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math/0010307
|
Fields, towers of function fields meeting asymptotic bounds, and basis
constructions for algebraic-geometric codes
|
math.NT cs.IT math.IT
|
In this work, we use the notion of ``symmetry'' of functions for an extension
$K/L$ of finite fields to produce extensions of a function field $F/K$ in which
almost all places of degree one split completely. Then we introduce the notion
of ``quasi-symmetry'' of functions for $K/L$, and demonstrate its use in
producing extensions of $F/K$ in which all places of degree one split
completely. Using these techniques, we are able to restrict the ramification
either to one chosen rational place, or entirely to non-rational places. We
then apply these methods to the related problem of building asymptotically good
towers of function fields. We construct examples of towers of function fields
in which all rational places split completely throughout the tower. We
construct Abelian towers with this property also.
Furthermore, all of the above are done explicitly, ie., we give generators
for the extensions, and equations that they satisfy.
We also construct an integral basis for a set of places in a tower of
function fields meeting the Drinfeld-Vladut bound using the discriminant of the
tower localized at each place. Thus we are able to obtain a basis for a
collection of functions that contains the set of regular functions in this
tower. Regular functions are of interest in the theory of error-correcting
codes as they lead to an explicit description of the code associated to the
tower by providing the code's generator matrix.
|
math/0012163
|
Learning Complexity Dimensions for a Continuous-Time Control System
|
math.OC cs.LG
|
This paper takes a computational learning theory approach to a problem of
linear systems identification. It is assumed that input signals have only a
finite number k of frequency components, and systems to be identified have
dimension no greater than n. The main result establishes that the sample
complexity needed for identification scales polynomially with n and
logarithmically with k.
|
math/0101092
|
Structure of $Z^2$ modulo selfsimilar sublattices
|
math.CO cs.IT math.IT
|
In this paper we show the combinatorial structure of $\mathbb{Z}^2$ modulo
sublattices selfsimilar to $\mathbb{Z}^2$. The tool we use for dealing with
this purpose is the notion of association scheme. We classify when the scheme
defined by the lattice is imprimitive and characterize its decomposition in
terms of the decomposition of the gaussian integer defining the lattice. This
arise in the classification of different forms of tiling $\mathbb{Z}^2$ by
lattices of this type.
|
math/0103007
|
Source Coding, Large Deviations, and Approximate Pattern Matching
|
math.PR cs.IT math.IT
|
We present a development of parts of rate-distortion theory and pattern-
matching algorithms for lossy data compression, centered around a lossy version
of the Asymptotic Equipartition Property (AEP). This treatment closely
parallels the corresponding development in lossless compression, a point of
view that was advanced in an important paper of Wyner and Ziv in 1989. In the
lossless case we review how the AEP underlies the analysis of the Lempel-Ziv
algorithm by viewing it as a random code and reducing it to the idealized
Shannon code. This also provides information about the redundancy of the
Lempel-Ziv algorithm and about the asymptotic behavior of several relevant
quantities. In the lossy case we give various versions of the statement of the
generalized AEP and we outline the general methodology of its proof via large
deviations. Its relationship with Barron's generalized AEP is also discussed.
The lossy AEP is applied to: (i) prove strengthened versions of Shannon's
source coding theorem and universal coding theorems; (ii) characterize the
performance of mismatched codebooks; (iii) analyze the performance of pattern-
matching algorithms for lossy compression; (iv) determine the first order
asymptotics of waiting times (with distortion) between stationary processes;
(v) characterize the best achievable rate of weighted codebooks as an optimal
sphere-covering exponent. We then present a refinement to the lossy AEP and use
it to: (i) prove second order coding theorems; (ii) characterize which sources
are easier to compress; (iii) determine the second order asymptotics of waiting
times; (iv) determine the precise asymptotic behavior of longest match-lengths.
Extensions to random fields are also given.
|
math/0103107
|
Explicit modular towers
|
math.NT cs.IT math.AG math.IT
|
We give a general recipe for explicitly constructing asymptotically optimal
towers of modular curves such as {X_0(l^n): n=1,2,3,...}. We illustrate the
method by giving equations for eight towers with various geometric features. We
conclude by observing that such towers are all of a specific recursive form,
and speculate that perhaps every tower of this form that attains the
Drinfeld-Vladut bound is modular.
|
math/0103109
|
In search of an evolutionary coding style
|
math.NA cs.IT math.DS math.IT q-bio
|
In the near future, all the human genes will be identified. But understanding
the functions coded in the genes is a much harder problem. For example, by
using block entropy, one has that the DNA code is closer to a random code then
written text, which in turn is less ordered then an ordinary computer code; see
\cite{schmitt}.
Instead of saying that the DNA is badly written, using our programming
standards, we might say that it is written in a different style -- an
evolutionary style.
We will suggest a way to search for such a style in a quantified manner by
using an artificial life program, and by giving a definition of general codes
and a definition of style for such codes.
|
math/0104016
|
Bounds for weight distribution of weakly self-dual codes
|
math.CO cs.IT math.IT quant-ph
|
Upper bounds are given for the weight distribution of binary weakly self-dual
codes. To get these new bounds, we introduce a novel method of utilizing
unitary operations on Hilbert spaces. This method is motivated by recent
progress on quantum computing. This new approach leads to much simpler proofs
for such genre of bounds on the weight distributions of certain classes of
codes. Moreover, in some cases, our bounds are improvements on the earlier
bounds. These improvements are achieved, either by extending the range of the
weights over which the bounds apply, or by extending the class of codes
subjected to these bounds.
|
math/0104115
|
Excellent nonlinear codes from modular curves
|
math.NT cs.IT math.AG math.IT
|
We introduce a new construction of error-correcting codes from algebraic
curves over finite fields. Modular curves of genus g -> infty over a field of
size q0^2 yield nