sequence implies that infinitely many Mersenne numbers have large prime\nfactors. Our argument proceeds in two steps. First we show that a k-nice sequence yields an infinite sequence of\nprimes {pi}i≥1 , where every Cpi contains a k-tuple of elements summing to zero. Next we show that Cp contains\na short additive dependence only if p is a large factor of a Mersenne number.\n5.1\nA nice sequence yields infinitely many primes p with short dependencies between p-th roots of unity\nWe start with some notation. Consider a a finite field Fq = Fpl, where p is prime. Fix a basis e1, . . . , el of Fq\nover Fp. In what follows we often write (α1, . . . , αl) ∈Fl\np to denote α = Pl\ni=1 αiei ∈Fq. Let R denote the ring\nF2[x1, . . . , xl]/(xp\n1 −1, . . . , xp\nl −1). Consider a natural one to one correspondence between subsets S1 of Fq and\npolynomials φS1(x1, . . . , xl) ∈R.\nφS1(x1, . . . , xl) =\nX\n(α1,...,αl)∈S1\nxα1\n1 . . . xαl\nl .\n\nIt is easy to see that for all sets S1 ⊆Fq and all α, β ∈Fq :\nφ(α1,...,αl)+βS1(x1, . . . , xl) = xα1\n1 . . . xαl\nl φβS1(x1, . . . , xl).\n(26)\nLet Γ be a family of subsets of Fq. It is straightforward to verify that a set S0 ⊆Fq has even intersections with\nevery element of Γ if and only if φS0 belongs to L⊥, where L is the linear subspace of R spanned by {φS1}S1∈Γ .\nCombining the last observation with formula (26) we conclude that a set S ⊆F∗\nq is k algebraically nice if and\nonly if there exists a set S1 ⊆Fq of odd size k′ ≤k such that the ideal generated by polynomials {φβS1}{β∈S}\nis a proper ideal of R. Note that polynomials {f1, . . . , fh} ∈R generate a proper ideal if an only if polynomials\n{f1, . . . , fh, xp\n1 −1, . . . , xp\nl −1} generate a proper ideal in F2[x1, . . . , xl]. Also note that a family of polynomials\ngenerates a proper ideal in F2[x1, . . . , xl] if and only if it generates a proper ideal in F2[x1, . . . , xl]. Now an\napplication of Hilbert’s Nullstellensatz [7, p. 168] implies that a set S ⊆F∗\nq is k algebraically nice if and only\nif there is a set S1 ⊆Fq of odd size k′ ≤k such that the polynomials {φβS1}{β∈S} and {xp\ni −1}1≤i≤l have a\ncommon root in F2.\nLemma 17 Let Fq = Fpl, where p is prime. Suppose Fq contains a nonempty k algebraically nice subset; then\nthere exist ζ1, . . . , ζk ∈Cp such that ζ1 + . . . + ζk = 0.\nProof:\nAssume S ⊆F∗\nq is nonempty and k algebraically nice. The discussion above implies that there exists\nS1 ⊆Fq of odd size k′ ≤k such that all polynomials {φβS1}{β∈S} vanish at some (ζ1, . . . , ζl) ∈Cl\np. Fix an\narbitrary β0 ∈S, and note that Cp is closed under multiplication. Thus,\nφβ0S1(ζ1, . . . , ζl) = 0\n(27)\nyields k′ p-th roots of unity that add up to zero. It is readily seen that one can extend (27) (by adding an appropriate\nnumber of pairs of identical roots) to obtain k p-th roots of unity that add up to zero for any odd k ≥k′.\nNote that lemma 17 does not suffice to prove that a k-nice sequence\n \nSi ⊆F∗\nqi\n \ni≥1 yields infinitely many primes p\nwith short (nontrivial) additive dependencies in Cp. We need to argue that the set {charFqi}i≥1 can not be finite. To\nproceed, we need some more notation. Recall that q = pl and p is prime. For x ∈Fq let Tr(x) = x+. . .+xpl−1 ∈\nFp denote the (absolute) trace of x. For γ ∈Fq, c ∈F∗\np we call the set πγ,c = {x ∈Fq | Tr(γx) = c} a proper\naffine hyperplane of Fq.\nLemma 18 Let Fq = Fpl, where p is prime. Suppose S ⊆F∗\nq is k algebraically nice; then there exist h ≤pk\nproper affine hyperplanes {πγi,ci}1≤i≤h of Fq such that S ⊆\nhS\ni=1\nπγi,ci.\nProof:\nDiscussion preceding lemma 17 implies that there exists a set S1 = {σ1, . . . , σk′} ⊆Fq of odd size\nk′ ≤k such that all polynomials {φβS1}{β∈S} vanish at some (ζ1, . . . , ζl) ∈Cl\np. Let ζ be a generator of Cp. For\nevery 1 ≤i ≤l pick ωi ∈Zp such that ζi = ζωi. For every β ∈S, φβS1(ζ1, . . . , ζl) = 0 yields\nX\nμ=(μ1,...,μl)∈βS1\nζ\nPl\ni=1 μiωi = 0.\n(28)\nObserve that for fixed values {ωi}1≤i≤l ∈Zp the map D(μ) = Pl\ni=1 μiωi is a linear map from Fq to Fp. It is\nnot hard to prove that every such map can be expressed as D(μ) = Tr(δμ) for an appropriate choice of δ ∈Fq.\nTherefore we can rewrite (28) as\nX\nμ∈βS1\nζTr(δμ) =\nX\nσ∈S1\nζTr(δβσ) = 0.\n(29)\n\nLet W =\nn\n(w1, . . . , wk′) ∈Zk′\np | ζw1 + . . . + ζwk′ = 0\no\ndenote the set of exponents of k′-dependencies be-\ntween powers of ζ. Clearly, |W| ≤pk. Identity (29) implies that every β ∈S satisfies\n \n \n \n \n \nTr((δσ1)β)\n=\nw1,\n...\nTr((δσk′)β)\n=\nwk′;\n(30)\nfor an appropriate choice of (w1, . . . , wk′) ∈W. Note that the all-zeros vector does not lie in W since k′ is odd.\nTherefore at least one of the identities in (30) has a non-zero right-hand side, and defines a proper affine hyperplane\nof Fq. Collecting one such hyperplane for every element of W we get a family of |W| proper affine hyperplanes\ncontaining every element of S.\nLemma 18 gives us some insight into the structure of algebraically nice subsets of Fq. Our next goal is to develop\nan insight into the structure of combinatorially nice subsets. We start by reviewing some relations between tensor\nand dot products of vectors. For vectors u ∈Fm\nq and v ∈Fn\nq let u⊗v ∈Fmn\nq\ndenote the tensor product of u and v.\nCoordinates of u ⊗v are labelled by all possible elements of [m] × [n] and (u ⊗v)i,j = uivj. Also, let u⊗l denote\nthe l-the tensor power of u and u ◦v denote the concatenation of u and v. The following identity is standard. For\nany u, x ∈Fm\nq and v, y ∈Fn\nq :\n(u ⊗v, x ⊗y) =\nX\ni∈[m],j∈[n]\nuivjxiyj =\n \n X\ni∈[m]\nuixi\n \n \n \n X\nj∈[n]\nvjyj\n \n = (u, x)(v, y).\n(31)\nIn what follows we need a generalization of identity (31). Let f(x1, . . . , xh) = P\ni cixαi\n1\n1 . . . x\nαi\nh\nh be a polynomial\nin Fq[x1, . . . , xh]. Given f we define ̄f ∈Fq[x1, . . . , xh] by ̄f = P\ni xαi\n1\n1 . . . x\nαi\nh\nh , i.e., we simply set all nonzero\ncoefficients of f to 1. For vectors u1, . . . , uh in Fm\nq define\nf(u1, . . . , uh) = ◦i ciu⊗αi\n1\n1\n⊗. . . ⊗u\n⊗αi\nh\nh\n.\n(32)\nNote that to obtain f(u1, . . . , uh) we replaced products in f by tensor products and addition by concatenation.\nClearly, f(u1, . . . , uh) is a vector whose length may be larger than m.\nClaim 19 For every f ∈Fq[x1, . . . , xh] and u1, . . . , uh, v1, . . . , vh ∈Fm\nq :\n f(u1, . . . , uh), ̄f(v1, . . . , vh)\n \n= f((u1, v1), . . . , (uh, vh)).\n(33)\nProof: Let u = (u1, . . . , uh) and v = (v1, . . . , vh). Observe that if (33) holds for polynomials f1 and f2 defined\nover disjoint sets of monomials then it also holds for f = f1 + f2 :\n f(u), ̄f(v)\n \n=\n (f1 + f2)(u), ( ̄f1 + ̄f2)(v)\n \n=\n f1(u) ◦f2(u), ̄f1(v) ◦ ̄f2(v)\n \n=\nf1 ((u1, v1), . . . , (uh, vh)) + f2 ((u1, v1), . . . , (uh, vh)) = f ((u1, v1), . . . , (uh, vh)) .\nTherefore it suffices to prove (33) for monomials f = cxα1\n1 . . . xαh\nh . It remains to notice identity (33) for monomi-\nals f = cxα1\n1 . . . xαh\nh follows immediately from formula (31) using induction on Ph\ni=1 αi.\nThe next lemma bounds combinatorial niceness of certain subsets of F∗\nq.\nLemma 20 Let Fq = Fpl, where p is prime. Let S ⊆F∗\nq. Suppose there exist h proper affine hyperplanes\n{πγr,cr}1≤r≤h of Fq such that S ⊆\nhS\nr=1\nπγr,cr; then S is at most h(p −1) combinatorially nice.\n\nProof:\nAssume S is t combinatorially nice. This implies that for some c > 0 and every m there exist two\nn = ⌊cmt⌋-sized collections of vectors {ui}i∈[n] and {vi}i∈[n] in Fm\nq , such that:\n• For all i ∈[n], (ui, vi) = 0;\n• For all i, j ∈[n] such that i ̸= j, (uj, vi) ∈S.\nFor a vector u ∈Fm\nq and integer e let ue denote a vector resulting from raising every coordinate of u to the power e.\nFor every i ∈[n] and r ∈[h] define vectors u(r)\ni\nand v(r)\ni\nin Fml\nq\nby\nu(r)\ni\n= (γrui) ◦(γrui)p ◦. . . ◦(γrui)pl−1\nand\nv(r)\ni\n= vi ◦vp\ni ◦. . . ◦vpl−1\ni\n.\n(34)\nNote that for every r1, r2 ∈[h], v(r1)\ni\n= v(r2)\ni\n. It is straightforward to verify that for every i, j ∈[n] and r ∈[h] :\n \nu(r)\nj , v(r)\ni\n \n= Tr(γr(uj, vi)).\n(35)\nCombining (35) with the fact that S is covered by proper affine hyperplanes πγi,ci we conclude that\n• For all i ∈[n] and r ∈[h],\n \nu(r)\ni , v(r)\ni\n \n= 0;\n• For all i, j ∈[n] such that i ̸= j, there exists r ∈[h] such that\n \nu(r)\nj , v(r)\ni\n \n∈F∗\np.\nPick g(x1, . . . , xh) ∈Fp[x1, . . . , xh] to be a homogeneous degree h polynomial such that for a = (a1, . . . , ah) ∈\nFh\np : g(a) = 0 if and only if a is the all-zeros vector. The existence of such a polynomial g follows from [17,\nExample 6.7]. Set f = gp−1. Note that for a ∈Fh\np : f(a) = 0 if a is the all-zeros vector, and f(a) = 1 otherwise.\nFor all i ∈[n] define\nu′\ni = f\n \nu(1)\ni , . . . , u(h)\ni\n \n◦(1)\nand\nv′\ni = ̄f\n \nv(1)\ni\n, . . . , v(h)\ni\n \n◦(−1).\n(36)\nNote that f and ̄f are homogeneous degree (p −1)h polynomials in h variables. Therefore (32) implies that\nfor all i vectors u′\ni and v′\ni have length m′ ≤h(p−1)h(ml)(p−1)h. Combining identities (36) and (33) and using the\nproperties of dot products between vectors\nn\nu(r)\ni\no\nand\nn\nv(r)\ni\no\ndiscussed above we conclude that for every m there\nexist two n = ⌊cmt⌋-sized collections of vectors {u′\ni}i∈[n] and {v′\ni}i∈[n] in Fm′\nq , such that:\n• For all i ∈[n], (u′\ni, v′\ni) = −1;\n• For all i, j ∈[n] such that i ̸= j, (uj, vi) = 0.\nIt remains to notice that a family of vectors with such properties exists only if n ≤m′, i.e., ⌊cmt⌋≤h(p−1)h(ml)(p−1)h.\nGiven that we can pick m to be arbitrarily large, this implies that t ≤(p −1)h.\nThe next lemma presents the main result of this section.\nLemma 21 Let k be an odd integer. Suppose there exists a k-nice sequence; then for infinitely many primes p\nsome k of elements of Cp add up to zero.\nProof:\nAssume\n \nSi ⊆F∗\nqi\n \ni≥1 is k-nice. Let p be a fixed prime. Combining lemmas 18 and 20 we conclude\nthat every k algebraically nice subset S ⊆F∗\npl is at most (p −1)pk combinatorially nice. Note that our bound on\ncombinatorial niceness is independent of l. Therefore there are only finitely many extensions of the field Fp in the\nsequence {Fqi}i≥1 , and the set P = {charFqi}i≥1 is infinite. It remains to notice that according to lemma 17 for\nevery p ∈P there exist k elements of Cp that add up to zero.\nIn what follows we present necessary conditions for the existence of k-tuples of p-th roots of unity in F2 that\nsum to zero. We treat the k = 3 case separately since in that case we can use a specialized argument to derive a\nslightly stronger conclusion.\n\n5.2\nA necessary condition for the existence of k p-th roots of unity summing to zero\nLemma 22 Let k ≥3 be odd and p be a prime. Suppose there exist ζ1, . . . , ζk ∈Cp such that Pk\ni=1 ζi = 0; then\nord2(p) ≤2p1−1/(k−1).\n(37)\nProof:\nLet t = ord2(p). Note that Cp ⊆F2t. Note also that all elements of Cp other than the multiplicative\nidentity are proper elements of F2t. Therefore for every ζ ∈Cp where ζ ̸= 1 and every f(x) ∈F2[x] such that\ndeg f ≤t −1 we have: f(ζ) ̸= 0.\nBy multiplying Pk\ni=1 ζi = 0 through by ζ−1\nk , we may reduce to the case ζk = 1. Let ζ be the generator of Cp.\nFor every i ∈[k −1] pick wi ∈Zp such that ζi = ζwi. We now have Pk−1\ni=1 ζwi + 1 = 0. Set h = ⌊(t −1)/2⌋.\nConsider the (k −1)-tuples:\n(mw1 + i1, . . . , mwk−1 + ik−1) ∈Zk−1\np\n, for m ∈Zp and i1, . . . , ik−1 ∈[0, h].\n(38)\nSuppose two of these coincide, say\n(mw1 + i1, . . . , mwk−1 + ik−1) = (m′w1 + i′\n1, . . . , m′wk−1 + i′\nk−1),\nwith (m, i1, . . . , ik−1) ̸= (m′, i′\n1, . . . , i′\nk−1). Set n = m −m′ and jl = i′\nl −il for l ∈[k −1]. We now have\n(nw1, . . . , nwk−1) = (j1, . . . , jl)\nwith −h ≤j1, . . . , jk−1 ≤h. Observe that n ̸= 0, and thus it has a multiplicative inverse g ∈Zp. Consider a\npolynomial\nP(z) = zj1+h + . . . + zjk−1+h + zh ∈F2[z].\nNote that deg P ≤2h ≤t −1. Note also that P(1) = 1 and P(ζg) = 0. The latter identity contradicts the fact\nthat ζg is a proper element of F2t. This contradiction implies that all (k −1)-tuples in (38) are distinct. This yields\npk−1 ≥p\n t\n2\n k−1\n,\nwhich is equivalent to (37).\n5.3\nA necessary condition for the existence of three p-th roots of unity summing to zero\nIn this section we slightly strengthen lemma 22 in the special case when k = 3. Our argument is loosely inspired\nby the Agrawal-Kayal-Saxena deterministic primality test [1].\nLemma 23 Let p be a prime. Suppose there exist ζ1, ζ2, ζ3 ∈Cp that sum up to zero; then\nord2(p) ≤((4/3)p)1/2 .\n(39)\nProof:\nLet t = ord2(p). Note that Cp ⊆F2t. Note also that all elements of Cp other than the multiplicative\nidentity are proper elements of F2t. Therefore for every ζ ∈Cp where ζ ̸= 1 and every f(x) ∈F2[x] such that\ndeg f ≤t −1 we have: f(ζ) ̸= 0.\nObserve that ζ1 + ζ2 + ζ3 = 0 implies ζ1ζ−1\n2\n+ 1 = ζ3ζ−1\n2 . This yields\n ζ1ζ−1\n2\n+ 1\n p = 1. Put ζ = ζ1ζ−1\n2 .\nNote that ζ ̸= 1 and ζ, 1 + ζ ∈Cp. Consider the products πi,j = ζi(1 + ζ)j ∈Cp for 0 ≤i, j ≤t −1. Note that\nπi,j, πk,l cannot be the same if i ≥k and l ≥j, as then\nζi−k −(1 + ζ)l−j = 0,\n\nbut the left side has degree less than t. In other words, if πi,j = πk,l and (i, j) ̸= (k, l), then the pairs (i, j) and\n(k, l) are comparable under termwise comparison. In particular, either (k, l) = (i+a, j+b) or (i, j) = (k+a, l+b)\nfor some pair (a, b) with πa,b = 1.\nWe next check that there cannot be two distinct nonzero pairs (a, b), (a′, b′) with πa,b = πa′,b′ = 1. As above,\nthese pairs must be comparable; we may assume without loss of generality that a ≤a′, b ≤b′. The equations\nπa,b = 1 and πa′−a,b′−b = 1 force a + b ≥t and (a′ −a) + (b′ −b) ≥t, so a′ + b′ ≥2t. But a′, b′ ≤t −1,\ncontradiction.\nIf there is no nonzero pair (a, b) with 0 ≤a, b ≤t −1 and πa,b = 1, then all πi,j are distinct, so p ≥t2.\nOtherwise, as above, the pair (a, b) is unique, and the pairs (i, j) with 0 ≤i, j ≤t −1 and (i, j) ̸≥(a, b) are\npairwise distinct. The number of pairs excluded by the condition (i, j) ̸≥(a, b) is (t −a)(t −b); since a + b ≥t,\n(t −a)(t −b) ≤t2/4. Hence p ≥t2 −t2/4 = 3t2/4 as desired.\nWhile the necessary condition given by lemma 23 is quite far away from the sufficient condition given by\nlemma 8, it nonetheless suffices for checking that for most primes p, there do not exist three p-th roots of unity\nsumming to zero. For instance, among the 664578 odd primes p ≤108, all but 550 are ruled out by Lemma 23.\n(There is an easy argument that t must be odd if p > 3; this cuts the list down to 273 primes.) Each remaining\np can be tested by computing gcd(xp + 1, (x + 1)p + 1); the only examples we found that did not satisfy the\ncondition of lemma 8 were (p, t) = (73, 9), (262657, 27), (599479, 33), (121369, 39).\n5.4\nSummary\nIn the beginning of this section 5 we argued that in order to use the method of [34], (i.e., proposition 5) to obtain\nk-query locally decodable codes of length exp(nǫ) for some fixed k and all ǫ > 0, one needs to exhibit a k-nice\nsequence of subsets of finite fields. In what follows we use technical results of the previous subsections to show\nthat the existence of a k-nice sequence implies that infinitely many Mersenne numbers have large prime factors.\nTheorem 24 Let k be odd. Suppose there exists a k-nice sequence of subsets of finite fields; then for infinitely\nmany values of t we have\nP(2t −1) ≥(t/2)1+1/(k−2).\n(40)\nProof:\nUsing lemmas 21 and 22 we conclude that a k-nice sequence yields infinitely many primes p such that\nord2(p) ≤2p1−1/(k−1). Let p be such a prime and t = ord2(p). Then P(2t −1) ≥(t/2)1+1/(k−2).\nA combination of lemmas 21 and 23 yields a slightly stronger bound for the special case of 3-nice sequences.\nTheorem 25 Suppose there exists a 3-nice sequence of subsets; then for infinitely many values of t we have\nP(2t −1) ≥(3/4)t2.\n(41)\nWe would like to remind the reader that although the lower bounds for P(2t −1) given by (40) and (41) are\nextremely weak light of the widely accepted conjecture saying that the number of Mersenne primes is infinite,\nthey are substantially stronger than what is currently known unconditionally (2).\n6\nConclusion\nRecently [34] came up with a novel technique for constructing locally decodable codes and obtained vast im-\nprovements upon the earlier work. The construction proceeds in two steps. First [34] shows that if there exist\nsubsets of finite fields with certain ’nice’ properties then there exist good codes. Next [34] constructs nice subsets\nof prime fields Fp for Mersenne primes p.\n\nIn this paper we have undertaken an in-depth study of nice subsets of general finite fields. We have shown\nthat constructing nice subsets is closely related to proving lower bounds on the size of largest prime factors of\nMersenne numbers. Specifically we extended the constructions of [34] to obtain nice subsets of prime fields Fp\nfor primes p that are large factors of Mersenne numbers. This implies that strong lower bounds for size of the\nlargest prime factors of Mersenne numbers yield better locally decodable codes. Conversely, we argued that if one\ncan obtain codes of subexponential length and constant query complexity through nice subsets of finite fields then\ninfinitely many Mersenne numbers have prime factors larger than known currently.\nAcknowledgements\nKiran Kedlaya’s research is supported by NSF CAREER grant DMS-0545904 and by the Sloan Research Fel-\nlowship. Sergey Yekhanin would like to thank Swastik Kopparty for providing the reference [5] and outlining\nthe proof of lemma 9. He would also like to thank Henryk Iwaniec, Carl Pomerance and Peter Sarnak for their\nfeedback regarding the number theory problems discussed in this paper.\nReferences\n[1] M. Agrawal, N. Kayal, N. Saxena, “PRIMES is in P,” Annals of Mathematics, vol. 160, pp. 781-793, 2004.\n[2] L. Babai, L. Fortnow, L. Levin, and M. Szegedy, “Checking computations in polylogarithmic time,”. In Proc.\nof the 23th ACM Symposium on Theory of Computing (STOC), pp. 21-31, 1991.\n[3] A. Beimel, Y. Ishai and E. Kushilevitz,“General constructions for information-theoretic private information\nretrieval,” Journal of Computer and System Sciences, vol. 71, pp. 213-247, 2005. Preliminary versions in\nSTOC 1999 and ICALP 2001.\n[4] A. Beimel, Y. Ishai, E. Kushilevitz, and J. F. Raymond. “Breaking the O\n n1/(2k−1) \nbarrier for information-\ntheoretic private information retrieval,” In Proc. of the 43rd IEEE Symposium on Foundations of Computer\nScience (FOCS), pp. 261-270, 2002.\n[5] J. Bourgain, M. Chang, “A Gauss sum estimate in arbitrary finite fields,” Comptes Rendus Mathematique,\nvol. 342, pp. 643-646, 2006.\n[6] B. Chor, O. Goldreich, E. Kushilevitz, and M. Sudan. “Private information retrieval,” In Proc. of the 36rd\nIEEE Symposium on Foundations of Computer Science (FOCS), pp. 41-50, 1995. Also, in Journal of the\nACM, vol. 45, 1998.\n[7] D. Cox, J. Little, D. O’Shea, Ideals, varieties, and algorithms: an introduction to computational algebraic\ngeometry and commutative algebra. Springer, 1996.\n[8] A. Deshpande, R. Jain, T. Kavitha, S. Lokam and J. Radhakrishnan, “Better lower bounds for locally decod-\nable codes,” In Proc. of the 20th IEEE Computational Complexity Conference (CCC), pp. 184-193, 2002.\n[9] Curtis Cooper, Steven Boone, http://www.mersenne.org/32582657.htm\n[10] P. Erdos and T. Shorey, “On the greatest prime factor of 2p −1 for a prime p and other expressions,” Acta.\nArith. vol. 30, pp. 257-265, 1976.\n[11] W. Gasarch, “A survey on private information retrieval,” The Bulletin of the EATCS, vol. 82, pp. 72-107,\n2004.\n\n[12] O. Goldreich, “Short locally testable codes and proofs,” Technical Report TR05-014, Electronic Colloquim\non Computational Complexity (ECCC), 2005.\n[13] O. Goldreich, H. Karloff, L. Schulman, L. Trevisan “Lower bounds for locally decodable codes and private\ninformation retrieval,” In Proc. of the 17th IEEE Computational Complexity Conference (CCC), pp. 175-183,\n2002.\n[14] B. Hemenway and R. Ostrovsky, “Public key encryption which is simultaneously a locally-decodable error-\ncorrecting code,” In Cryptology ePrint Archive, Report 2007/083.\n[15] J. Katz and L. Trevisan, “On the efficiency of local decoding procedures for error-correcting codes,” In Proc.\nof the 32th ACM Symposium on Theory of Computing (STOC), pp. 80-86, 2000.\n[16] I. Kerenidis, R. de Wolf, “Exponential lower bound for 2-query locally decodable codes via a quantum\nargument,” Journal of Computer and System Sciences, 69(3), pp. 395-420. Earlier version in STOC’03.\nquant-ph/0208062.\n[17] R. Lidl and H. Niederreiter, Finite Fields. Cambridge: Cambridge University Press, 1983.\n[18] L. Murata, C. Pomerance, “On the largest prime factor of a Mersenne number,” Number theory, CRM Proc.\nLecture Notes of American Mathematical Society vol. 36, pp. 209-218, 2004.\n[19] M. Murty and S. Wong, “The ABC conjecture and prime divisors of the Lucas and Lehmer sequences,” In\nProc. of Milennial Conference on Number Theory III, (Urbana, IL, 2000) (A. K. Peters, Natick, MA, 2002)\npp. 43-54.\n[20] K. Obata, “Optimal lower bounds for 2-query locally decodable linear codes,” In Proc. of the 6th RANDOM,\nvol. 2483 of Lecture Notes in Computer Science, pp. 39-50, 2002.\n[21] A. Polishchuk and D. Spielman, ”Nearly-linear size holographic proofs,” In Proc. of the 26th ACM Sympo-\nsium on Theory of Computing (STOC), pp. 194-203, 1994.\n[22] C. Pomerance, “Recent developments in primality testing,” Math. Intelligencer, 3:3, pp. 97-105, (1980/81).\n[23] P. Raghavendra, “A Note on Yekhanin’s locally decodable codes,” In Electronic Colloquium on Computa-\ntional Complexity Report TR07-016, 2007.\n[24] A. Romashchenko, “Reliable computations based on locally decodable codes,” In Proc. of the 23rd Inter-\nnational Symposium on Theoretical Aspects of Computer Science (STACS), vol. 3884 of Lecture Notes in\nComputer Science, pp. 537-548, 2006.\n[25] A. Schinzel, “On primitive factors of an −bn,” In Proc. of Cambridge Philos. Soc. vol. 58, pp. 555-562,\n1962.\n[26] C. Stewart, “The greatest prime factor of an −bn,” Acta Arith. vol. 26, pp. 427-433, 1974/75.\n[27] C. Stewart, “On divisors of Fermat, Fibonacci, Lucas, and Lehmer numbers,” In Proc. of London Math. Soc.\nvol. 35 (3), pp. 425-447, 1977.\n[28] M. Sudan, Efficient checking of polynomials and proofs and the hardness of approximation problems. PhD\nthesis, University of California at Berkeley, 1992.\n[29] L. Trevisan, “Some applications of coding theory in computational complexity,” Quaderni di Matematica,\nvol. 13, pp. 347-424, 2004.\n\n[30] S. Wehner and R. de Wolf, “Improved lower bounds for locally decodable codes and private information re-\ntrieval,” In Proc. of 32nd International Colloquium on Automata, Languages and Programming (ICALP’05),\nLNCS 3580, pp. 1424-1436.\n[31] Lenstra-Pomerance-Wagstaff conjecture. (2006, May 22). In Wikipedia, The Free Encyclopedia. Re-\ntrieved 00:18,\nOctober 3,\n2006,\nfrom http://en.wikipedia.org/w/index.php?title=Lenstra-Pomerance-\nWagstaff conjecture&oldid=54506577\n[32] S. Wagstaff, “Divisors of Mersenne numbers,” Math. Comp., 40:161, pp. 385-397, 1983.\n[33] D. Woodruff, “New lower bounds for general locally decodable codes,” Electronic Colloquium on Computa-\ntional Complexity, TR07-006, 2007.\n[34] S. Yekhanin, “Towards 3-query locally decodable codes of subexponential length,” In Proc. of the 39th ACM\nSymposium on Theory of Computing (STOC), 2007.\n[35] S. Yekhanin, Locally decodable codes and private information retrieval schemes. PhD thesis, MIT, to appear.","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0704.1694v1 [cs.CC] 13 Apr 2007\nLocally Decodable Codes From Nice Subsets of Finite Fields\nand Prime Factors of Mersenne Numbers\nKiran S. Kedlaya\nMIT\nkedlaya@mit.edu\nSergey Yekhanin\nMIT\nyekhanin@mit.edu\nAbstract\nA k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that\none can probabilistically recover any bit xi of the message by querying only k bits of the codeword C(x), even\nafter some constant fraction of codeword bits has been corrupted. The major goal of LDC related research is to\nestablish the optimal trade-off between length and query complexity of such codes.\nRecently [34] introduced a novel technique for constructing locally decodable codes and vastly improved the\nupper bounds for code length. The technique is based on Mersenne primes. In this paper we extend the work\nof [34] and argue that further progress via these methods is tied to progress on an old number theory question\nregarding the size of the largest prime factors of Mersenne numbers.\nSpecifically, we show that every Mersenne number m = 2t −1 that has a prime factor p > mγ yields a family\nof k(γ)-query locally decodable codes of length exp\n n1/t \n. Conversely, if for some fixed k and all ǫ > 0 one can\nuse the technique of [34] to obtain a family of k-query LDCs of length exp (nǫ) ; then infinitely many Mersenne\nnumbers have prime factors larger than known currently.\n1\nIntroduction\nClassical error-correcting codes allow one to encode an n-bit string x into in N-bit codeword C(x), in such\na way that x can still be recovered even if C(x) gets corrupted in a number of coordinates. It is well-known\nthat codewords C(x) of length N = O(n) already suffice to correct errors in up to δN locations of C(x) for\nany constant δ < 1/4. The disadvantage of classical error-correction is that one needs to consider all or most\nof the (corrupted) codeword to recover anything about x. Now suppose that one is only interested in recovering\none or a few bits of x. In such case more efficient schemes are possible. Such schemes are known as locally\ndecodable codes (LDCs). Locally decodable codes allow reconstruction of an arbitrary bit xi, from looking only\nat k randomly chosen coordinates of C(x), where k can be as small as 2. Locally decodable codes have numerous\napplications in complexity theory [15, 29], cryptography [6, 11] and the theory of fault tolerant computation [24].\nBelow is a slightly informal definition of LDCs:\nA (k, δ, ǫ)-locally decodable code encodes n-bit strings to N-bit codewords C(x), such that for every i ∈[n],\nthe bit xi can be recovered with probability 1−ǫ, by a randomized decoding procedure that makes only k queries,\neven if the codeword C(x) is corrupted in up to δN locations.\nOne should think of δ > 0 and ǫ < 1/2 as constants. The main parameters of interest in LDCs are the length\nN and the query complexity k. Ideally we would like to have both of them as small as possible. The concept\nof locally decodable codes was explicitly discussed in various papers in the early 1990s [2, 28, 21]. Katz and"},{"paragraph_id":"p2","order":2,"text":"Trevisan [15] were the first to provide a formal definition of LDCs. Further work on locally decodable codes\nincludes [3, 8, 20, 4, 16, 30, 34, 33, 14, 23].\nBelow is a brief summary of what was known regarding the length of LDCs prior to [34]. The length of optimal\n2-query LDCs was settled by Kerenidis and de Wolf in [16] and is exp(n).1 The best upper bound for the length\nof 3-query LDCs was exp\n n1/2 \ndue to Beimel et al. [3], and the best lower bound is ̃Ω(n2) [33]. For general\n(constant) k the best upper bound was exp\n nO(log log k/(k log k)) \ndue to Beimel et al. [4] and the best lower bound\nis ̃Ω\n n1+1/(⌈k/2⌉−1) \n[33].\nThe recent work [34] improved the upper bounds to the extent that it changed the common perception of what\nmay be achievable [12, 11]. [34] introduced a novel technique to construct codes from so-called nice subsets\nof finite fields and showed that every Mersenne prime p = 2t −1 yields a family of 3-query LDCs of length\nexp\n n1/t \n. Based on the largest known Mersenne prime [9], this translates to a length of less than exp"},{"paragraph_id":"p3","order":3,"text":"n10−7 \n.\nCombined with the recursive construction from [4], this result yields vast improvements for all values of k > 2. It\nhas often been conjectured that the number of Mersenne primes is infinite. If indeed this conjecture holds, [34] gets\nthree query locally decodable codes of length N = exp"},{"paragraph_id":"p4","order":4,"text":"nO\n“\n1\nlog log n\n” \nfor infinitely many n. Finally, assuming\nthat the conjecture of Lenstra, Pomerance and Wagstaff [31, 22, 32] regarding the density of Mersenne primes\nholds, [34] gets three query locally decodable codes of length N = exp"},{"paragraph_id":"p5","order":5,"text":"n\nO\n“\n1\nlog1−ǫ log n\n” \nfor all n, for every ǫ >\n0.\n1.1\nOur results\nIn this paper we address two natural questions left open by [34]:\n1. Are Mersenne primes necessary for the constructions of [34]?\n2. Has the technique of [34] been pushed to its limits, or one can construct better codes through a more clever\nchoice of nice subsets of finite fields?\nWe extend the work of [34] and answer both of the questions above. In what follows let P(m) denote the\nlargest prime factor of m. We show that one does not necessarily need to use Mersenne primes. It suffices to have\nMersenne numbers with polynomially large prime factors. Specifically, every Mersenne number m = 2t −1 such\nthat P(m) ≥mγ yields a family of k(γ)-query locally decodable codes of length exp\n n1/t \n. A partial converse\nalso holds. Namely, if for some fixed k ≥3 and all ǫ > 0 one can use the technique of [34] to (unconditionally)\nobtain a family of k-query LDCs of length exp (nǫ) ; then for infinitely many t we have\nP(2t −1) ≥(t/2)1+1/(k−2).\n(1)\nThe bound (1) may seem quite weak in light of the widely accepted conjecture saying that the number of\nMersenne primes is infinite. However (for any k ≥3) this bound is substantially stronger than what is currently\nknown unconditionally. Lower bounds for P(2t −1) have received a considerable amount of attention in the\nnumber theory literature [25, 26, 10, 27, 19, 18]. The strongest result to date is due to Stewart [27]. It says that\nfor all integers t ignoring a set of asymptotic density zero, and for all functions ǫ(t) > 0 where ǫ(t) tends to zero\nmonotonically and arbitrarily slowly:\nP(2t −1) > ǫ(t)t (log t)2 / log log t.\n(2)\n1Throughout the paper we use the standard notation exp(x)\ndef\n= eO(x)."},{"paragraph_id":"p6","order":6,"text":"There are no better bounds known to hold for infinitely many values of t, unless one is willing to accept some\nnumber theoretic conjectures [19, 18]. We hope that our work will further stimulate the interest in proving lower\nbounds for P(2t −1) in the number theory community.\nIn summary, we show that one may be able to improve the unconditional bounds of [34] (say, by discovering a\nnew Mersenne number with a very large prime factor) using the same technique. However any attempts to reach\nthe exp (nǫ) length for some fixed query complexity and all ǫ > 0 require either progress on an old number theory\nproblem or some radically new ideas.\nIn this paper we deal only with binary codes for the sake of clarity of presentation. We remark however that\nour results as well as the results of [34] can be easily generalized to larger alphabets. Such generalization will be\ndiscussed in detail in [35].\n1.2\nOutline\nIn section 3 we introduce the key concepts of [34], namely that of combinatorial and algebraic niceness of\nsubsets of finite fields. We also briefly review the construction of locally decodable codes from nice subsets. In\nsection 4 we show how Mersenne numbers with large prime factors yield nice subsets of prime fields. In section 5\nwe prove a partial converse. Namely, we show that every finite field Fq containing a sufficiently nice subset, is an\nextension of a prime field Fp, where p is a large prime factor of a large Mersenne number. Our main results are\nsummarized in sections 4.3 and 5.4.\n2\nNotation\nWe use the following standard mathematical notation:\n• [s] = {1, . . . , s};\n• Zn denotes integers modulo n;\n• Fq is a finite field of q elements;\n• dH(x, y) denotes the Hamming distance between binary vectors x and y;\n• (u, v) stands for the dot product of vectors u and v;\n• For a linear space L ⊆Fm\n2 , L⊥denotes the dual space. That is, L⊥= {u ∈Fm\n2 | ∀v ∈L, (u, v) = 0};\n• For an odd prime p, ord2(p) denotes the smallest integer t such that p | 2t −1.\n3\nNice subsets of finite fields and locally decodable codes\nIn this section we introduce the key technical concepts of [34], namely that of combinatorial and algebraic\nniceness of subsets of finite fields. We briefly review the construction of locally decodable codes from nice\nsubsets. Our review is concise although self-contained. We refer the reader interested in a more detailed and\nintuitive treatment of the construction to the original paper [34]. We start by formally defining locally decodable\ncodes.\nDefinition 1 A binary code C : {0, 1}n →{0, 1}N is said to be (k, δ, ǫ)-locally decodable if there exists a\nrandomized decoding algorithm A such that"},{"paragraph_id":"p7","order":7,"text":"1. For all x ∈{0, 1}n, i ∈[n] and y ∈{0, 1}N such that dH(C(x), y) ≤δN : Pr[Ay(i) = xi] ≥1−ǫ, where\nthe probability is taken over the random coin tosses of the algorithm A.\n2. A makes at most k queries to y.\nWe now introduce the concepts of combinatorial and algebraic niceness of subsets of finite fields. Our defini-\ntions are syntactically slightly different from the original definitions in [34]. We prefer these formulations since\nthey are more appropriate for the purposes of the current paper. In what follows let F∗\nq denote the multiplicative\ngroup of Fq.\nDefinition 2 A set S ⊆F∗\nq is called t combinatorially nice if for some constant c > 0 and every positive integer\nm there exist two n = ⌊cmt⌋-sized collections of vectors {u1, . . . , un} and {v1, . . . , vn} in Fm\nq , such that\n• For all i ∈[n], (ui, vi) = 0;\n• For all i, j ∈[n] such that i ̸= j, (uj, vi) ∈S.\nDefinition 3 A set S ⊆F∗\nq is called k algebraically nice if k is odd and there exists an odd k′ ≤k and two sets\nS0, S1 ⊆Fq such that\n• S0 is not empty;\n• |S1| = k′;\n• For all α ∈Fq and β ∈S : |S0 ∩(α + βS1)| ≡0 mod (2).\nThe following lemma shows that for an algebraically nice set S, the set S0 can always be chosen to be large. It\nis a straightforward generalization of [34, lemma 15].\nLemma 4 Let S ⊆F∗\nq be a k algebraically nice set. Let S0, S1 ⊆Fq be sets from the definition of algebraic\nniceness of S. One can always redefine the set S0 to satisfy |S0| ≥⌈q/2⌉.\nProof:\nLet L be the linear subspace of Fq\n2 spanned by the incidence vectors of the sets α + βS1, for α ∈Fq and\nβ ∈S. Observe that L is invariant under the actions of a 1-transitive permutation group (permuting the coordinates\nin accordance with addition in Fq). This implies that the space L⊥is also invariant under the actions of the same\ngroup. Note that L⊥has positive dimension since it contains the incidence vector of the set S0. The last two\nobservations imply that L⊥has full support, i.e., for every i ∈[q] there exists a vector v ∈L⊥such that vi ̸= 0. It\nis easy to verify that any linear subspace of Fq\n2 that has full support contains a vector of Hamming weight at least\n⌈q/2⌉. Let v ∈L⊥be such a vector. Redefining the set S0 to be the set of nonzero coordinates of v we conclude\nthe proof.\nWe now proceed to the core proposition of [34] that shows how sets exhibiting both combinatorial and algebraic\nniceness yield locally decodable codes.\nProposition 5 Suppose S ⊆F∗\nq is t combinatorially nice and k algebraically nice; then for every positive integer\nn there exists a code of length exp(n1/t) that is (k, δ, 2kδ) locally decodable for all δ > 0.\nProof:\nOur proof comes in three steps. We specify encoding and local decoding procedures for our codes and\nthen argue the lower bound for the probability of correct decoding. We use the notation from definitions 2 and 3.\nEncoding: We assume that our message has length n = ⌊cmt⌋for some value of m. (Otherwise we pad the\nmessage with zeros. It is easy to see that such padding does not not affect the asymptotic length of the code.) Our"},{"paragraph_id":"p8","order":8,"text":"code will be linear. Therefore it suffices to specify the encoding of unit vectors e1, . . . , en, where ej has length n\nand a unique non-zero coordinate j. We define the encoding of ej to be a qm long vector, whose coordinates are\nlabelled by elements of Fm\nq . For all w ∈Fm\nq we set:\nEnc(ej)w =\n 1,\nif (uj, w) ∈S0;\n0,\notherwise.\n(3)\nIt is straightforward to verify that we defined a code encoding n bits to exp(n1/t) bits.\nLocal decoding: Given a (possibly corrupted) codeword y and an index i ∈[n], the decoding algorithm A picks\nw ∈Fm\nq , such that (ui, w) ∈S0 uniformly at random, reads k′ ≤k coordinates of y, and outputs the sum:\nX\nλ∈S1\nyw+λvi.\n(4)\nProbability of correct decoding: First we argue that decoding is always correct if A picks w ∈Fm\nq such that\nall bits of y in locations {w + λvi}λ∈S1 are not corrupted. We need to show that for all i ∈[n], x ∈{0, 1}n and\nw ∈Fm\nq , such that (ui, w) ∈S0:\nX\nλ∈S1"},{"paragraph_id":"p9","order":9,"text":"n\nX\nj=1\nxj Enc(ej)"},{"paragraph_id":"p10","order":10,"text":"w+λvi\n= xi.\n(5)\nNote that\nX\nλ∈S1"},{"paragraph_id":"p11","order":11,"text":"n\nX\nj=1\nxj Enc(ej)"},{"paragraph_id":"p12","order":12,"text":"w+λvi\n=\nn\nX\nj=1\nxj\nX\nλ∈S1\nEnc(ej)w+λvi =\nn\nX\nj=1\nxj\nX\nλ∈S1\nI [(uj, w + λvi) ∈S0] ,\n(6)\nwhere I[γ ∈S0] = 1 if γ ∈S0 and zero otherwise. Now note that\nX\nλ∈S1\nI [(uj, w + λvi) ∈S0] =\nX\nλ∈S1\nI [(uj, w) + λ(uj, vi) ∈S0] =\n 1,\nif i = j,\n0,\notherwise.\n(7)\nThe last identity in (7) for i = j follows from: (ui, vi) = 0, (ui, w) ∈S0 and k′ = |S1| is odd. The last identity\nfor i ̸= j follows from (uj, vi) ∈S and the algebraic niceness of S. Combining identities (6) and (7) we get (5).\nNow assume that up to δ fraction of bits of y are corrupted. Let Ti denote the set of coordinates whose labels\nbelong to"},{"paragraph_id":"p13","order":13,"text":"w ∈Fm\nq | (ui, w) ∈S0"},{"paragraph_id":"p14","order":14,"text":". Recall that by lemma 4, |Ti| ≥qm/2. Thus at most 2δ fraction of coor-\ndinates in Ti contain corrupted bits. Let Qi ="},{"paragraph_id":"p15","order":15,"text":"{w + λvi}λ∈S1 | w : (ui, w) ∈S0"},{"paragraph_id":"p16","order":16,"text":"be the family of k′-tuples\nof coordinates that may be queried by A. (ui, vi) = 0 implies that elements of Qi uniformly cover the set Ti.\nCombining the last two observations we conclude that with probability at least 1 −2kδ A picks an uncorrupted\nk′-tuple and outputs the correct value of xi.\nAll locally decodable codes constructed in this paper are obtained by applying proposition 5 to certain nice\nsets. Thus all our codes have the same dependence of ǫ (the probability of the decoding error) on δ (the fraction\nof corrupted bits). In what follows we often ignore these parameters and consider only the length and query\ncomplexity of codes."},{"paragraph_id":"p17","order":17,"text":"4\nMersenne numbers with large prime factors yield nice subsets of prime fields\nIn what follows let ⟨2⟩⊆F∗\np denote the multiplicative subgroup of F∗\np generated by 2. In [34] it is shown\nthat for every Mersenne prime p = 2t −1 the set ⟨2⟩⊆F∗\np is simultaneously 3 algebraically nice and ord2(p)\ncombinatorially nice. In this section we prove the same conclusion for a substantially broader class of primes.\nLemma 6 Suppose p is an odd prime; then ⟨2⟩⊆F∗\np is ord2(p) combinatorially nice.\nProof: Let t = ord2(p). Clearly, t divides p −1. We need to specify a constant c > 0 such that for every positive\ninteger m there exist two n = ⌊cmt⌋-sized collections of m long vectors over Fp satisfying:\n• For all i ∈[n], (ui, vi) = 0;\n• For all i, j ∈[n] such that i ̸= j, (uj, vi) ∈⟨2⟩.\nFirst assume that m has the shape m ="},{"paragraph_id":"p18","order":18,"text":"m′−1+(p−1)/t\n(p−1)/t"},{"paragraph_id":"p19","order":19,"text":", for some integer m′ ≥p −1. In this case [34, lemma\n13] gives us a collection of n ="},{"paragraph_id":"p20","order":20,"text":"m′\np−1"},{"paragraph_id":"p21","order":21,"text":"vectors with the right properties. Observe that n ≥cmt for a constant\nc that depends only on p and t. Now assume m does not have the right shape, and let m1 be the largest integer\nsmaller than m that does have it. In order to get vectors of length m we use vectors of length m1 coming from [34,\nlemma 13] padded with zeros. It is not hard to verify such a construction still gives us n ≥cmt large families of\nvectors for a suitably chosen constant c.\nWe use the standard notation F to denote the algebraic closure of the field F. Also let Cp ⊆F\n∗\n2 denote the\nmultiplicative subgroup of p-th roots of unity in F2. The next lemma generalizes [34, lemma 14].\nLemma 7 Let p be a prime and k be odd. Suppose there exist ζ1, . . . , ζk ∈Cp such that\nζ1 + . . . + ζk = 0;\n(8)\nthen ⟨2⟩⊆F∗\np is k algebraically nice.\nProof:\nIn what follows we define the set S1 ⊆Fp and prove the existence of a set S0 such that that together S0\nand S1 yield k algebraic niceness of ⟨2⟩. Identity 8 implies that there exists an odd integer k′ ≤k and k′ distinct\np-th roots of unity ζ′\n1, . . . , ζ′\nk ∈Cp such that\nζ′\n1 + . . . + ζ′\nk′ = 0.\n(9)\nLet t = ord2(p). Observe that Cp ⊆F2t. Let g be a generator of Cp. Identity (9) yields gγ1 + . . . + gγk′ = 0, for\nsome distinct values of {γi}i∈[k′]. Set S1 = {γ1, . . . , γk′}.\nConsider a natural one to one correspondence between subsets S′ of Fp and polynomials φS′(x) in the ring\nF2[x]/(xp −1) : φS′(x) = P\ns∈S′ xs. It is easy to see that for all sets S′ ⊆Fp and all α, β ∈Fp, such that β ̸= 0 :\nφα+βS′(x) = xαφS′(xβ).\nLet α be a variable ranging over Fp and β be a variable ranging over ⟨2⟩. We are going to argue the existence of a\nset S0 that has even intersections with all sets of the form α+βS1, by showing that all polynomials φα+βS1 belong\nto a certain linear space L ∈F2[x]/(xp −1) of dimension less than p. In this case any nonempty set T ⊆Fp such\nthat φT ∈L⊥can be used as the set S0. Let τ(x) = gcd(xp −1, φS1(x)). Note that τ(x) ̸= 1 since g is a common\nroot of xp −1 and φS1(x). Let L be the space of polynomials in F2[x]/(xp −1) that are multiples of τ(x). Clearly,\ndim L = p −deg τ. Fix some α ∈Fp and β ∈⟨2⟩. Let us prove that φα+βS1(x) is in L :\nφα+βS1(x) = xαφS1(xβ) = xα(φS1(x))β.\nThe last identity above follows from the fact that for any f ∈F2[x] and any integer i : f(x2i) = (f(x))2i."},{"paragraph_id":"p22","order":22,"text":"In what follows we present sufficient conditions for the existence of k-tuples of p-th roots of unity in F2 that\nsum to zero. We treat the k = 3 case separately since in that case we can use a specialized argument to derive a\nmore explicit conclusion.\n4.1\nA sufficient condition for the existence of three p-th roots of unity summing to zero\nLemma 8 Let p be an odd prime. Suppose ord2(p) < (4/3) log2 p; then there exist three p-th roots of unity in F2\nthat sum to zero.\nProof: We start with a brief review of some basic concepts of projective algebraic geometry. Let F be a field, and\nf ∈F[x, y, z] be a homogeneous polynomial. A triple (x0, y0, z0) ∈F3 is called a zero of f if f(x0, y0, z0) = 0.\nA zero is called nontrivial if it is different from the origin. An equation f = 0 defines a projective plane curve χf.\nNontrivial zeros of f considered up to multiplication by a scalars are called F-rational points of χf. If F is a finite\nfield it makes sense to talk about the number of F-rational points on a curve.\nLet t = ord2(p). Note that Cp ⊆F2t. Consider a projective plane Fermat curve χ defined by\nx(2t−1)/p + y(2t−1)/p + z(2t−1)/p = 0.\n(10)\nLet us call a point a on χ trivial if one of the coordinates of a is zero. Cyclicity of F∗\n2t implies that χ contains\nexactly 3(2t −1)/p trivial F2t-rational points. Note that every nontrivial point of χ yields a triple of elements of\nCp that sum to zero. The classical Weil bound [17, p. 330] provides an estimate\n|Nq −(q + 1)| ≤(d −1)(d −2)√q\n(11)\nfor the number Nq of Fq-rational points on an arbitrary smooth projective plane curve of degree d. (11) implies\nthat in case\n2t + 1 >\n 2t −1\np\n−1\n 2t −1\np\n−2"},{"paragraph_id":"p23","order":23,"text":"2t/2 + 32t −1\np\n(12)\nthere exists a nontrivial point on the curve (10). Note that (12) follows from\n2t + 1 >\n 2t\np\n 2t\np"},{"paragraph_id":"p24","order":24,"text":"2t/2 −23t/2+1\np\n+ 3 ∗2t\np\n,\n(13)\nand (13) follows from\n2t > 22t+t/2/p2\nand\n2t/2+1 > 3.\nNow note that the first inequality above follows from t < (4/3) log2 p and the second follows from t > 1.\nNote that the constant 4/3 in lemma 8 cannot be improved to 2: there are no three elements of C13264529 that\nsum to zero, even though ord2(13264529) = 47 < 2 ∗log2 13264529 ≈47.3.\n4.2\nA sufficient condition for the existence of k p-th roots of unity summing to zero\nOur argument in this section comes in three steps. First we briefly review the notion of (additive) Fourier\ncoefficients of subsets of F2t. Next, we invoke a folklore argument to show that subsets of F2t with appropriately\nsmall nontrivial Fourier coefficients contain k-tuples of elements that sum to zero. Finally, we use a recent result of\nBourgain and Chang [5] (generalizing the classical estimate for Gauss sums) to argue that (under certain constraints\non p) all nontrivial Fourier coefficients of Cp are small.\nFor x ∈F2t let Tr(x) = x + x2 + . . . + x2t−1 denote the trace of x. It is not hard to verify that for all x,\nTr(x) ∈F2. Characters of F2t are homomorphisms from the additive group of F2t into the multiplicative group"},{"paragraph_id":"p25","order":25,"text":"{±1}. There exist 2t characters. We denote characters by χa, where a ranges in F2t, and set χa(x) = (−1)Tr(ax).\nLet C(x) denote the incidence function of a set C ⊆F2t. For arbitrary a ∈Ft\n2 the Fourier coefficient χa(C) is\ndefined by χa(C) = P χa(x)C(x), where the sum is over all x ∈F2t. Fourier coefficient χ0(C) = |C| is called\ntrivial, and other Fourier coefficients are called nontrivial. In what follows P\nχ stands for summation over all 2t\ncharacters of F2t. We need the following two standard properties of characters and Fourier coefficients.\nX\nχ\nχ(x) =\n 2t,\nif x = 0;\n0,\notherwise.\n(14)\nX\nχ\nχ2(C) = 2t|C|.\n(15)\nThe following lemma is a folklore.\nLemma 9 Let C ⊆F2t and k ≥3 be a positive integer. Let F be the largest absolute value of a nontrivial Fourier\ncoefficient of C. Suppose\nF\n|C| <\n |C|\n2t\n 1/(k−2)\n(16)\nthen there exist k elements of C that sum to zero.\nProof:\nLet M(C) = # {ζ1, . . . , ζk ∈C | ζ1 + . . . + ζk = 0} . (14) yields\nM(C) = 1\n2t\nX\nx1,...,xk∈F2t\nC(x1) . . . C(xk)\nX\nχ\nχ(x1 + . . . + xk).\n(17)\nNote that χ(x1 + . . . + xk) = χ(x1) . . . χ(xk). Changing the order of summation in (17) we get\nM(C) = 1\n2t\nX\nχ\nX\nx1,...,xk∈F2t\nC(x1) . . . C(xk)χ(x1) . . . χ(xk) = 1\n2t\nX\nχ\nχk(C).\n(18)\nNote that\n1\n2t\nX\nχ\nχk(C) = |C|k\n2t\n+ 1\n2t\nX\nχ̸=χ0\nχk(C) ≥|C|k\n2t\n−F k−2 1\n2t\nX\nχ\nχ2(C) = |C|k\n2t\n−F k−2|C|,\n(19)\nwhere the last identity follows from (15). Combining (18) and (19) we conclude that (16) implies M(C) > 0.\nThe following lemma is a special case of [5, theorem 1].\nLemma 10 Assume that n | 2t −1 and satisfies the condition\ngcd"},{"paragraph_id":"p26","order":26,"text":"n, 2t −1\n2t′ −1"},{"paragraph_id":"p27","order":27,"text":"< 2t(1−ǫ)−t′,\nfor all 1 ≤t′ < t, t′ | t,\nwhere ǫ > 0 is arbitrary and fixed. Then for all a ∈F∗\n2t"},{"paragraph_id":"p28","order":28,"text":"X\nx∈F2t\n(−1)Tr(axn)"},{"paragraph_id":"p29","order":29,"text":"< c12t(1−δ),\n(20)\nwhere δ = δ(ǫ) > 0 and c1 = c1(ǫ) are absolute constants."},{"paragraph_id":"p30","order":30,"text":"Below is the main result of this section. Recall that Cp denotes the set of p-th roots of unity in F2.\nLemma 11 For every c > 0 there exists an odd integer k = k(c) such that the following implication holds. If p is\nan odd prime and ord2(p) < c log2 p then some k elements of Cp sum to zero.\nProof:\nNote that if there exist k′ elements of a set C ⊆F2 that sum to zero, where k′ is odd; then there exist\nk elements of C that sum to zero for every odd k ≥k′. Also note that the sum of all p-th roots of unity is\nzero. Therefore given c it suffices to prove the existence of an odd k = k(c) that works for all sufficiently large\np. Let t = ord2(p). Observe that p > 2t/c. Assume p is sufficiently large so that t > 2c. Next we show that\nthe precondition of lemma 10 holds for n = (2t −1)/p and ǫ = 1/(2c). Let t′ | t and 1 ≤t′ < t. Clearly\ngcd(2t′ −1, p) = 1. Therefore\ngcd\n 2t −1\np\n, 2t −1\n2t′ −1"},{"paragraph_id":"p31","order":31,"text":"=\n2t −1\np(2t′ −1) < 2t(1−1/c)\n2t′ −1 ,\n(21)\nwhere the inequality follows from p > 2t/c. Clearly, t > 2c yields 2t/(2c)/2 > 1. Multiplying the right hand side\nof (21) by 2t/(2c)/2 and using 2(2t′ −1) > 2t′ we get\ngcd\n 2t −1\np\n, 2t −1\n2t′ −1"},{"paragraph_id":"p32","order":32,"text":"< 2t(1−1/(2c))−t′.\n(22)\nCombining (22) with lemma 10 we conclude that there exist δ > 0 and c1 such that for all a ∈F∗\n2t"},{"paragraph_id":"p33","order":33,"text":"X\nx∈F2t\n(−1)Tr\n“\nax(2t−1)/p” \n< c12t(1−δ).\n(23)\nObserve that x(2t−1)/p takes every value in Cp exactly (2t −1)/p times when x ranges over F∗\n2t. Thus (23) implies\n(2t −1)(F/p) < c12t(1−δ),\n(24)\nwhere F denotes that largest nontrivial Fourier coefficient of Cp. (24) yields F/p < (2c1)2−δt. Pick k ≥3 to be\nthe smallest odd integer such that (1 −1/c)/(k −2) < δ. We now have\nF\np < 2−(1−1/c)t\n(k−2)\n(25)\nfor all sufficiently large values of p. Combining p > 2t/c with (25) we get\nF\n|Cp| <\n |Cp|\n2t\n 1/(k−2)\n,\nand the application of lemma 9 concludes the proof.\n4.3\nSummary\nIn this section we summarize our positive results and show that one does not necessarily need to use Mersenne\nprimes to construct locally decodable codes via the methods of [34]. It suffices to have Mersenne numbers with\npolynomially large prime factors. Recall that P(m) denotes the largest prime factor of an integer m. Our first\ntheorem gets 3-query LDCs from Mersenne numbers m with prime factors larger than m3/4."},{"paragraph_id":"p34","order":34,"text":"Theorem 12 Suppose P(2t −1) > 20.75t; then for every message length n there exists a three query locally\ndecodable code of length exp(n1/t).\nProof:\nLet P(2t −1) = p. Observe that p | 2t −1 and p > 20.75t yield ord2(p) < (4/3) log2 p. Combining\nlemmas 8,7 and 6 with proposition 5 we obtain the statement of the theorem.\nAs an example application of theorem 12 one can observe that P(223 −1) = 178481 > 2(3/4)∗23 ≈155872 yields\na family of three query locally decodable codes of length exp(n1/23). Theorem 12 immediately yields:\nTheorem 13 Suppose for infinitely many t we have P(2t −1) > 20.75t; then for every ǫ > 0 there exists a family\nof three query locally decodable codes of length exp(nǫ).\nThe next theorem gets constant query LDCs from Mersenne numbers m with prime factors larger than mγ for\nevery value of γ.\nTheorem 14 For every γ > 0 there exists an odd integer k = k(γ) such that the following implication holds.\nSuppose P(2t −1) > 2γt; then for every message length n there exists a k query locally decodable code of length\nexp(n1/t).\nProof:\nLet P(2t −1) = p. Observe that p | 2t −1 and p > 2γt yield ord2(p) < (1/γ) log2 p. Combining\nlemmas 22,7 and 6 with proposition 5 we obtain the statement of the theorem.\nAs an immediate corollary we get:\nTheorem 15 Suppose for some γ > 0 and infinitely many t we have P(2t −1) > 2γt; then there is a fixed k such\nthat for every ǫ > 0 there exists a family of k query locally decodable codes of length exp(nǫ).\n5\nNice subsets of finite fields yield Mersenne numbers with large prime factors\nDefinition 16 We say that a sequence"},{"paragraph_id":"p35","order":35,"text":"Si ⊆F∗\nqi"},{"paragraph_id":"p36","order":36,"text":"i≥1 of subsets of finite fields is k-nice if every Si is k alge-\nbraically nice and t(i) combinatorially nice, for some integer valued monotonically increasing function t.\nThe core proposition 5 asserts that a subset S ⊆F∗\nq that is k algebraically nice and t combinatorially nice yields\na family of k-query locally decodable codes of length exp(n1/t). Clearly, to get k-query LDCs of length exp(nǫ)\nfor some fixed k and every ǫ > 0 via this proposition, one needs to exhibit a k-nice sequence. In this section\nwe show how the existence of a k-nice sequence implies that infinitely many Mersenne numbers have large prime\nfactors. Our argument proceeds in two steps. First we show that a k-nice sequence yields an infinite sequence of\nprimes {pi}i≥1 , where every Cpi contains a k-tuple of elements summing to zero. Next we show that Cp contains\na short additive dependence only if p is a large factor of a Mersenne number.\n5.1\nA nice sequence yields infinitely many primes p with short dependencies between p-th roots of unity\nWe start with some notation. Consider a a finite field Fq = Fpl, where p is prime. Fix a basis e1, . . . , el of Fq\nover Fp. In what follows we often write (α1, . . . , αl) ∈Fl\np to denote α = Pl\ni=1 αiei ∈Fq. Let R denote the ring\nF2[x1, . . . , xl]/(xp\n1 −1, . . . , xp\nl −1). Consider a natural one to one correspondence between subsets S1 of Fq and\npolynomials φS1(x1, . . . , xl) ∈R.\nφS1(x1, . . . , xl) =\nX\n(α1,...,αl)∈S1\nxα1\n1 . . . xαl\nl ."},{"paragraph_id":"p37","order":37,"text":"It is easy to see that for all sets S1 ⊆Fq and all α, β ∈Fq :\nφ(α1,...,αl)+βS1(x1, . . . , xl) = xα1\n1 . . . xαl\nl φβS1(x1, . . . , xl).\n(26)\nLet Γ be a family of subsets of Fq. It is straightforward to verify that a set S0 ⊆Fq has even intersections with\nevery element of Γ if and only if φS0 belongs to L⊥, where L is the linear subspace of R spanned by {φS1}S1∈Γ .\nCombining the last observation with formula (26) we conclude that a set S ⊆F∗\nq is k algebraically nice if and\nonly if there exists a set S1 ⊆Fq of odd size k′ ≤k such that the ideal generated by polynomials {φβS1}{β∈S}\nis a proper ideal of R. Note that polynomials {f1, . . . , fh} ∈R generate a proper ideal if an only if polynomials\n{f1, . . . , fh, xp\n1 −1, . . . , xp\nl −1} generate a proper ideal in F2[x1, . . . , xl]. Also note that a family of polynomials\ngenerates a proper ideal in F2[x1, . . . , xl] if and only if it generates a proper ideal in F2[x1, . . . , xl]. Now an\napplication of Hilbert’s Nullstellensatz [7, p. 168] implies that a set S ⊆F∗\nq is k algebraically nice if and only\nif there is a set S1 ⊆Fq of odd size k′ ≤k such that the polynomials {φβS1}{β∈S} and {xp\ni −1}1≤i≤l have a\ncommon root in F2.\nLemma 17 Let Fq = Fpl, where p is prime. Suppose Fq contains a nonempty k algebraically nice subset; then\nthere exist ζ1, . . . , ζk ∈Cp such that ζ1 + . . . + ζk = 0.\nProof:\nAssume S ⊆F∗\nq is nonempty and k algebraically nice. The discussion above implies that there exists\nS1 ⊆Fq of odd size k′ ≤k such that all polynomials {φβS1}{β∈S} vanish at some (ζ1, . . . , ζl) ∈Cl\np. Fix an\narbitrary β0 ∈S, and note that Cp is closed under multiplication. Thus,\nφβ0S1(ζ1, . . . , ζl) = 0\n(27)\nyields k′ p-th roots of unity that add up to zero. It is readily seen that one can extend (27) (by adding an appropriate\nnumber of pairs of identical roots) to obtain k p-th roots of unity that add up to zero for any odd k ≥k′.\nNote that lemma 17 does not suffice to prove that a k-nice sequence"},{"paragraph_id":"p38","order":38,"text":"Si ⊆F∗\nqi"},{"paragraph_id":"p39","order":39,"text":"i≥1 yields infinitely many primes p\nwith short (nontrivial) additive dependencies in Cp. We need to argue that the set {charFqi}i≥1 can not be finite. To\nproceed, we need some more notation. Recall that q = pl and p is prime. For x ∈Fq let Tr(x) = x+. . .+xpl−1 ∈\nFp denote the (absolute) trace of x. For γ ∈Fq, c ∈F∗\np we call the set πγ,c = {x ∈Fq | Tr(γx) = c} a proper\naffine hyperplane of Fq.\nLemma 18 Let Fq = Fpl, where p is prime. Suppose S ⊆F∗\nq is k algebraically nice; then there exist h ≤pk\nproper affine hyperplanes {πγi,ci}1≤i≤h of Fq such that S ⊆\nhS\ni=1\nπγi,ci.\nProof:\nDiscussion preceding lemma 17 implies that there exists a set S1 = {σ1, . . . , σk′} ⊆Fq of odd size\nk′ ≤k such that all polynomials {φβS1}{β∈S} vanish at some (ζ1, . . . , ζl) ∈Cl\np. Let ζ be a generator of Cp. For\nevery 1 ≤i ≤l pick ωi ∈Zp such that ζi = ζωi. For every β ∈S, φβS1(ζ1, . . . , ζl) = 0 yields\nX\nμ=(μ1,...,μl)∈βS1\nζ\nPl\ni=1 μiωi = 0.\n(28)\nObserve that for fixed values {ωi}1≤i≤l ∈Zp the map D(μ) = Pl\ni=1 μiωi is a linear map from Fq to Fp. It is\nnot hard to prove that every such map can be expressed as D(μ) = Tr(δμ) for an appropriate choice of δ ∈Fq.\nTherefore we can rewrite (28) as\nX\nμ∈βS1\nζTr(δμ) =\nX\nσ∈S1\nζTr(δβσ) = 0.\n(29)"},{"paragraph_id":"p40","order":40,"text":"Let W =\nn\n(w1, . . . , wk′) ∈Zk′\np | ζw1 + . . . + ζwk′ = 0\no\ndenote the set of exponents of k′-dependencies be-\ntween powers of ζ. Clearly, |W| ≤pk. Identity (29) implies that every β ∈S satisfies"},{"paragraph_id":"p41","order":41,"text":"Tr((δσ1)β)\n=\nw1,\n...\nTr((δσk′)β)\n=\nwk′;\n(30)\nfor an appropriate choice of (w1, . . . , wk′) ∈W. Note that the all-zeros vector does not lie in W since k′ is odd.\nTherefore at least one of the identities in (30) has a non-zero right-hand side, and defines a proper affine hyperplane\nof Fq. Collecting one such hyperplane for every element of W we get a family of |W| proper affine hyperplanes\ncontaining every element of S.\nLemma 18 gives us some insight into the structure of algebraically nice subsets of Fq. Our next goal is to develop\nan insight into the structure of combinatorially nice subsets. We start by reviewing some relations between tensor\nand dot products of vectors. For vectors u ∈Fm\nq and v ∈Fn\nq let u⊗v ∈Fmn\nq\ndenote the tensor product of u and v.\nCoordinates of u ⊗v are labelled by all possible elements of [m] × [n] and (u ⊗v)i,j = uivj. Also, let u⊗l denote\nthe l-the tensor power of u and u ◦v denote the concatenation of u and v. The following identity is standard. For\nany u, x ∈Fm\nq and v, y ∈Fn\nq :\n(u ⊗v, x ⊗y) =\nX\ni∈[m],j∈[n]\nuivjxiyj ="},{"paragraph_id":"p42","order":42,"text":"X\ni∈[m]\nuixi"},{"paragraph_id":"p43","order":43,"text":"X\nj∈[n]\nvjyj"},{"paragraph_id":"p44","order":44,"text":"= (u, x)(v, y).\n(31)\nIn what follows we need a generalization of identity (31). Let f(x1, . . . , xh) = P\ni cixαi\n1\n1 . . . x\nαi\nh\nh be a polynomial\nin Fq[x1, . . . , xh]. Given f we define ̄f ∈Fq[x1, . . . , xh] by ̄f = P\ni xαi\n1\n1 . . . x\nαi\nh\nh , i.e., we simply set all nonzero\ncoefficients of f to 1. For vectors u1, . . . , uh in Fm\nq define\nf(u1, . . . , uh) = ◦i ciu⊗αi\n1\n1\n⊗. . . ⊗u\n⊗αi\nh\nh\n.\n(32)\nNote that to obtain f(u1, . . . , uh) we replaced products in f by tensor products and addition by concatenation.\nClearly, f(u1, . . . , uh) is a vector whose length may be larger than m.\nClaim 19 For every f ∈Fq[x1, . . . , xh] and u1, . . . , uh, v1, . . . , vh ∈Fm\nq :\n f(u1, . . . , uh), ̄f(v1, . . . , vh)"},{"paragraph_id":"p45","order":45,"text":"= f((u1, v1), . . . , (uh, vh)).\n(33)\nProof: Let u = (u1, . . . , uh) and v = (v1, . . . , vh). Observe that if (33) holds for polynomials f1 and f2 defined\nover disjoint sets of monomials then it also holds for f = f1 + f2 :\n f(u), ̄f(v)"},{"paragraph_id":"p46","order":46,"text":"=\n (f1 + f2)(u), ( ̄f1 + ̄f2)(v)"},{"paragraph_id":"p47","order":47,"text":"=\n f1(u) ◦f2(u), ̄f1(v) ◦ ̄f2(v)"},{"paragraph_id":"p48","order":48,"text":"=\nf1 ((u1, v1), . . . , (uh, vh)) + f2 ((u1, v1), . . . , (uh, vh)) = f ((u1, v1), . . . , (uh, vh)) .\nTherefore it suffices to prove (33) for monomials f = cxα1\n1 . . . xαh\nh . It remains to notice identity (33) for monomi-\nals f = cxα1\n1 . . . xαh\nh follows immediately from formula (31) using induction on Ph\ni=1 αi.\nThe next lemma bounds combinatorial niceness of certain subsets of F∗\nq.\nLemma 20 Let Fq = Fpl, where p is prime. Let S ⊆F∗\nq. Suppose there exist h proper affine hyperplanes\n{πγr,cr}1≤r≤h of Fq such that S ⊆\nhS\nr=1\nπγr,cr; then S is at most h(p −1) combinatorially nice."},{"paragraph_id":"p49","order":49,"text":"Proof:\nAssume S is t combinatorially nice. This implies that for some c > 0 and every m there exist two\nn = ⌊cmt⌋-sized collections of vectors {ui}i∈[n] and {vi}i∈[n] in Fm\nq , such that:\n• For all i ∈[n], (ui, vi) = 0;\n• For all i, j ∈[n] such that i ̸= j, (uj, vi) ∈S.\nFor a vector u ∈Fm\nq and integer e let ue denote a vector resulting from raising every coordinate of u to the power e.\nFor every i ∈[n] and r ∈[h] define vectors u(r)\ni\nand v(r)\ni\nin Fml\nq\nby\nu(r)\ni\n= (γrui) ◦(γrui)p ◦. . . ◦(γrui)pl−1\nand\nv(r)\ni\n= vi ◦vp\ni ◦. . . ◦vpl−1\ni\n.\n(34)\nNote that for every r1, r2 ∈[h], v(r1)\ni\n= v(r2)\ni\n. It is straightforward to verify that for every i, j ∈[n] and r ∈[h] :"},{"paragraph_id":"p50","order":50,"text":"u(r)\nj , v(r)\ni"},{"paragraph_id":"p51","order":51,"text":"= Tr(γr(uj, vi)).\n(35)\nCombining (35) with the fact that S is covered by proper affine hyperplanes πγi,ci we conclude that\n• For all i ∈[n] and r ∈[h],"},{"paragraph_id":"p52","order":52,"text":"u(r)\ni , v(r)\ni"},{"paragraph_id":"p53","order":53,"text":"= 0;\n• For all i, j ∈[n] such that i ̸= j, there exists r ∈[h] such that"},{"paragraph_id":"p54","order":54,"text":"u(r)\nj , v(r)\ni"},{"paragraph_id":"p55","order":55,"text":"∈F∗\np.\nPick g(x1, . . . , xh) ∈Fp[x1, . . . , xh] to be a homogeneous degree h polynomial such that for a = (a1, . . . , ah) ∈\nFh\np : g(a) = 0 if and only if a is the all-zeros vector. The existence of such a polynomial g follows from [17,\nExample 6.7]. Set f = gp−1. Note that for a ∈Fh\np : f(a) = 0 if a is the all-zeros vector, and f(a) = 1 otherwise.\nFor all i ∈[n] define\nu′\ni = f"},{"paragraph_id":"p56","order":56,"text":"u(1)\ni , . . . , u(h)\ni"},{"paragraph_id":"p57","order":57,"text":"◦(1)\nand\nv′\ni = ̄f"},{"paragraph_id":"p58","order":58,"text":"v(1)\ni\n, . . . , v(h)\ni"},{"paragraph_id":"p59","order":59,"text":"◦(−1).\n(36)\nNote that f and ̄f are homogeneous degree (p −1)h polynomials in h variables. Therefore (32) implies that\nfor all i vectors u′\ni and v′\ni have length m′ ≤h(p−1)h(ml)(p−1)h. Combining identities (36) and (33) and using the\nproperties of dot products between vectors\nn\nu(r)\ni\no\nand\nn\nv(r)\ni\no\ndiscussed above we conclude that for every m there\nexist two n = ⌊cmt⌋-sized collections of vectors {u′\ni}i∈[n] and {v′\ni}i∈[n] in Fm′\nq , such that:\n• For all i ∈[n], (u′\ni, v′\ni) = −1;\n• For all i, j ∈[n] such that i ̸= j, (uj, vi) = 0.\nIt remains to notice that a family of vectors with such properties exists only if n ≤m′, i.e., ⌊cmt⌋≤h(p−1)h(ml)(p−1)h.\nGiven that we can pick m to be arbitrarily large, this implies that t ≤(p −1)h.\nThe next lemma presents the main result of this section.\nLemma 21 Let k be an odd integer. Suppose there exists a k-nice sequence; then for infinitely many primes p\nsome k of elements of Cp add up to zero.\nProof:\nAssume"},{"paragraph_id":"p60","order":60,"text":"Si ⊆F∗\nqi"},{"paragraph_id":"p61","order":61,"text":"i≥1 is k-nice. Let p be a fixed prime. Combining lemmas 18 and 20 we conclude\nthat every k algebraically nice subset S ⊆F∗\npl is at most (p −1)pk combinatorially nice. Note that our bound on\ncombinatorial niceness is independent of l. Therefore there are only finitely many extensions of the field Fp in the\nsequence {Fqi}i≥1 , and the set P = {charFqi}i≥1 is infinite. It remains to notice that according to lemma 17 for\nevery p ∈P there exist k elements of Cp that add up to zero.\nIn what follows we present necessary conditions for the existence of k-tuples of p-th roots of unity in F2 that\nsum to zero. We treat the k = 3 case separately since in that case we can use a specialized argument to derive a\nslightly stronger conclusion."},{"paragraph_id":"p62","order":62,"text":"5.2\nA necessary condition for the existence of k p-th roots of unity summing to zero\nLemma 22 Let k ≥3 be odd and p be a prime. Suppose there exist ζ1, . . . , ζk ∈Cp such that Pk\ni=1 ζi = 0; then\nord2(p) ≤2p1−1/(k−1).\n(37)\nProof:\nLet t = ord2(p). Note that Cp ⊆F2t. Note also that all elements of Cp other than the multiplicative\nidentity are proper elements of F2t. Therefore for every ζ ∈Cp where ζ ̸= 1 and every f(x) ∈F2[x] such that\ndeg f ≤t −1 we have: f(ζ) ̸= 0.\nBy multiplying Pk\ni=1 ζi = 0 through by ζ−1\nk , we may reduce to the case ζk = 1. Let ζ be the generator of Cp.\nFor every i ∈[k −1] pick wi ∈Zp such that ζi = ζwi. We now have Pk−1\ni=1 ζwi + 1 = 0. Set h = ⌊(t −1)/2⌋.\nConsider the (k −1)-tuples:\n(mw1 + i1, . . . , mwk−1 + ik−1) ∈Zk−1\np\n, for m ∈Zp and i1, . . . , ik−1 ∈[0, h].\n(38)\nSuppose two of these coincide, say\n(mw1 + i1, . . . , mwk−1 + ik−1) = (m′w1 + i′\n1, . . . , m′wk−1 + i′\nk−1),\nwith (m, i1, . . . , ik−1) ̸= (m′, i′\n1, . . . , i′\nk−1). Set n = m −m′ and jl = i′\nl −il for l ∈[k −1]. We now have\n(nw1, . . . , nwk−1) = (j1, . . . , jl)\nwith −h ≤j1, . . . , jk−1 ≤h. Observe that n ̸= 0, and thus it has a multiplicative inverse g ∈Zp. Consider a\npolynomial\nP(z) = zj1+h + . . . + zjk−1+h + zh ∈F2[z].\nNote that deg P ≤2h ≤t −1. Note also that P(1) = 1 and P(ζg) = 0. The latter identity contradicts the fact\nthat ζg is a proper element of F2t. This contradiction implies that all (k −1)-tuples in (38) are distinct. This yields\npk−1 ≥p\n t\n2\n k−1\n,\nwhich is equivalent to (37).\n5.3\nA necessary condition for the existence of three p-th roots of unity summing to zero\nIn this section we slightly strengthen lemma 22 in the special case when k = 3. Our argument is loosely inspired\nby the Agrawal-Kayal-Saxena deterministic primality test [1].\nLemma 23 Let p be a prime. Suppose there exist ζ1, ζ2, ζ3 ∈Cp that sum up to zero; then\nord2(p) ≤((4/3)p)1/2 .\n(39)\nProof:\nLet t = ord2(p). Note that Cp ⊆F2t. Note also that all elements of Cp other than the multiplicative\nidentity are proper elements of F2t. Therefore for every ζ ∈Cp where ζ ̸= 1 and every f(x) ∈F2[x] such that\ndeg f ≤t −1 we have: f(ζ) ̸= 0.\nObserve that ζ1 + ζ2 + ζ3 = 0 implies ζ1ζ−1\n2\n+ 1 = ζ3ζ−1\n2 . This yields\n ζ1ζ−1\n2\n+ 1\n p = 1. Put ζ = ζ1ζ−1\n2 .\nNote that ζ ̸= 1 and ζ, 1 + ζ ∈Cp. Consider the products πi,j = ζi(1 + ζ)j ∈Cp for 0 ≤i, j ≤t −1. Note that\nπi,j, πk,l cannot be the same if i ≥k and l ≥j, as then\nζi−k −(1 + ζ)l−j = 0,"},{"paragraph_id":"p63","order":63,"text":"but the left side has degree less than t. In other words, if πi,j = πk,l and (i, j) ̸= (k, l), then the pairs (i, j) and\n(k, l) are comparable under termwise comparison. In particular, either (k, l) = (i+a, j+b) or (i, j) = (k+a, l+b)\nfor some pair (a, b) with πa,b = 1.\nWe next check that there cannot be two distinct nonzero pairs (a, b), (a′, b′) with πa,b = πa′,b′ = 1. As above,\nthese pairs must be comparable; we may assume without loss of generality that a ≤a′, b ≤b′. The equations\nπa,b = 1 and πa′−a,b′−b = 1 force a + b ≥t and (a′ −a) + (b′ −b) ≥t, so a′ + b′ ≥2t. But a′, b′ ≤t −1,\ncontradiction.\nIf there is no nonzero pair (a, b) with 0 ≤a, b ≤t −1 and πa,b = 1, then all πi,j are distinct, so p ≥t2.\nOtherwise, as above, the pair (a, b) is unique, and the pairs (i, j) with 0 ≤i, j ≤t −1 and (i, j) ̸≥(a, b) are\npairwise distinct. The number of pairs excluded by the condition (i, j) ̸≥(a, b) is (t −a)(t −b); since a + b ≥t,\n(t −a)(t −b) ≤t2/4. Hence p ≥t2 −t2/4 = 3t2/4 as desired.\nWhile the necessary condition given by lemma 23 is quite far away from the sufficient condition given by\nlemma 8, it nonetheless suffices for checking that for most primes p, there do not exist three p-th roots of unity\nsumming to zero. For instance, among the 664578 odd primes p ≤108, all but 550 are ruled out by Lemma 23.\n(There is an easy argument that t must be odd if p > 3; this cuts the list down to 273 primes.) Each remaining\np can be tested by computing gcd(xp + 1, (x + 1)p + 1); the only examples we found that did not satisfy the\ncondition of lemma 8 were (p, t) = (73, 9), (262657, 27), (599479, 33), (121369, 39).\n5.4\nSummary\nIn the beginning of this section 5 we argued that in order to use the method of [34], (i.e., proposition 5) to obtain\nk-query locally decodable codes of length exp(nǫ) for some fixed k and all ǫ > 0, one needs to exhibit a k-nice\nsequence of subsets of finite fields. In what follows we use technical results of the previous subsections to show\nthat the existence of a k-nice sequence implies that infinitely many Mersenne numbers have large prime factors.\nTheorem 24 Let k be odd. Suppose there exists a k-nice sequence of subsets of finite fields; then for infinitely\nmany values of t we have\nP(2t −1) ≥(t/2)1+1/(k−2).\n(40)\nProof:\nUsing lemmas 21 and 22 we conclude that a k-nice sequence yields infinitely many primes p such that\nord2(p) ≤2p1−1/(k−1). Let p be such a prime and t = ord2(p). Then P(2t −1) ≥(t/2)1+1/(k−2).\nA combination of lemmas 21 and 23 yields a slightly stronger bound for the special case of 3-nice sequences.\nTheorem 25 Suppose there exists a 3-nice sequence of subsets; then for infinitely many values of t we have\nP(2t −1) ≥(3/4)t2.\n(41)\nWe would like to remind the reader that although the lower bounds for P(2t −1) given by (40) and (41) are\nextremely weak light of the widely accepted conjecture saying that the number of Mersenne primes is infinite,\nthey are substantially stronger than what is currently known unconditionally (2).\n6\nConclusion\nRecently [34] came up with a novel technique for constructing locally decodable codes and obtained vast im-\nprovements upon the earlier work. The construction proceeds in two steps. First [34] shows that if there exist\nsubsets of finite fields with certain ’nice’ properties then there exist good codes. Next [34] constructs nice subsets\nof prime fields Fp for Mersenne primes p."},{"paragraph_id":"p64","order":64,"text":"In this paper we have undertaken an in-depth study of nice subsets of general finite fields. We have shown\nthat constructing nice subsets is closely related to proving lower bounds on the size of largest prime factors of\nMersenne numbers. Specifically we extended the constructions of [34] to obtain nice subsets of prime fields Fp\nfor primes p that are large factors of Mersenne numbers. This implies that strong lower bounds for size of the\nlargest prime factors of Mersenne numbers yield better locally decodable codes. Conversely, we argued that if one\ncan obtain codes of subexponential length and constant query complexity through nice subsets of finite fields then\ninfinitely many Mersenne numbers have prime factors larger than known currently.\nAcknowledgements\nKiran Kedlaya’s research is supported by NSF CAREER grant DMS-0545904 and by the Sloan Research Fel-\nlowship. Sergey Yekhanin would like to thank Swastik Kopparty for providing the reference [5] and outlining\nthe proof of lemma 9. He would also like to thank Henryk Iwaniec, Carl Pomerance and Peter Sarnak for their\nfeedback regarding the number theory problems discussed in this paper.\nReferences\n[1] M. Agrawal, N. Kayal, N. Saxena, “PRIMES is in P,” Annals of Mathematics, vol. 160, pp. 781-793, 2004.\n[2] L. Babai, L. Fortnow, L. Levin, and M. Szegedy, “Checking computations in polylogarithmic time,”. In Proc.\nof the 23th ACM Symposium on Theory of Computing (STOC), pp. 21-31, 1991.\n[3] A. Beimel, Y. Ishai and E. Kushilevitz,“General constructions for information-theoretic private information\nretrieval,” Journal of Computer and System Sciences, vol. 71, pp. 213-247, 2005. Preliminary versions in\nSTOC 1999 and ICALP 2001.\n[4] A. Beimel, Y. Ishai, E. Kushilevitz, and J. F. Raymond. “Breaking the O\n n1/(2k−1) \nbarrier for information-\ntheoretic private information retrieval,” In Proc. of the 43rd IEEE Symposium on Foundations of Computer\nScience (FOCS), pp. 261-270, 2002.\n[5] J. Bourgain, M. Chang, “A Gauss sum estimate in arbitrary finite fields,” Comptes Rendus Mathematique,\nvol. 342, pp. 643-646, 2006.\n[6] B. Chor, O. Goldreich, E. Kushilevitz, and M. Sudan. “Private information retrieval,” In Proc. of the 36rd\nIEEE Symposium on Foundations of Computer Science (FOCS), pp. 41-50, 1995. Also, in Journal of the\nACM, vol. 45, 1998.\n[7] D. Cox, J. Little, D. O’Shea, Ideals, varieties, and algorithms: an introduction to computational algebraic\ngeometry and commutative algebra. Springer, 1996.\n[8] A. Deshpande, R. Jain, T. Kavitha, S. Lokam and J. Radhakrishnan, “Better lower bounds for locally decod-\nable codes,” In Proc. of the 20th IEEE Computational Complexity Conference (CCC), pp. 184-193, 2002.\n[9] Curtis Cooper, Steven Boone, http://www.mersenne.org/32582657.htm\n[10] P. Erdos and T. Shorey, “On the greatest prime factor of 2p −1 for a prime p and other expressions,” Acta.\nArith. vol. 30, pp. 257-265, 1976.\n[11] W. Gasarch, “A survey on private information retrieval,” The Bulletin of the EATCS, vol. 82, pp. 72-107,\n2004."},{"paragraph_id":"p65","order":65,"text":"[12] O. Goldreich, “Short locally testable codes and proofs,” Technical Report TR05-014, Electronic Colloquim\non Computational Complexity (ECCC), 2005.\n[13] O. Goldreich, H. Karloff, L. Schulman, L. Trevisan “Lower bounds for locally decodable codes and private\ninformation retrieval,” In Proc. of the 17th IEEE Computational Complexity Conference (CCC), pp. 175-183,\n2002.\n[14] B. Hemenway and R. Ostrovsky, “Public key encryption which is simultaneously a locally-decodable error-\ncorrecting code,” In Cryptology ePrint Archive, Report 2007/083.\n[15] J. Katz and L. Trevisan, “On the efficiency of local decoding procedures for error-correcting codes,” In Proc.\nof the 32th ACM Symposium on Theory of Computing (STOC), pp. 80-86, 2000.\n[16] I. Kerenidis, R. de Wolf, “Exponential lower bound for 2-query locally decodable codes via a quantum\nargument,” Journal of Computer and System Sciences, 69(3), pp. 395-420. Earlier version in STOC’03.\nquant-ph/0208062.\n[17] R. Lidl and H. Niederreiter, Finite Fields. Cambridge: Cambridge University Press, 1983.\n[18] L. Murata, C. Pomerance, “On the largest prime factor of a Mersenne number,” Number theory, CRM Proc.\nLecture Notes of American Mathematical Society vol. 36, pp. 209-218, 2004.\n[19] M. Murty and S. Wong, “The ABC conjecture and prime divisors of the Lucas and Lehmer sequences,” In\nProc. of Milennial Conference on Number Theory III, (Urbana, IL, 2000) (A. K. Peters, Natick, MA, 2002)\npp. 43-54.\n[20] K. Obata, “Optimal lower bounds for 2-query locally decodable linear codes,” In Proc. of the 6th RANDOM,\nvol. 2483 of Lecture Notes in Computer Science, pp. 39-50, 2002.\n[21] A. Polishchuk and D. Spielman, ”Nearly-linear size holographic proofs,” In Proc. of the 26th ACM Sympo-\nsium on Theory of Computing (STOC), pp. 194-203, 1994.\n[22] C. Pomerance, “Recent developments in primality testing,” Math. Intelligencer, 3:3, pp. 97-105, (1980/81).\n[23] P. Raghavendra, “A Note on Yekhanin’s locally decodable codes,” In Electronic Colloquium on Computa-\ntional Complexity Report TR07-016, 2007.\n[24] A. Romashchenko, “Reliable computations based on locally decodable codes,” In Proc. of the 23rd Inter-\nnational Symposium on Theoretical Aspects of Computer Science (STACS), vol. 3884 of Lecture Notes in\nComputer Science, pp. 537-548, 2006.\n[25] A. Schinzel, “On primitive factors of an −bn,” In Proc. of Cambridge Philos. Soc. vol. 58, pp. 555-562,\n1962.\n[26] C. Stewart, “The greatest prime factor of an −bn,” Acta Arith. vol. 26, pp. 427-433, 1974/75.\n[27] C. Stewart, “On divisors of Fermat, Fibonacci, Lucas, and Lehmer numbers,” In Proc. of London Math. Soc.\nvol. 35 (3), pp. 425-447, 1977.\n[28] M. Sudan, Efficient checking of polynomials and proofs and the hardness of approximation problems. PhD\nthesis, University of California at Berkeley, 1992.\n[29] L. Trevisan, “Some applications of coding theory in computational complexity,” Quaderni di Matematica,\nvol. 13, pp. 347-424, 2004."},{"paragraph_id":"p66","order":66,"text":"[30] S. Wehner and R. de Wolf, “Improved lower bounds for locally decodable codes and private information re-\ntrieval,” In Proc. of 32nd International Colloquium on Automata, Languages and Programming (ICALP’05),\nLNCS 3580, pp. 1424-1436.\n[31] Lenstra-Pomerance-Wagstaff conjecture. (2006, May 22). In Wikipedia, The Free Encyclopedia. Re-\ntrieved 00:18,\nOctober 3,\n2006,\nfrom http://en.wikipedia.org/w/index.php?title=Lenstra-Pomerance-\nWagstaff conjecture&oldid=54506577\n[32] S. Wagstaff, “Divisors of Mersenne numbers,” Math. Comp., 40:161, pp. 385-397, 1983.\n[33] D. Woodruff, “New lower bounds for general locally decodable codes,” Electronic Colloquium on Computa-\ntional Complexity, TR07-006, 2007.\n[34] S. Yekhanin, “Towards 3-query locally decodable codes of subexponential length,” In Proc. of the 39th ACM\nSymposium on Theory of Computing (STOC), 2007.\n[35] S. Yekhanin, Locally decodable codes and private information retrieval schemes. PhD thesis, MIT, to appear."}],"pages":[{"page":1,"text":"arXiv:0704.1694v1 [cs.CC] 13 Apr 2007\nLocally Decodable Codes From Nice Subsets of Finite Fields\nand Prime Factors of Mersenne Numbers\nKiran S. Kedlaya\nMIT\nkedlaya@mit.edu\nSergey Yekhanin\nMIT\nyekhanin@mit.edu\nAbstract\nA k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that\none can probabilistically recover any bit xi of the message by querying only k bits of the codeword C(x), even\nafter some constant fraction of codeword bits has been corrupted. The major goal of LDC related research is to\nestablish the optimal trade-off between length and query complexity of such codes.\nRecently [34] introduced a novel technique for constructing locally decodable codes and vastly improved the\nupper bounds for code length. The technique is based on Mersenne primes. In this paper we extend the work\nof [34] and argue that further progress via these methods is tied to progress on an old number theory question\nregarding the size of the largest prime factors of Mersenne numbers.\nSpecifically, we show that every Mersenne number m = 2t −1 that has a prime factor p > mγ yields a family\nof k(γ)-query locally decodable codes of length exp\n n1/t \n. Conversely, if for some fixed k and all ǫ > 0 one can\nuse the technique of [34] to obtain a family of k-query LDCs of length exp (nǫ) ; then infinitely many Mersenne\nnumbers have prime factors larger than known currently.\n1\nIntroduction\nClassical error-correcting codes allow one to encode an n-bit string x into in N-bit codeword C(x), in such\na way that x can still be recovered even if C(x) gets corrupted in a number of coordinates. It is well-known\nthat codewords C(x) of length N = O(n) already suffice to correct errors in up to δN locations of C(x) for\nany constant δ < 1/4. The disadvantage of classical error-correction is that one needs to consider all or most\nof the (corrupted) codeword to recover anything about x. Now suppose that one is only interested in recovering\none or a few bits of x. In such case more efficient schemes are possible. Such schemes are known as locally\ndecodable codes (LDCs). Locally decodable codes allow reconstruction of an arbitrary bit xi, from looking only\nat k randomly chosen coordinates of C(x), where k can be as small as 2. Locally decodable codes have numerous\napplications in complexity theory [15, 29], cryptography [6, 11] and the theory of fault tolerant computation [24].\nBelow is a slightly informal definition of LDCs:\nA (k, δ, ǫ)-locally decodable code encodes n-bit strings to N-bit codewords C(x), such that for every i ∈[n],\nthe bit xi can be recovered with probability 1−ǫ, by a randomized decoding procedure that makes only k queries,\neven if the codeword C(x) is corrupted in up to δN locations.\nOne should think of δ > 0 and ǫ < 1/2 as constants. The main parameters of interest in LDCs are the length\nN and the query complexity k. Ideally we would like to have both of them as small as possible. The concept\nof locally decodable codes was explicitly discussed in various papers in the early 1990s [2, 28, 21]. Katz and"},{"page":2,"text":"Trevisan [15] were the first to provide a formal definition of LDCs. Further work on locally decodable codes\nincludes [3, 8, 20, 4, 16, 30, 34, 33, 14, 23].\nBelow is a brief summary of what was known regarding the length of LDCs prior to [34]. The length of optimal\n2-query LDCs was settled by Kerenidis and de Wolf in [16] and is exp(n).1 The best upper bound for the length\nof 3-query LDCs was exp\n n1/2 \ndue to Beimel et al. [3], and the best lower bound is ̃Ω(n2) [33]. For general\n(constant) k the best upper bound was exp\n nO(log log k/(k log k)) \ndue to Beimel et al. [4] and the best lower bound\nis ̃Ω\n n1+1/(⌈k/2⌉−1) \n[33].\nThe recent work [34] improved the upper bounds to the extent that it changed the common perception of what\nmay be achievable [12, 11]. [34] introduced a novel technique to construct codes from so-called nice subsets\nof finite fields and showed that every Mersenne prime p = 2t −1 yields a family of 3-query LDCs of length\nexp\n n1/t \n. Based on the largest known Mersenne prime [9], this translates to a length of less than exp\n \nn10−7 \n.\nCombined with the recursive construction from [4], this result yields vast improvements for all values of k > 2. It\nhas often been conjectured that the number of Mersenne primes is infinite. If indeed this conjecture holds, [34] gets\nthree query locally decodable codes of length N = exp\n \nnO\n“\n1\nlog log n\n” \nfor infinitely many n. Finally, assuming\nthat the conjecture of Lenstra, Pomerance and Wagstaff [31, 22, 32] regarding the density of Mersenne primes\nholds, [34] gets three query locally decodable codes of length N = exp\n \nn\nO\n“\n1\nlog1−ǫ log n\n” \nfor all n, for every ǫ >\n0.\n1.1\nOur results\nIn this paper we address two natural questions left open by [34]:\n1. Are Mersenne primes necessary for the constructions of [34]?\n2. Has the technique of [34] been pushed to its limits, or one can construct better codes through a more clever\nchoice of nice subsets of finite fields?\nWe extend the work of [34] and answer both of the questions above. In what follows let P(m) denote the\nlargest prime factor of m. We show that one does not necessarily need to use Mersenne primes. It suffices to have\nMersenne numbers with polynomially large prime factors. Specifically, every Mersenne number m = 2t −1 such\nthat P(m) ≥mγ yields a family of k(γ)-query locally decodable codes of length exp\n n1/t \n. A partial converse\nalso holds. Namely, if for some fixed k ≥3 and all ǫ > 0 one can use the technique of [34] to (unconditionally)\nobtain a family of k-query LDCs of length exp (nǫ) ; then for infinitely many t we have\nP(2t −1) ≥(t/2)1+1/(k−2).\n(1)\nThe bound (1) may seem quite weak in light of the widely accepted conjecture saying that the number of\nMersenne primes is infinite. However (for any k ≥3) this bound is substantially stronger than what is currently\nknown unconditionally. Lower bounds for P(2t −1) have received a considerable amount of attention in the\nnumber theory literature [25, 26, 10, 27, 19, 18]. The strongest result to date is due to Stewart [27]. It says that\nfor all integers t ignoring a set of asymptotic density zero, and for all functions ǫ(t) > 0 where ǫ(t) tends to zero\nmonotonically and arbitrarily slowly:\nP(2t −1) > ǫ(t)t (log t)2 / log log t.\n(2)\n1Throughout the paper we use the standard notation exp(x)\ndef\n= eO(x)."},{"page":3,"text":"There are no better bounds known to hold for infinitely many values of t, unless one is willing to accept some\nnumber theoretic conjectures [19, 18]. We hope that our work will further stimulate the interest in proving lower\nbounds for P(2t −1) in the number theory community.\nIn summary, we show that one may be able to improve the unconditional bounds of [34] (say, by discovering a\nnew Mersenne number with a very large prime factor) using the same technique. However any attempts to reach\nthe exp (nǫ) length for some fixed query complexity and all ǫ > 0 require either progress on an old number theory\nproblem or some radically new ideas.\nIn this paper we deal only with binary codes for the sake of clarity of presentation. We remark however that\nour results as well as the results of [34] can be easily generalized to larger alphabets. Such generalization will be\ndiscussed in detail in [35].\n1.2\nOutline\nIn section 3 we introduce the key concepts of [34], namely that of combinatorial and algebraic niceness of\nsubsets of finite fields. We also briefly review the construction of locally decodable codes from nice subsets. In\nsection 4 we show how Mersenne numbers with large prime factors yield nice subsets of prime fields. In section 5\nwe prove a partial converse. Namely, we show that every finite field Fq containing a sufficiently nice subset, is an\nextension of a prime field Fp, where p is a large prime factor of a large Mersenne number. Our main results are\nsummarized in sections 4.3 and 5.4.\n2\nNotation\nWe use the following standard mathematical notation:\n• [s] = {1, . . . , s};\n• Zn denotes integers modulo n;\n• Fq is a finite field of q elements;\n• dH(x, y) denotes the Hamming distance between binary vectors x and y;\n• (u, v) stands for the dot product of vectors u and v;\n• For a linear space L ⊆Fm\n2 , L⊥denotes the dual space. That is, L⊥= {u ∈Fm\n2 | ∀v ∈L, (u, v) = 0};\n• For an odd prime p, ord2(p) denotes the smallest integer t such that p | 2t −1.\n3\nNice subsets of finite fields and locally decodable codes\nIn this section we introduce the key technical concepts of [34], namely that of combinatorial and algebraic\nniceness of subsets of finite fields. We briefly review the construction of locally decodable codes from nice\nsubsets. Our review is concise although self-contained. We refer the reader interested in a more detailed and\nintuitive treatment of the construction to the original paper [34]. We start by formally defining locally decodable\ncodes.\nDefinition 1 A binary code C : {0, 1}n →{0, 1}N is said to be (k, δ, ǫ)-locally decodable if there exists a\nrandomized decoding algorithm A such that"},{"page":4,"text":"1. For all x ∈{0, 1}n, i ∈[n] and y ∈{0, 1}N such that dH(C(x), y) ≤δN : Pr[Ay(i) = xi] ≥1−ǫ, where\nthe probability is taken over the random coin tosses of the algorithm A.\n2. A makes at most k queries to y.\nWe now introduce the concepts of combinatorial and algebraic niceness of subsets of finite fields. Our defini-\ntions are syntactically slightly different from the original definitions in [34]. We prefer these formulations since\nthey are more appropriate for the purposes of the current paper. In what follows let F∗\nq denote the multiplicative\ngroup of Fq.\nDefinition 2 A set S ⊆F∗\nq is called t combinatorially nice if for some constant c > 0 and every positive integer\nm there exist two n = ⌊cmt⌋-sized collections of vectors {u1, . . . , un} and {v1, . . . , vn} in Fm\nq , such that\n• For all i ∈[n], (ui, vi) = 0;\n• For all i, j ∈[n] such that i ̸= j, (uj, vi) ∈S.\nDefinition 3 A set S ⊆F∗\nq is called k algebraically nice if k is odd and there exists an odd k′ ≤k and two sets\nS0, S1 ⊆Fq such that\n• S0 is not empty;\n• |S1| = k′;\n• For all α ∈Fq and β ∈S : |S0 ∩(α + βS1)| ≡0 mod (2).\nThe following lemma shows that for an algebraically nice set S, the set S0 can always be chosen to be large. It\nis a straightforward generalization of [34, lemma 15].\nLemma 4 Let S ⊆F∗\nq be a k algebraically nice set. Let S0, S1 ⊆Fq be sets from the definition of algebraic\nniceness of S. One can always redefine the set S0 to satisfy |S0| ≥⌈q/2⌉.\nProof:\nLet L be the linear subspace of Fq\n2 spanned by the incidence vectors of the sets α + βS1, for α ∈Fq and\nβ ∈S. Observe that L is invariant under the actions of a 1-transitive permutation group (permuting the coordinates\nin accordance with addition in Fq). This implies that the space L⊥is also invariant under the actions of the same\ngroup. Note that L⊥has positive dimension since it contains the incidence vector of the set S0. The last two\nobservations imply that L⊥has full support, i.e., for every i ∈[q] there exists a vector v ∈L⊥such that vi ̸= 0. It\nis easy to verify that any linear subspace of Fq\n2 that has full support contains a vector of Hamming weight at least\n⌈q/2⌉. Let v ∈L⊥be such a vector. Redefining the set S0 to be the set of nonzero coordinates of v we conclude\nthe proof.\nWe now proceed to the core proposition of [34] that shows how sets exhibiting both combinatorial and algebraic\nniceness yield locally decodable codes.\nProposition 5 Suppose S ⊆F∗\nq is t combinatorially nice and k algebraically nice; then for every positive integer\nn there exists a code of length exp(n1/t) that is (k, δ, 2kδ) locally decodable for all δ > 0.\nProof:\nOur proof comes in three steps. We specify encoding and local decoding procedures for our codes and\nthen argue the lower bound for the probability of correct decoding. We use the notation from definitions 2 and 3.\nEncoding: We assume that our message has length n = ⌊cmt⌋for some value of m. (Otherwise we pad the\nmessage with zeros. It is easy to see that such padding does not not affect the asymptotic length of the code.) Our"},{"page":5,"text":"code will be linear. Therefore it suffices to specify the encoding of unit vectors e1, . . . , en, where ej has length n\nand a unique non-zero coordinate j. We define the encoding of ej to be a qm long vector, whose coordinates are\nlabelled by elements of Fm\nq . For all w ∈Fm\nq we set:\nEnc(ej)w =\n 1,\nif (uj, w) ∈S0;\n0,\notherwise.\n(3)\nIt is straightforward to verify that we defined a code encoding n bits to exp(n1/t) bits.\nLocal decoding: Given a (possibly corrupted) codeword y and an index i ∈[n], the decoding algorithm A picks\nw ∈Fm\nq , such that (ui, w) ∈S0 uniformly at random, reads k′ ≤k coordinates of y, and outputs the sum:\nX\nλ∈S1\nyw+λvi.\n(4)\nProbability of correct decoding: First we argue that decoding is always correct if A picks w ∈Fm\nq such that\nall bits of y in locations {w + λvi}λ∈S1 are not corrupted. We need to show that for all i ∈[n], x ∈{0, 1}n and\nw ∈Fm\nq , such that (ui, w) ∈S0:\nX\nλ∈S1\n \n \nn\nX\nj=1\nxj Enc(ej)\n \n \nw+λvi\n= xi.\n(5)\nNote that\nX\nλ∈S1\n \n \nn\nX\nj=1\nxj Enc(ej)\n \n \nw+λvi\n=\nn\nX\nj=1\nxj\nX\nλ∈S1\nEnc(ej)w+λvi =\nn\nX\nj=1\nxj\nX\nλ∈S1\nI [(uj, w + λvi) ∈S0] ,\n(6)\nwhere I[γ ∈S0] = 1 if γ ∈S0 and zero otherwise. Now note that\nX\nλ∈S1\nI [(uj, w + λvi) ∈S0] =\nX\nλ∈S1\nI [(uj, w) + λ(uj, vi) ∈S0] =\n 1,\nif i = j,\n0,\notherwise.\n(7)\nThe last identity in (7) for i = j follows from: (ui, vi) = 0, (ui, w) ∈S0 and k′ = |S1| is odd. The last identity\nfor i ̸= j follows from (uj, vi) ∈S and the algebraic niceness of S. Combining identities (6) and (7) we get (5).\nNow assume that up to δ fraction of bits of y are corrupted. Let Ti denote the set of coordinates whose labels\nbelong to\n \nw ∈Fm\nq | (ui, w) ∈S0\n \n. Recall that by lemma 4, |Ti| ≥qm/2. Thus at most 2δ fraction of coor-\ndinates in Ti contain corrupted bits. Let Qi =\n \n{w + λvi}λ∈S1 | w : (ui, w) ∈S0\n \nbe the family of k′-tuples\nof coordinates that may be queried by A. (ui, vi) = 0 implies that elements of Qi uniformly cover the set Ti.\nCombining the last two observations we conclude that with probability at least 1 −2kδ A picks an uncorrupted\nk′-tuple and outputs the correct value of xi.\nAll locally decodable codes constructed in this paper are obtained by applying proposition 5 to certain nice\nsets. Thus all our codes have the same dependence of ǫ (the probability of the decoding error) on δ (the fraction\nof corrupted bits). In what follows we often ignore these parameters and consider only the length and query\ncomplexity of codes."},{"page":6,"text":"4\nMersenne numbers with large prime factors yield nice subsets of prime fields\nIn what follows let ⟨2⟩⊆F∗\np denote the multiplicative subgroup of F∗\np generated by 2. In [34] it is shown\nthat for every Mersenne prime p = 2t −1 the set ⟨2⟩⊆F∗\np is simultaneously 3 algebraically nice and ord2(p)\ncombinatorially nice. In this section we prove the same conclusion for a substantially broader class of primes.\nLemma 6 Suppose p is an odd prime; then ⟨2⟩⊆F∗\np is ord2(p) combinatorially nice.\nProof: Let t = ord2(p). Clearly, t divides p −1. We need to specify a constant c > 0 such that for every positive\ninteger m there exist two n = ⌊cmt⌋-sized collections of m long vectors over Fp satisfying:\n• For all i ∈[n], (ui, vi) = 0;\n• For all i, j ∈[n] such that i ̸= j, (uj, vi) ∈⟨2⟩.\nFirst assume that m has the shape m =\n \nm′−1+(p−1)/t\n(p−1)/t\n \n, for some integer m′ ≥p −1. In this case [34, lemma\n13] gives us a collection of n =\n \nm′\np−1\n \nvectors with the right properties. Observe that n ≥cmt for a constant\nc that depends only on p and t. Now assume m does not have the right shape, and let m1 be the largest integer\nsmaller than m that does have it. In order to get vectors of length m we use vectors of length m1 coming from [34,\nlemma 13] padded with zeros. It is not hard to verify such a construction still gives us n ≥cmt large families of\nvectors for a suitably chosen constant c.\nWe use the standard notation F to denote the algebraic closure of the field F. Also let Cp ⊆F\n∗\n2 denote the\nmultiplicative subgroup of p-th roots of unity in F2. The next lemma generalizes [34, lemma 14].\nLemma 7 Let p be a prime and k be odd. Suppose there exist ζ1, . . . , ζk ∈Cp such that\nζ1 + . . . + ζk = 0;\n(8)\nthen ⟨2⟩⊆F∗\np is k algebraically nice.\nProof:\nIn what follows we define the set S1 ⊆Fp and prove the existence of a set S0 such that that together S0\nand S1 yield k algebraic niceness of ⟨2⟩. Identity 8 implies that there exists an odd integer k′ ≤k and k′ distinct\np-th roots of unity ζ′\n1, . . . , ζ′\nk ∈Cp such that\nζ′\n1 + . . . + ζ′\nk′ = 0.\n(9)\nLet t = ord2(p). Observe that Cp ⊆F2t. Let g be a generator of Cp. Identity (9) yields gγ1 + . . . + gγk′ = 0, for\nsome distinct values of {γi}i∈[k′]. Set S1 = {γ1, . . . , γk′}.\nConsider a natural one to one correspondence between subsets S′ of Fp and polynomials φS′(x) in the ring\nF2[x]/(xp −1) : φS′(x) = P\ns∈S′ xs. It is easy to see that for all sets S′ ⊆Fp and all α, β ∈Fp, such that β ̸= 0 :\nφα+βS′(x) = xαφS′(xβ).\nLet α be a variable ranging over Fp and β be a variable ranging over ⟨2⟩. We are going to argue the existence of a\nset S0 that has even intersections with all sets of the form α+βS1, by showing that all polynomials φα+βS1 belong\nto a certain linear space L ∈F2[x]/(xp −1) of dimension less than p. In this case any nonempty set T ⊆Fp such\nthat φT ∈L⊥can be used as the set S0. Let τ(x) = gcd(xp −1, φS1(x)). Note that τ(x) ̸= 1 since g is a common\nroot of xp −1 and φS1(x). Let L be the space of polynomials in F2[x]/(xp −1) that are multiples of τ(x). Clearly,\ndim L = p −deg τ. Fix some α ∈Fp and β ∈⟨2⟩. Let us prove that φα+βS1(x) is in L :\nφα+βS1(x) = xαφS1(xβ) = xα(φS1(x))β.\nThe last identity above follows from the fact that for any f ∈F2[x] and any integer i : f(x2i) = (f(x))2i."},{"page":7,"text":"In what follows we present sufficient conditions for the existence of k-tuples of p-th roots of unity in F2 that\nsum to zero. We treat the k = 3 case separately since in that case we can use a specialized argument to derive a\nmore explicit conclusion.\n4.1\nA sufficient condition for the existence of three p-th roots of unity summing to zero\nLemma 8 Let p be an odd prime. Suppose ord2(p) < (4/3) log2 p; then there exist three p-th roots of unity in F2\nthat sum to zero.\nProof: We start with a brief review of some basic concepts of projective algebraic geometry. Let F be a field, and\nf ∈F[x, y, z] be a homogeneous polynomial. A triple (x0, y0, z0) ∈F3 is called a zero of f if f(x0, y0, z0) = 0.\nA zero is called nontrivial if it is different from the origin. An equation f = 0 defines a projective plane curve χf.\nNontrivial zeros of f considered up to multiplication by a scalars are called F-rational points of χf. If F is a finite\nfield it makes sense to talk about the number of F-rational points on a curve.\nLet t = ord2(p). Note that Cp ⊆F2t. Consider a projective plane Fermat curve χ defined by\nx(2t−1)/p + y(2t−1)/p + z(2t−1)/p = 0.\n(10)\nLet us call a point a on χ trivial if one of the coordinates of a is zero. Cyclicity of F∗\n2t implies that χ contains\nexactly 3(2t −1)/p trivial F2t-rational points. Note that every nontrivial point of χ yields a triple of elements of\nCp that sum to zero. The classical Weil bound [17, p. 330] provides an estimate\n|Nq −(q + 1)| ≤(d −1)(d −2)√q\n(11)\nfor the number Nq of Fq-rational points on an arbitrary smooth projective plane curve of degree d. (11) implies\nthat in case\n2t + 1 >\n 2t −1\np\n−1\n 2t −1\np\n−2\n \n2t/2 + 32t −1\np\n(12)\nthere exists a nontrivial point on the curve (10). Note that (12) follows from\n2t + 1 >\n 2t\np\n 2t\np\n \n2t/2 −23t/2+1\np\n+ 3 ∗2t\np\n,\n(13)\nand (13) follows from\n2t > 22t+t/2/p2\nand\n2t/2+1 > 3.\nNow note that the first inequality above follows from t < (4/3) log2 p and the second follows from t > 1.\nNote that the constant 4/3 in lemma 8 cannot be improved to 2: there are no three elements of C13264529 that\nsum to zero, even though ord2(13264529) = 47 < 2 ∗log2 13264529 ≈47.3.\n4.2\nA sufficient condition for the existence of k p-th roots of unity summing to zero\nOur argument in this section comes in three steps. First we briefly review the notion of (additive) Fourier\ncoefficients of subsets of F2t. Next, we invoke a folklore argument to show that subsets of F2t with appropriately\nsmall nontrivial Fourier coefficients contain k-tuples of elements that sum to zero. Finally, we use a recent result of\nBourgain and Chang [5] (generalizing the classical estimate for Gauss sums) to argue that (under certain constraints\non p) all nontrivial Fourier coefficients of Cp are small.\nFor x ∈F2t let Tr(x) = x + x2 + . . . + x2t−1 denote the trace of x. It is not hard to verify that for all x,\nTr(x) ∈F2. Characters of F2t are homomorphisms from the additive group of F2t into the multiplicative group"},{"page":8,"text":"{±1}. There exist 2t characters. We denote characters by χa, where a ranges in F2t, and set χa(x) = (−1)Tr(ax).\nLet C(x) denote the incidence function of a set C ⊆F2t. For arbitrary a ∈Ft\n2 the Fourier coefficient χa(C) is\ndefined by χa(C) = P χa(x)C(x), where the sum is over all x ∈F2t. Fourier coefficient χ0(C) = |C| is called\ntrivial, and other Fourier coefficients are called nontrivial. In what follows P\nχ stands for summation over all 2t\ncharacters of F2t. We need the following two standard properties of characters and Fourier coefficients.\nX\nχ\nχ(x) =\n 2t,\nif x = 0;\n0,\notherwise.\n(14)\nX\nχ\nχ2(C) = 2t|C|.\n(15)\nThe following lemma is a folklore.\nLemma 9 Let C ⊆F2t and k ≥3 be a positive integer. Let F be the largest absolute value of a nontrivial Fourier\ncoefficient of C. Suppose\nF\n|C| <\n |C|\n2t\n 1/(k−2)\n(16)\nthen there exist k elements of C that sum to zero.\nProof:\nLet M(C) = # {ζ1, . . . , ζk ∈C | ζ1 + . . . + ζk = 0} . (14) yields\nM(C) = 1\n2t\nX\nx1,...,xk∈F2t\nC(x1) . . . C(xk)\nX\nχ\nχ(x1 + . . . + xk).\n(17)\nNote that χ(x1 + . . . + xk) = χ(x1) . . . χ(xk). Changing the order of summation in (17) we get\nM(C) = 1\n2t\nX\nχ\nX\nx1,...,xk∈F2t\nC(x1) . . . C(xk)χ(x1) . . . χ(xk) = 1\n2t\nX\nχ\nχk(C).\n(18)\nNote that\n1\n2t\nX\nχ\nχk(C) = |C|k\n2t\n+ 1\n2t\nX\nχ̸=χ0\nχk(C) ≥|C|k\n2t\n−F k−2 1\n2t\nX\nχ\nχ2(C) = |C|k\n2t\n−F k−2|C|,\n(19)\nwhere the last identity follows from (15). Combining (18) and (19) we conclude that (16) implies M(C) > 0.\nThe following lemma is a special case of [5, theorem 1].\nLemma 10 Assume that n | 2t −1 and satisfies the condition\ngcd\n \nn, 2t −1\n2t′ −1\n \n< 2t(1−ǫ)−t′,\nfor all 1 ≤t′ < t, t′ | t,\nwhere ǫ > 0 is arbitrary and fixed. Then for all a ∈F∗\n2t\n \nX\nx∈F2t\n(−1)Tr(axn)\n \n< c12t(1−δ),\n(20)\nwhere δ = δ(ǫ) > 0 and c1 = c1(ǫ) are absolute constants."},{"page":9,"text":"Below is the main result of this section. Recall that Cp denotes the set of p-th roots of unity in F2.\nLemma 11 For every c > 0 there exists an odd integer k = k(c) such that the following implication holds. If p is\nan odd prime and ord2(p) < c log2 p then some k elements of Cp sum to zero.\nProof:\nNote that if there exist k′ elements of a set C ⊆F2 that sum to zero, where k′ is odd; then there exist\nk elements of C that sum to zero for every odd k ≥k′. Also note that the sum of all p-th roots of unity is\nzero. Therefore given c it suffices to prove the existence of an odd k = k(c) that works for all sufficiently large\np. Let t = ord2(p). Observe that p > 2t/c. Assume p is sufficiently large so that t > 2c. Next we show that\nthe precondition of lemma 10 holds for n = (2t −1)/p and ǫ = 1/(2c). Let t′ | t and 1 ≤t′ < t. Clearly\ngcd(2t′ −1, p) = 1. Therefore\ngcd\n 2t −1\np\n, 2t −1\n2t′ −1\n \n=\n2t −1\np(2t′ −1) < 2t(1−1/c)\n2t′ −1 ,\n(21)\nwhere the inequality follows from p > 2t/c. Clearly, t > 2c yields 2t/(2c)/2 > 1. Multiplying the right hand side\nof (21) by 2t/(2c)/2 and using 2(2t′ −1) > 2t′ we get\ngcd\n 2t −1\np\n, 2t −1\n2t′ −1\n \n< 2t(1−1/(2c))−t′.\n(22)\nCombining (22) with lemma 10 we conclude that there exist δ > 0 and c1 such that for all a ∈F∗\n2t\n \nX\nx∈F2t\n(−1)Tr\n“\nax(2t−1)/p” \n< c12t(1−δ).\n(23)\nObserve that x(2t−1)/p takes every value in Cp exactly (2t −1)/p times when x ranges over F∗\n2t. Thus (23) implies\n(2t −1)(F/p) < c12t(1−δ),\n(24)\nwhere F denotes that largest nontrivial Fourier coefficient of Cp. (24) yields F/p < (2c1)2−δt. Pick k ≥3 to be\nthe smallest odd integer such that (1 −1/c)/(k −2) < δ. We now have\nF\np < 2−(1−1/c)t\n(k−2)\n(25)\nfor all sufficiently large values of p. Combining p > 2t/c with (25) we get\nF\n|Cp| <\n |Cp|\n2t\n 1/(k−2)\n,\nand the application of lemma 9 concludes the proof.\n4.3\nSummary\nIn this section we summarize our positive results and show that one does not necessarily need to use Mersenne\nprimes to construct locally decodable codes via the methods of [34]. It suffices to have Mersenne numbers with\npolynomially large prime factors. Recall that P(m) denotes the largest prime factor of an integer m. Our first\ntheorem gets 3-query LDCs from Mersenne numbers m with prime factors larger than m3/4."},{"page":10,"text":"Theorem 12 Suppose P(2t −1) > 20.75t; then for every message length n there exists a three query locally\ndecodable code of length exp(n1/t).\nProof:\nLet P(2t −1) = p. Observe that p | 2t −1 and p > 20.75t yield ord2(p) < (4/3) log2 p. Combining\nlemmas 8,7 and 6 with proposition 5 we obtain the statement of the theorem.\nAs an example application of theorem 12 one can observe that P(223 −1) = 178481 > 2(3/4)∗23 ≈155872 yields\na family of three query locally decodable codes of length exp(n1/23). Theorem 12 immediately yields:\nTheorem 13 Suppose for infinitely many t we have P(2t −1) > 20.75t; then for every ǫ > 0 there exists a family\nof three query locally decodable codes of length exp(nǫ).\nThe next theorem gets constant query LDCs from Mersenne numbers m with prime factors larger than mγ for\nevery value of γ.\nTheorem 14 For every γ > 0 there exists an odd integer k = k(γ) such that the following implication holds.\nSuppose P(2t −1) > 2γt; then for every message length n there exists a k query locally decodable code of length\nexp(n1/t).\nProof:\nLet P(2t −1) = p. Observe that p | 2t −1 and p > 2γt yield ord2(p) < (1/γ) log2 p. Combining\nlemmas 22,7 and 6 with proposition 5 we obtain the statement of the theorem.\nAs an immediate corollary we get:\nTheorem 15 Suppose for some γ > 0 and infinitely many t we have P(2t −1) > 2γt; then there is a fixed k such\nthat for every ǫ > 0 there exists a family of k query locally decodable codes of length exp(nǫ).\n5\nNice subsets of finite fields yield Mersenne numbers with large prime factors\nDefinition 16 We say that a sequence\n \nSi ⊆F∗\nqi\n \ni≥1 of subsets of finite fields is k-nice if every Si is k alge-\nbraically nice and t(i) combinatorially nice, for some integer valued monotonically increasing function t.\nThe core proposition 5 asserts that a subset S ⊆F∗\nq that is k algebraically nice and t combinatorially nice yields\na family of k-query locally decodable codes of length exp(n1/t). Clearly, to get k-query LDCs of length exp(nǫ)\nfor some fixed k and every ǫ > 0 via this proposition, one needs to exhibit a k-nice sequence. In this section\nwe show how the existence of a k-nice sequence implies that infinitely many Mersenne numbers have large prime\nfactors. Our argument proceeds in two steps. First we show that a k-nice sequence yields an infinite sequence of\nprimes {pi}i≥1 , where every Cpi contains a k-tuple of elements summing to zero. Next we show that Cp contains\na short additive dependence only if p is a large factor of a Mersenne number.\n5.1\nA nice sequence yields infinitely many primes p with short dependencies between p-th roots of unity\nWe start with some notation. Consider a a finite field Fq = Fpl, where p is prime. Fix a basis e1, . . . , el of Fq\nover Fp. In what follows we often write (α1, . . . , αl) ∈Fl\np to denote α = Pl\ni=1 αiei ∈Fq. Let R denote the ring\nF2[x1, . . . , xl]/(xp\n1 −1, . . . , xp\nl −1). Consider a natural one to one correspondence between subsets S1 of Fq and\npolynomials φS1(x1, . . . , xl) ∈R.\nφS1(x1, . . . , xl) =\nX\n(α1,...,αl)∈S1\nxα1\n1 . . . xαl\nl ."},{"page":11,"text":"It is easy to see that for all sets S1 ⊆Fq and all α, β ∈Fq :\nφ(α1,...,αl)+βS1(x1, . . . , xl) = xα1\n1 . . . xαl\nl φβS1(x1, . . . , xl).\n(26)\nLet Γ be a family of subsets of Fq. It is straightforward to verify that a set S0 ⊆Fq has even intersections with\nevery element of Γ if and only if φS0 belongs to L⊥, where L is the linear subspace of R spanned by {φS1}S1∈Γ .\nCombining the last observation with formula (26) we conclude that a set S ⊆F∗\nq is k algebraically nice if and\nonly if there exists a set S1 ⊆Fq of odd size k′ ≤k such that the ideal generated by polynomials {φβS1}{β∈S}\nis a proper ideal of R. Note that polynomials {f1, . . . , fh} ∈R generate a proper ideal if an only if polynomials\n{f1, . . . , fh, xp\n1 −1, . . . , xp\nl −1} generate a proper ideal in F2[x1, . . . , xl]. Also note that a family of polynomials\ngenerates a proper ideal in F2[x1, . . . , xl] if and only if it generates a proper ideal in F2[x1, . . . , xl]. Now an\napplication of Hilbert’s Nullstellensatz [7, p. 168] implies that a set S ⊆F∗\nq is k algebraically nice if and only\nif there is a set S1 ⊆Fq of odd size k′ ≤k such that the polynomials {φβS1}{β∈S} and {xp\ni −1}1≤i≤l have a\ncommon root in F2.\nLemma 17 Let Fq = Fpl, where p is prime. Suppose Fq contains a nonempty k algebraically nice subset; then\nthere exist ζ1, . . . , ζk ∈Cp such that ζ1 + . . . + ζk = 0.\nProof:\nAssume S ⊆F∗\nq is nonempty and k algebraically nice. The discussion above implies that there exists\nS1 ⊆Fq of odd size k′ ≤k such that all polynomials {φβS1}{β∈S} vanish at some (ζ1, . . . , ζl) ∈Cl\np. Fix an\narbitrary β0 ∈S, and note that Cp is closed under multiplication. Thus,\nφβ0S1(ζ1, . . . , ζl) = 0\n(27)\nyields k′ p-th roots of unity that add up to zero. It is readily seen that one can extend (27) (by adding an appropriate\nnumber of pairs of identical roots) to obtain k p-th roots of unity that add up to zero for any odd k ≥k′.\nNote that lemma 17 does not suffice to prove that a k-nice sequence\n \nSi ⊆F∗\nqi\n \ni≥1 yields infinitely many primes p\nwith short (nontrivial) additive dependencies in Cp. We need to argue that the set {charFqi}i≥1 can not be finite. To\nproceed, we need some more notation. Recall that q = pl and p is prime. For x ∈Fq let Tr(x) = x+. . .+xpl−1 ∈\nFp denote the (absolute) trace of x. For γ ∈Fq, c ∈F∗\np we call the set πγ,c = {x ∈Fq | Tr(γx) = c} a proper\naffine hyperplane of Fq.\nLemma 18 Let Fq = Fpl, where p is prime. Suppose S ⊆F∗\nq is k algebraically nice; then there exist h ≤pk\nproper affine hyperplanes {πγi,ci}1≤i≤h of Fq such that S ⊆\nhS\ni=1\nπγi,ci.\nProof:\nDiscussion preceding lemma 17 implies that there exists a set S1 = {σ1, . . . , σk′} ⊆Fq of odd size\nk′ ≤k such that all polynomials {φβS1}{β∈S} vanish at some (ζ1, . . . , ζl) ∈Cl\np. Let ζ be a generator of Cp. For\nevery 1 ≤i ≤l pick ωi ∈Zp such that ζi = ζωi. For every β ∈S, φβS1(ζ1, . . . , ζl) = 0 yields\nX\nμ=(μ1,...,μl)∈βS1\nζ\nPl\ni=1 μiωi = 0.\n(28)\nObserve that for fixed values {ωi}1≤i≤l ∈Zp the map D(μ) = Pl\ni=1 μiωi is a linear map from Fq to Fp. It is\nnot hard to prove that every such map can be expressed as D(μ) = Tr(δμ) for an appropriate choice of δ ∈Fq.\nTherefore we can rewrite (28) as\nX\nμ∈βS1\nζTr(δμ) =\nX\nσ∈S1\nζTr(δβσ) = 0.\n(29)"},{"page":12,"text":"Let W =\nn\n(w1, . . . , wk′) ∈Zk′\np | ζw1 + . . . + ζwk′ = 0\no\ndenote the set of exponents of k′-dependencies be-\ntween powers of ζ. Clearly, |W| ≤pk. Identity (29) implies that every β ∈S satisfies\n \n \n \n \n \nTr((δσ1)β)\n=\nw1,\n...\nTr((δσk′)β)\n=\nwk′;\n(30)\nfor an appropriate choice of (w1, . . . , wk′) ∈W. Note that the all-zeros vector does not lie in W since k′ is odd.\nTherefore at least one of the identities in (30) has a non-zero right-hand side, and defines a proper affine hyperplane\nof Fq. Collecting one such hyperplane for every element of W we get a family of |W| proper affine hyperplanes\ncontaining every element of S.\nLemma 18 gives us some insight into the structure of algebraically nice subsets of Fq. Our next goal is to develop\nan insight into the structure of combinatorially nice subsets. We start by reviewing some relations between tensor\nand dot products of vectors. For vectors u ∈Fm\nq and v ∈Fn\nq let u⊗v ∈Fmn\nq\ndenote the tensor product of u and v.\nCoordinates of u ⊗v are labelled by all possible elements of [m] × [n] and (u ⊗v)i,j = uivj. Also, let u⊗l denote\nthe l-the tensor power of u and u ◦v denote the concatenation of u and v. The following identity is standard. For\nany u, x ∈Fm\nq and v, y ∈Fn\nq :\n(u ⊗v, x ⊗y) =\nX\ni∈[m],j∈[n]\nuivjxiyj =\n \n X\ni∈[m]\nuixi\n \n \n \n X\nj∈[n]\nvjyj\n \n = (u, x)(v, y).\n(31)\nIn what follows we need a generalization of identity (31). Let f(x1, . . . , xh) = P\ni cixαi\n1\n1 . . . x\nαi\nh\nh be a polynomial\nin Fq[x1, . . . , xh]. Given f we define ̄f ∈Fq[x1, . . . , xh] by ̄f = P\ni xαi\n1\n1 . . . x\nαi\nh\nh , i.e., we simply set all nonzero\ncoefficients of f to 1. For vectors u1, . . . , uh in Fm\nq define\nf(u1, . . . , uh) = ◦i ciu⊗αi\n1\n1\n⊗. . . ⊗u\n⊗αi\nh\nh\n.\n(32)\nNote that to obtain f(u1, . . . , uh) we replaced products in f by tensor products and addition by concatenation.\nClearly, f(u1, . . . , uh) is a vector whose length may be larger than m.\nClaim 19 For every f ∈Fq[x1, . . . , xh] and u1, . . . , uh, v1, . . . , vh ∈Fm\nq :\n f(u1, . . . , uh), ̄f(v1, . . . , vh)\n \n= f((u1, v1), . . . , (uh, vh)).\n(33)\nProof: Let u = (u1, . . . , uh) and v = (v1, . . . , vh). Observe that if (33) holds for polynomials f1 and f2 defined\nover disjoint sets of monomials then it also holds for f = f1 + f2 :\n f(u), ̄f(v)\n \n=\n (f1 + f2)(u), ( ̄f1 + ̄f2)(v)\n \n=\n f1(u) ◦f2(u), ̄f1(v) ◦ ̄f2(v)\n \n=\nf1 ((u1, v1), . . . , (uh, vh)) + f2 ((u1, v1), . . . , (uh, vh)) = f ((u1, v1), . . . , (uh, vh)) .\nTherefore it suffices to prove (33) for monomials f = cxα1\n1 . . . xαh\nh . It remains to notice identity (33) for monomi-\nals f = cxα1\n1 . . . xαh\nh follows immediately from formula (31) using induction on Ph\ni=1 αi.\nThe next lemma bounds combinatorial niceness of certain subsets of F∗\nq.\nLemma 20 Let Fq = Fpl, where p is prime. Let S ⊆F∗\nq. Suppose there exist h proper affine hyperplanes\n{πγr,cr}1≤r≤h of Fq such that S ⊆\nhS\nr=1\nπγr,cr; then S is at most h(p −1) combinatorially nice."},{"page":13,"text":"Proof:\nAssume S is t combinatorially nice. This implies that for some c > 0 and every m there exist two\nn = ⌊cmt⌋-sized collections of vectors {ui}i∈[n] and {vi}i∈[n] in Fm\nq , such that:\n• For all i ∈[n], (ui, vi) = 0;\n• For all i, j ∈[n] such that i ̸= j, (uj, vi) ∈S.\nFor a vector u ∈Fm\nq and integer e let ue denote a vector resulting from raising every coordinate of u to the power e.\nFor every i ∈[n] and r ∈[h] define vectors u(r)\ni\nand v(r)\ni\nin Fml\nq\nby\nu(r)\ni\n= (γrui) ◦(γrui)p ◦. . . ◦(γrui)pl−1\nand\nv(r)\ni\n= vi ◦vp\ni ◦. . . ◦vpl−1\ni\n.\n(34)\nNote that for every r1, r2 ∈[h], v(r1)\ni\n= v(r2)\ni\n. It is straightforward to verify that for every i, j ∈[n] and r ∈[h] :\n \nu(r)\nj , v(r)\ni\n \n= Tr(γr(uj, vi)).\n(35)\nCombining (35) with the fact that S is covered by proper affine hyperplanes πγi,ci we conclude that\n• For all i ∈[n] and r ∈[h],\n \nu(r)\ni , v(r)\ni\n \n= 0;\n• For all i, j ∈[n] such that i ̸= j, there exists r ∈[h] such that\n \nu(r)\nj , v(r)\ni\n \n∈F∗\np.\nPick g(x1, . . . , xh) ∈Fp[x1, . . . , xh] to be a homogeneous degree h polynomial such that for a = (a1, . . . , ah) ∈\nFh\np : g(a) = 0 if and only if a is the all-zeros vector. The existence of such a polynomial g follows from [17,\nExample 6.7]. Set f = gp−1. Note that for a ∈Fh\np : f(a) = 0 if a is the all-zeros vector, and f(a) = 1 otherwise.\nFor all i ∈[n] define\nu′\ni = f\n \nu(1)\ni , . . . , u(h)\ni\n \n◦(1)\nand\nv′\ni = ̄f\n \nv(1)\ni\n, . . . , v(h)\ni\n \n◦(−1).\n(36)\nNote that f and ̄f are homogeneous degree (p −1)h polynomials in h variables. Therefore (32) implies that\nfor all i vectors u′\ni and v′\ni have length m′ ≤h(p−1)h(ml)(p−1)h. Combining identities (36) and (33) and using the\nproperties of dot products between vectors\nn\nu(r)\ni\no\nand\nn\nv(r)\ni\no\ndiscussed above we conclude that for every m there\nexist two n = ⌊cmt⌋-sized collections of vectors {u′\ni}i∈[n] and {v′\ni}i∈[n] in Fm′\nq , such that:\n• For all i ∈[n], (u′\ni, v′\ni) = −1;\n• For all i, j ∈[n] such that i ̸= j, (uj, vi) = 0.\nIt remains to notice that a family of vectors with such properties exists only if n ≤m′, i.e., ⌊cmt⌋≤h(p−1)h(ml)(p−1)h.\nGiven that we can pick m to be arbitrarily large, this implies that t ≤(p −1)h.\nThe next lemma presents the main result of this section.\nLemma 21 Let k be an odd integer. Suppose there exists a k-nice sequence; then for infinitely many primes p\nsome k of elements of Cp add up to zero.\nProof:\nAssume\n \nSi ⊆F∗\nqi\n \ni≥1 is k-nice. Let p be a fixed prime. Combining lemmas 18 and 20 we conclude\nthat every k algebraically nice subset S ⊆F∗\npl is at most (p −1)pk combinatorially nice. Note that our bound on\ncombinatorial niceness is independent of l. Therefore there are only finitely many extensions of the field Fp in the\nsequence {Fqi}i≥1 , and the set P = {charFqi}i≥1 is infinite. It remains to notice that according to lemma 17 for\nevery p ∈P there exist k elements of Cp that add up to zero.\nIn what follows we present necessary conditions for the existence of k-tuples of p-th roots of unity in F2 that\nsum to zero. We treat the k = 3 case separately since in that case we can use a specialized argument to derive a\nslightly stronger conclusion."},{"page":14,"text":"5.2\nA necessary condition for the existence of k p-th roots of unity summing to zero\nLemma 22 Let k ≥3 be odd and p be a prime. Suppose there exist ζ1, . . . , ζk ∈Cp such that Pk\ni=1 ζi = 0; then\nord2(p) ≤2p1−1/(k−1).\n(37)\nProof:\nLet t = ord2(p). Note that Cp ⊆F2t. Note also that all elements of Cp other than the multiplicative\nidentity are proper elements of F2t. Therefore for every ζ ∈Cp where ζ ̸= 1 and every f(x) ∈F2[x] such that\ndeg f ≤t −1 we have: f(ζ) ̸= 0.\nBy multiplying Pk\ni=1 ζi = 0 through by ζ−1\nk , we may reduce to the case ζk = 1. Let ζ be the generator of Cp.\nFor every i ∈[k −1] pick wi ∈Zp such that ζi = ζwi. We now have Pk−1\ni=1 ζwi + 1 = 0. Set h = ⌊(t −1)/2⌋.\nConsider the (k −1)-tuples:\n(mw1 + i1, . . . , mwk−1 + ik−1) ∈Zk−1\np\n, for m ∈Zp and i1, . . . , ik−1 ∈[0, h].\n(38)\nSuppose two of these coincide, say\n(mw1 + i1, . . . , mwk−1 + ik−1) = (m′w1 + i′\n1, . . . , m′wk−1 + i′\nk−1),\nwith (m, i1, . . . , ik−1) ̸= (m′, i′\n1, . . . , i′\nk−1). Set n = m −m′ and jl = i′\nl −il for l ∈[k −1]. We now have\n(nw1, . . . , nwk−1) = (j1, . . . , jl)\nwith −h ≤j1, . . . , jk−1 ≤h. Observe that n ̸= 0, and thus it has a multiplicative inverse g ∈Zp. Consider a\npolynomial\nP(z) = zj1+h + . . . + zjk−1+h + zh ∈F2[z].\nNote that deg P ≤2h ≤t −1. Note also that P(1) = 1 and P(ζg) = 0. The latter identity contradicts the fact\nthat ζg is a proper element of F2t. This contradiction implies that all (k −1)-tuples in (38) are distinct. This yields\npk−1 ≥p\n t\n2\n k−1\n,\nwhich is equivalent to (37).\n5.3\nA necessary condition for the existence of three p-th roots of unity summing to zero\nIn this section we slightly strengthen lemma 22 in the special case when k = 3. Our argument is loosely inspired\nby the Agrawal-Kayal-Saxena deterministic primality test [1].\nLemma 23 Let p be a prime. Suppose there exist ζ1, ζ2, ζ3 ∈Cp that sum up to zero; then\nord2(p) ≤((4/3)p)1/2 .\n(39)\nProof:\nLet t = ord2(p). Note that Cp ⊆F2t. Note also that all elements of Cp other than the multiplicative\nidentity are proper elements of F2t. Therefore for every ζ ∈Cp where ζ ̸= 1 and every f(x) ∈F2[x] such that\ndeg f ≤t −1 we have: f(ζ) ̸= 0.\nObserve that ζ1 + ζ2 + ζ3 = 0 implies ζ1ζ−1\n2\n+ 1 = ζ3ζ−1\n2 . This yields\n ζ1ζ−1\n2\n+ 1\n p = 1. Put ζ = ζ1ζ−1\n2 .\nNote that ζ ̸= 1 and ζ, 1 + ζ ∈Cp. Consider the products πi,j = ζi(1 + ζ)j ∈Cp for 0 ≤i, j ≤t −1. Note that\nπi,j, πk,l cannot be the same if i ≥k and l ≥j, as then\nζi−k −(1 + ζ)l−j = 0,"},{"page":15,"text":"but the left side has degree less than t. In other words, if πi,j = πk,l and (i, j) ̸= (k, l), then the pairs (i, j) and\n(k, l) are comparable under termwise comparison. In particular, either (k, l) = (i+a, j+b) or (i, j) = (k+a, l+b)\nfor some pair (a, b) with πa,b = 1.\nWe next check that there cannot be two distinct nonzero pairs (a, b), (a′, b′) with πa,b = πa′,b′ = 1. As above,\nthese pairs must be comparable; we may assume without loss of generality that a ≤a′, b ≤b′. The equations\nπa,b = 1 and πa′−a,b′−b = 1 force a + b ≥t and (a′ −a) + (b′ −b) ≥t, so a′ + b′ ≥2t. But a′, b′ ≤t −1,\ncontradiction.\nIf there is no nonzero pair (a, b) with 0 ≤a, b ≤t −1 and πa,b = 1, then all πi,j are distinct, so p ≥t2.\nOtherwise, as above, the pair (a, b) is unique, and the pairs (i, j) with 0 ≤i, j ≤t −1 and (i, j) ̸≥(a, b) are\npairwise distinct. The number of pairs excluded by the condition (i, j) ̸≥(a, b) is (t −a)(t −b); since a + b ≥t,\n(t −a)(t −b) ≤t2/4. Hence p ≥t2 −t2/4 = 3t2/4 as desired.\nWhile the necessary condition given by lemma 23 is quite far away from the sufficient condition given by\nlemma 8, it nonetheless suffices for checking that for most primes p, there do not exist three p-th roots of unity\nsumming to zero. For instance, among the 664578 odd primes p ≤108, all but 550 are ruled out by Lemma 23.\n(There is an easy argument that t must be odd if p > 3; this cuts the list down to 273 primes.) Each remaining\np can be tested by computing gcd(xp + 1, (x + 1)p + 1); the only examples we found that did not satisfy the\ncondition of lemma 8 were (p, t) = (73, 9), (262657, 27), (599479, 33), (121369, 39).\n5.4\nSummary\nIn the beginning of this section 5 we argued that in order to use the method of [34], (i.e., proposition 5) to obtain\nk-query locally decodable codes of length exp(nǫ) for some fixed k and all ǫ > 0, one needs to exhibit a k-nice\nsequence of subsets of finite fields. In what follows we use technical results of the previous subsections to show\nthat the existence of a k-nice sequence implies that infinitely many Mersenne numbers have large prime factors.\nTheorem 24 Let k be odd. Suppose there exists a k-nice sequence of subsets of finite fields; then for infinitely\nmany values of t we have\nP(2t −1) ≥(t/2)1+1/(k−2).\n(40)\nProof:\nUsing lemmas 21 and 22 we conclude that a k-nice sequence yields infinitely many primes p such that\nord2(p) ≤2p1−1/(k−1). Let p be such a prime and t = ord2(p). Then P(2t −1) ≥(t/2)1+1/(k−2).\nA combination of lemmas 21 and 23 yields a slightly stronger bound for the special case of 3-nice sequences.\nTheorem 25 Suppose there exists a 3-nice sequence of subsets; then for infinitely many values of t we have\nP(2t −1) ≥(3/4)t2.\n(41)\nWe would like to remind the reader that although the lower bounds for P(2t −1) given by (40) and (41) are\nextremely weak light of the widely accepted conjecture saying that the number of Mersenne primes is infinite,\nthey are substantially stronger than what is currently known unconditionally (2).\n6\nConclusion\nRecently [34] came up with a novel technique for constructing locally decodable codes and obtained vast im-\nprovements upon the earlier work. The construction proceeds in two steps. First [34] shows that if there exist\nsubsets of finite fields with certain ’nice’ properties then there exist good codes. Next [34] constructs nice subsets\nof prime fields Fp for Mersenne primes p."},{"page":16,"text":"In this paper we have undertaken an in-depth study of nice subsets of general finite fields. We have shown\nthat constructing nice subsets is closely related to proving lower bounds on the size of largest prime factors of\nMersenne numbers. Specifically we extended the constructions of [34] to obtain nice subsets of prime fields Fp\nfor primes p that are large factors of Mersenne numbers. This implies that strong lower bounds for size of the\nlargest prime factors of Mersenne numbers yield better locally decodable codes. Conversely, we argued that if one\ncan obtain codes of subexponential length and constant query complexity through nice subsets of finite fields then\ninfinitely many Mersenne numbers have prime factors larger than known currently.\nAcknowledgements\nKiran Kedlaya’s research is supported by NSF CAREER grant DMS-0545904 and by the Sloan Research Fel-\nlowship. Sergey Yekhanin would like to thank Swastik Kopparty for providing the reference [5] and outlining\nthe proof of lemma 9. He would also like to thank Henryk Iwaniec, Carl Pomerance and Peter Sarnak for their\nfeedback regarding the number theory problems discussed in this paper.\nReferences\n[1] M. Agrawal, N. Kayal, N. Saxena, “PRIMES is in P,” Annals of Mathematics, vol. 160, pp. 781-793, 2004.\n[2] L. Babai, L. Fortnow, L. Levin, and M. Szegedy, “Checking computations in polylogarithmic time,”. In Proc.\nof the 23th ACM Symposium on Theory of Computing (STOC), pp. 21-31, 1991.\n[3] A. Beimel, Y. Ishai and E. Kushilevitz,“General constructions for information-theoretic private information\nretrieval,” Journal of Computer and System Sciences, vol. 71, pp. 213-247, 2005. Preliminary versions in\nSTOC 1999 and ICALP 2001.\n[4] A. Beimel, Y. Ishai, E. Kushilevitz, and J. F. Raymond. “Breaking the O\n n1/(2k−1) \nbarrier for information-\ntheoretic private information retrieval,” In Proc. of the 43rd IEEE Symposium on Foundations of Computer\nScience (FOCS), pp. 261-270, 2002.\n[5] J. Bourgain, M. Chang, “A Gauss sum estimate in arbitrary finite fields,” Comptes Rendus Mathematique,\nvol. 342, pp. 643-646, 2006.\n[6] B. Chor, O. Goldreich, E. Kushilevitz, and M. Sudan. “Private information retrieval,” In Proc. of the 36rd\nIEEE Symposium on Foundations of Computer Science (FOCS), pp. 41-50, 1995. Also, in Journal of the\nACM, vol. 45, 1998.\n[7] D. Cox, J. Little, D. O’Shea, Ideals, varieties, and algorithms: an introduction to computational algebraic\ngeometry and commutative algebra. Springer, 1996.\n[8] A. Deshpande, R. Jain, T. Kavitha, S. Lokam and J. Radhakrishnan, “Better lower bounds for locally decod-\nable codes,” In Proc. of the 20th IEEE Computational Complexity Conference (CCC), pp. 184-193, 2002.\n[9] Curtis Cooper, Steven Boone, http://www.mersenne.org/32582657.htm\n[10] P. Erdos and T. Shorey, “On the greatest prime factor of 2p −1 for a prime p and other expressions,” Acta.\nArith. vol. 30, pp. 257-265, 1976.\n[11] W. Gasarch, “A survey on private information retrieval,” The Bulletin of the EATCS, vol. 82, pp. 72-107,\n2004."},{"page":17,"text":"[12] O. Goldreich, “Short locally testable codes and proofs,” Technical Report TR05-014, Electronic Colloquim\non Computational Complexity (ECCC), 2005.\n[13] O. Goldreich, H. Karloff, L. Schulman, L. Trevisan “Lower bounds for locally decodable codes and private\ninformation retrieval,” In Proc. of the 17th IEEE Computational Complexity Conference (CCC), pp. 175-183,\n2002.\n[14] B. Hemenway and R. Ostrovsky, “Public key encryption which is simultaneously a locally-decodable error-\ncorrecting code,” In Cryptology ePrint Archive, Report 2007/083.\n[15] J. Katz and L. Trevisan, “On the efficiency of local decoding procedures for error-correcting codes,” In Proc.\nof the 32th ACM Symposium on Theory of Computing (STOC), pp. 80-86, 2000.\n[16] I. Kerenidis, R. de Wolf, “Exponential lower bound for 2-query locally decodable codes via a quantum\nargument,” Journal of Computer and System Sciences, 69(3), pp. 395-420. Earlier version in STOC’03.\nquant-ph/0208062.\n[17] R. Lidl and H. Niederreiter, Finite Fields. Cambridge: Cambridge University Press, 1983.\n[18] L. Murata, C. Pomerance, “On the largest prime factor of a Mersenne number,” Number theory, CRM Proc.\nLecture Notes of American Mathematical Society vol. 36, pp. 209-218, 2004.\n[19] M. Murty and S. Wong, “The ABC conjecture and prime divisors of the Lucas and Lehmer sequences,” In\nProc. of Milennial Conference on Number Theory III, (Urbana, IL, 2000) (A. K. Peters, Natick, MA, 2002)\npp. 43-54.\n[20] K. Obata, “Optimal lower bounds for 2-query locally decodable linear codes,” In Proc. of the 6th RANDOM,\nvol. 2483 of Lecture Notes in Computer Science, pp. 39-50, 2002.\n[21] A. Polishchuk and D. Spielman, ”Nearly-linear size holographic proofs,” In Proc. of the 26th ACM Sympo-\nsium on Theory of Computing (STOC), pp. 194-203, 1994.\n[22] C. Pomerance, “Recent developments in primality testing,” Math. Intelligencer, 3:3, pp. 97-105, (1980/81).\n[23] P. Raghavendra, “A Note on Yekhanin’s locally decodable codes,” In Electronic Colloquium on Computa-\ntional Complexity Report TR07-016, 2007.\n[24] A. Romashchenko, “Reliable computations based on locally decodable codes,” In Proc. of the 23rd Inter-\nnational Symposium on Theoretical Aspects of Computer Science (STACS), vol. 3884 of Lecture Notes in\nComputer Science, pp. 537-548, 2006.\n[25] A. Schinzel, “On primitive factors of an −bn,” In Proc. of Cambridge Philos. Soc. vol. 58, pp. 555-562,\n1962.\n[26] C. Stewart, “The greatest prime factor of an −bn,” Acta Arith. vol. 26, pp. 427-433, 1974/75.\n[27] C. Stewart, “On divisors of Fermat, Fibonacci, Lucas, and Lehmer numbers,” In Proc. of London Math. Soc.\nvol. 35 (3), pp. 425-447, 1977.\n[28] M. Sudan, Efficient checking of polynomials and proofs and the hardness of approximation problems. PhD\nthesis, University of California at Berkeley, 1992.\n[29] L. Trevisan, “Some applications of coding theory in computational complexity,” Quaderni di Matematica,\nvol. 13, pp. 347-424, 2004."},{"page":18,"text":"[30] S. Wehner and R. de Wolf, “Improved lower bounds for locally decodable codes and private information re-\ntrieval,” In Proc. of 32nd International Colloquium on Automata, Languages and Programming (ICALP’05),\nLNCS 3580, pp. 1424-1436.\n[31] Lenstra-Pomerance-Wagstaff conjecture. (2006, May 22). In Wikipedia, The Free Encyclopedia. Re-\ntrieved 00:18,\nOctober 3,\n2006,\nfrom http://en.wikipedia.org/w/index.php?title=Lenstra-Pomerance-\nWagstaff conjecture&oldid=54506577\n[32] S. Wagstaff, “Divisors of Mersenne numbers,” Math. Comp., 40:161, pp. 385-397, 1983.\n[33] D. Woodruff, “New lower bounds for general locally decodable codes,” Electronic Colloquium on Computa-\ntional Complexity, TR07-006, 2007.\n[34] S. Yekhanin, “Towards 3-query locally decodable codes of subexponential length,” In Proc. of the 39th ACM\nSymposium on Theory of Computing (STOC), 2007.\n[35] S. Yekhanin, Locally decodable codes and private information retrieval schemes. PhD thesis, MIT, to appear."}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"Specifically, we show that every Mersenne number m = 2t −1 that has a prime factor p > mγ yields a family","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"that codewords C(x) of length N = O(n) already suffice to correct errors in up to δN locations of C(x) for","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"of finite fields and showed that every Mersenne prime p = 2t −1 yields a family of 3-query LDCs of length","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"three query locally decodable codes of length N = exp","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"holds, [34] gets three query locally decodable codes of length N = exp","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"Mersenne numbers with polynomially large prime factors. Specifically, every Mersenne number m = 2t −1 such","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"= eO(x).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"• [s] = {1, . . . , s};","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"2 , L⊥denotes the dual space. That is, L⊥= {u ∈Fm","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"2 | ∀v ∈L, (u, v) = 0};","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"1. For all x ∈{0, 1}n, i ∈[n] and y ∈{0, 1}N such that dH(C(x), y) ≤δN : Pr[Ay(i) = xi] ≥1−ǫ, where","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"m there exist two n = ⌊cmt⌋-sized collections of vectors {u1, . . . , un} and {v1, . . . , vn} in Fm","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"• For all i ∈[n], (ui, vi) = 0;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"• For all i, j ∈[n] such that i ̸= j, (uj, vi) ∈S.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"• |S1| = k′;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"observations imply that L⊥has full support, i.e., for every i ∈[q] there exists a vector v ∈L⊥such that vi ̸= 0. It","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"Encoding: We assume that our message has length n = ⌊cmt⌋for some value of m. (Otherwise we pad the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"Enc(ej)w =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"= xi.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"Enc(ej)w+λvi =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"where I[γ ∈S0] = 1 if γ ∈S0 and zero otherwise. Now note that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"I [(uj, w + λvi) ∈S0] =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"I [(uj, w) + λ(uj, vi) ∈S0] =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"if i = j,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"The last identity in (7) for i = j follows from: (ui, vi) = 0, (ui, w) ∈S0 and k′ = |S1| is odd. The last identity","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"for i ̸= j follows from (uj, vi) ∈S and the algebraic niceness of S. Combining identities (6) and (7) we get (5).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"dinates in Ti contain corrupted bits. Let Qi =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"of coordinates that may be queried by A. (ui, vi) = 0 implies that elements of Qi uniformly cover the set Ti.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"that for every Mersenne prime p = 2t −1 the set ⟨2⟩⊆F∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"Proof: Let t = ord2(p). Clearly, t divides p −1. We need to specify a constant c > 0 such that for every positive","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"integer m there exist two n = ⌊cmt⌋-sized collections of m long vectors over Fp satisfying:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"• For all i ∈[n], (ui, vi) = 0;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"• For all i, j ∈[n] such that i ̸= j, (uj, vi) ∈⟨2⟩.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"First assume that m has the shape m =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"13] gives us a collection of n =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"ζ1 + . . . + ζk = 0;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"k′ = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"Let t = ord2(p). Observe that Cp ⊆F2t. Let g be a generator of Cp. Identity (9) yields gγ1 + . . . + gγk′ = 0, for","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"some distinct values of {γi}i∈[k′]. Set S1 = {γ1, . . . , γk′}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"F2[x]/(xp −1) : φS′(x) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"s∈S′ xs. It is easy to see that for all sets S′ ⊆Fp and all α, β ∈Fp, such that β ̸= 0 :","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"φα+βS′(x) = xαφS′(xβ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"that φT ∈L⊥can be used as the set S0. Let τ(x) = gcd(xp −1, φS1(x)). Note that τ(x) ̸= 1 since g is a common","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"dim L = p −deg τ. Fix some α ∈Fp and β ∈⟨2⟩. Let us prove that φα+βS1(x) is in L :","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"φα+βS1(x) = xαφS1(xβ) = xα(φS1(x))β.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"The last identity above follows from the fact that for any f ∈F2[x] and any integer i : f(x2i) = (f(x))2i.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"sum to zero. We treat the k = 3 case separately since in that case we can use a specialized argument to derive a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"f ∈F[x, y, z] be a homogeneous polynomial. A triple (x0, y0, z0) ∈F3 is called a zero of f if f(x0, y0, z0) = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"A zero is called nontrivial if it is different from the origin. An equation f = 0 defines a projective plane curve χf.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"Let t = ord2(p). Note that Cp ⊆F2t. Consider a projective plane Fermat curve χ defined by","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"x(2t−1)/p + y(2t−1)/p + z(2t−1)/p = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"sum to zero, even though ord2(13264529) = 47 < 2 ∗log2 13264529 ≈47.3.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"For x ∈F2t let Tr(x) = x + x2 + . . . + x2t−1 denote the trace of x. It is not hard to verify that for all x,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"{±1}. There exist 2t characters. We denote characters by χa, where a ranges in F2t, and set χa(x) = (−1)Tr(ax).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"defined by χa(C) = P χa(x)C(x), where the sum is over all x ∈F2t. Fourier coefficient χ0(C) = |C| is called","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"χ(x) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"if x = 0;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"χ2(C) = 2t|C|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"Let M(C) = # {ζ1, . . . , ζk ∈C | ζ1 + . . . + ζk = 0} . (14) yields","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"M(C) = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"Note that χ(x1 + . . . + xk) = χ(x1) . . . χ(xk). Changing the order of summation in (17) we get","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq66","equation_number":null,"raw_text":"M(C) = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq67","equation_number":null,"raw_text":"C(x1) . . . C(xk)χ(x1) . . . χ(xk) = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq68","equation_number":null,"raw_text":"χk(C) = |C|k","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq69","equation_number":null,"raw_text":"χ2(C) = |C|k","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq70","equation_number":null,"raw_text":"where δ = δ(ǫ) > 0 and c1 = c1(ǫ) are absolute constants.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq71","equation_number":null,"raw_text":"Lemma 11 For every c > 0 there exists an odd integer k = k(c) such that the following implication holds. If p is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq72","equation_number":null,"raw_text":"zero. Therefore given c it suffices to prove the existence of an odd k = k(c) that works for all sufficiently large","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq73","equation_number":null,"raw_text":"p. Let t = ord2(p). Observe that p > 2t/c. Assume p is sufficiently large so that t > 2c. Next we show that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq74","equation_number":null,"raw_text":"the precondition of lemma 10 holds for n = (2t −1)/p and ǫ = 1/(2c). Let t′ | t and 1 ≤t′ < t. Clearly","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq75","equation_number":null,"raw_text":"gcd(2t′ −1, p) = 1. Therefore","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq76","equation_number":null,"raw_text":"Let P(2t −1) = p. Observe that p | 2t −1 and p > 20.75t yield ord2(p) < (4/3) log2 p. Combining","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq77","equation_number":null,"raw_text":"As an example application of theorem 12 one can observe that P(223 −1) = 178481 > 2(3/4)∗23 ≈155872 yields","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq78","equation_number":null,"raw_text":"Theorem 14 For every γ > 0 there exists an odd integer k = k(γ) such that the following implication holds.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq79","equation_number":null,"raw_text":"Let P(2t −1) = p. Observe that p | 2t −1 and p > 2γt yield ord2(p) < (1/γ) log2 p. Combining","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq80","equation_number":null,"raw_text":"We start with some notation. Consider a a finite field Fq = Fpl, where p is prime. Fix a basis e1, . . . , el of Fq","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq81","equation_number":null,"raw_text":"p to denote α = Pl","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq82","equation_number":null,"raw_text":"i=1 αiei ∈Fq. Let R denote the ring","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq83","equation_number":null,"raw_text":"φS1(x1, . . . , xl) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq84","equation_number":null,"raw_text":"φ(α1,...,αl)+βS1(x1, . . . , xl) = xα1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq85","equation_number":null,"raw_text":"Lemma 17 Let Fq = Fpl, where p is prime. Suppose Fq contains a nonempty k algebraically nice subset; then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq86","equation_number":null,"raw_text":"there exist ζ1, . . . , ζk ∈Cp such that ζ1 + . . . + ζk = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq87","equation_number":null,"raw_text":"φβ0S1(ζ1, . . . , ζl) = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq88","equation_number":null,"raw_text":"proceed, we need some more notation. Recall that q = pl and p is prime. For x ∈Fq let Tr(x) = x+. . .+xpl−1 ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq89","equation_number":null,"raw_text":"p we call the set πγ,c = {x ∈Fq | Tr(γx) = c} a proper","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq90","equation_number":null,"raw_text":"Lemma 18 Let Fq = Fpl, where p is prime. Suppose S ⊆F∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq91","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq92","equation_number":null,"raw_text":"Discussion preceding lemma 17 implies that there exists a set S1 = {σ1, . . . , σk′} ⊆Fq of odd size","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq93","equation_number":null,"raw_text":"every 1 ≤i ≤l pick ωi ∈Zp such that ζi = ζωi. For every β ∈S, φβS1(ζ1, . . . , ζl) = 0 yields","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq94","equation_number":null,"raw_text":"μ=(μ1,...,μl)∈βS1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq95","equation_number":null,"raw_text":"i=1 μiωi = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq96","equation_number":null,"raw_text":"Observe that for fixed values {ωi}1≤i≤l ∈Zp the map D(μ) = Pl","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq97","equation_number":null,"raw_text":"i=1 μiωi is a linear map from Fq to Fp. It is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq98","equation_number":null,"raw_text":"not hard to prove that every such map can be expressed as D(μ) = Tr(δμ) for an appropriate choice of δ ∈Fq.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq99","equation_number":null,"raw_text":"ζTr(δμ) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq100","equation_number":null,"raw_text":"ζTr(δβσ) = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq101","equation_number":null,"raw_text":"Let W =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq102","equation_number":null,"raw_text":"p | ζw1 + . . . + ζwk′ = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq103","equation_number":null,"raw_text":"Coordinates of u ⊗v are labelled by all possible elements of [m] × [n] and (u ⊗v)i,j = uivj. Also, let u⊗l denote","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq104","equation_number":null,"raw_text":"(u ⊗v, x ⊗y) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq105","equation_number":null,"raw_text":"uivjxiyj =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq106","equation_number":null,"raw_text":"= (u, x)(v, y).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq107","equation_number":null,"raw_text":"In what follows we need a generalization of identity (31). Let f(x1, . . . , xh) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq108","equation_number":null,"raw_text":"in Fq[x1, . . . , xh]. Given f we define ̄f ∈Fq[x1, . . . , xh] by ̄f = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq109","equation_number":null,"raw_text":"f(u1, . . . , uh) = ◦i ciu⊗αi","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq110","equation_number":null,"raw_text":"= f((u1, v1), . . . , (uh, vh)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq111","equation_number":null,"raw_text":"Proof: Let u = (u1, . . . , uh) and v = (v1, . . . , vh). Observe that if (33) holds for polynomials f1 and f2 defined","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq112","equation_number":null,"raw_text":"over disjoint sets of monomials then it also holds for f = f1 + f2 :","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq113","equation_number":null,"raw_text":"f1 ((u1, v1), . . . , (uh, vh)) + f2 ((u1, v1), . . . , (uh, vh)) = f ((u1, v1), . . . , (uh, vh)) .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq114","equation_number":null,"raw_text":"Therefore it suffices to prove (33) for monomials f = cxα1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq115","equation_number":null,"raw_text":"als f = cxα1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq116","equation_number":null,"raw_text":"i=1 αi.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq117","equation_number":null,"raw_text":"Lemma 20 Let Fq = Fpl, where p is prime. Let S ⊆F∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq118","equation_number":null,"raw_text":"r=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq119","equation_number":null,"raw_text":"n = ⌊cmt⌋-sized collections of vectors {ui}i∈[n] and {vi}i∈[n] in Fm","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq120","equation_number":null,"raw_text":"• For all i ∈[n], (ui, vi) = 0;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq121","equation_number":null,"raw_text":"• For all i, j ∈[n] such that i ̸= j, (uj, vi) ∈S.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq122","equation_number":null,"raw_text":"= (γrui) ◦(γrui)p ◦. . . ◦(γrui)pl−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq123","equation_number":null,"raw_text":"= vi ◦vp","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq124","equation_number":null,"raw_text":"= v(r2)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq125","equation_number":null,"raw_text":"= Tr(γr(uj, vi)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq126","equation_number":null,"raw_text":"• For all i, j ∈[n] such that i ̸= j, there exists r ∈[h] such that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq127","equation_number":null,"raw_text":"Pick g(x1, . . . , xh) ∈Fp[x1, . . . , xh] to be a homogeneous degree h polynomial such that for a = (a1, . . . , ah) ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq128","equation_number":null,"raw_text":"p : g(a) = 0 if and only if a is the all-zeros vector. The existence of such a polynomial g follows from [17,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq129","equation_number":null,"raw_text":"Example 6.7]. Set f = gp−1. Note that for a ∈Fh","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq130","equation_number":null,"raw_text":"p : f(a) = 0 if a is the all-zeros vector, and f(a) = 1 otherwise.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq131","equation_number":null,"raw_text":"i = f","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq132","equation_number":null,"raw_text":"i = ̄f","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq133","equation_number":null,"raw_text":"exist two n = ⌊cmt⌋-sized collections of vectors {u′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq134","equation_number":null,"raw_text":"i) = −1;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq135","equation_number":null,"raw_text":"• For all i, j ∈[n] such that i ̸= j, (uj, vi) = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq136","equation_number":null,"raw_text":"sequence {Fqi}i≥1 , and the set P = {charFqi}i≥1 is infinite. It remains to notice that according to lemma 17 for","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq137","equation_number":null,"raw_text":"sum to zero. We treat the k = 3 case separately since in that case we can use a specialized argument to derive a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq138","equation_number":null,"raw_text":"i=1 ζi = 0; then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq139","equation_number":null,"raw_text":"Let t = ord2(p). Note that Cp ⊆F2t. Note also that all elements of Cp other than the multiplicative","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq140","equation_number":null,"raw_text":"identity are proper elements of F2t. Therefore for every ζ ∈Cp where ζ ̸= 1 and every f(x) ∈F2[x] such that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq141","equation_number":null,"raw_text":"deg f ≤t −1 we have: f(ζ) ̸= 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq142","equation_number":null,"raw_text":"i=1 ζi = 0 through by ζ−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq143","equation_number":null,"raw_text":"k , we may reduce to the case ζk = 1. Let ζ be the generator of Cp.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq144","equation_number":null,"raw_text":"For every i ∈[k −1] pick wi ∈Zp such that ζi = ζwi. We now have Pk−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq145","equation_number":null,"raw_text":"i=1 ζwi + 1 = 0. Set h = ⌊(t −1)/2⌋.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq146","equation_number":null,"raw_text":"(mw1 + i1, . . . , mwk−1 + ik−1) = (m′w1 + i′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq147","equation_number":null,"raw_text":"with (m, i1, . . . , ik−1) ̸= (m′, i′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq148","equation_number":null,"raw_text":"k−1). Set n = m −m′ and jl = i′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq149","equation_number":null,"raw_text":"(nw1, . . . , nwk−1) = (j1, . . . , jl)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq150","equation_number":null,"raw_text":"with −h ≤j1, . . . , jk−1 ≤h. Observe that n ̸= 0, and thus it has a multiplicative inverse g ∈Zp. Consider a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq151","equation_number":null,"raw_text":"P(z) = zj1+h + . . . + zjk−1+h + zh ∈F2[z].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq152","equation_number":null,"raw_text":"Note that deg P ≤2h ≤t −1. Note also that P(1) = 1 and P(ζg) = 0. The latter identity contradicts the fact","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq153","equation_number":null,"raw_text":"In this section we slightly strengthen lemma 22 in the special case when k = 3. Our argument is loosely inspired","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq154","equation_number":null,"raw_text":"Let t = ord2(p). Note that Cp ⊆F2t. Note also that all elements of Cp other than the multiplicative","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq155","equation_number":null,"raw_text":"identity are proper elements of F2t. Therefore for every ζ ∈Cp where ζ ̸= 1 and every f(x) ∈F2[x] such that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq156","equation_number":null,"raw_text":"deg f ≤t −1 we have: f(ζ) ̸= 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq157","equation_number":null,"raw_text":"Observe that ζ1 + ζ2 + ζ3 = 0 implies ζ1ζ−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq158","equation_number":null,"raw_text":"p = 1. Put ζ = ζ1ζ−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq159","equation_number":null,"raw_text":"Note that ζ ̸= 1 and ζ, 1 + ζ ∈Cp. Consider the products πi,j = ζi(1 + ζ)j ∈Cp for 0 ≤i, j ≤t −1. Note that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq160","equation_number":null,"raw_text":"ζi−k −(1 + ζ)l−j = 0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq161","equation_number":null,"raw_text":"but the left side has degree less than t. In other words, if πi,j = πk,l and (i, j) ̸= (k, l), then the pairs (i, j) and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq162","equation_number":null,"raw_text":"(k, l) are comparable under termwise comparison. In particular, either (k, l) = (i+a, j+b) or (i, j) = (k+a, l+b)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq163","equation_number":null,"raw_text":"for some pair (a, b) with πa,b = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq164","equation_number":null,"raw_text":"We next check that there cannot be two distinct nonzero pairs (a, b), (a′, b′) with πa,b = πa′,b′ = 1. As above,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq165","equation_number":null,"raw_text":"πa,b = 1 and πa′−a,b′−b = 1 force a + b ≥t and (a′ −a) + (b′ −b) ≥t, so a′ + b′ ≥2t. But a′, b′ ≤t −1,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq166","equation_number":null,"raw_text":"If there is no nonzero pair (a, b) with 0 ≤a, b ≤t −1 and πa,b = 1, then all πi,j are distinct, so p ≥t2.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq167","equation_number":null,"raw_text":"(t −a)(t −b) ≤t2/4. Hence p ≥t2 −t2/4 = 3t2/4 as desired.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq168","equation_number":null,"raw_text":"condition of lemma 8 were (p, t) = (73, 9), (262657, 27), (599479, 33), (121369, 39).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq169","equation_number":null,"raw_text":"ord2(p) ≤2p1−1/(k−1). Let p be such a prime and t = ord2(p). Then P(2t −1) ≥(t/2)1+1/(k−2).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq170","equation_number":null,"raw_text":"Wagstaff conjecture&oldid=54506577","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":49974,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}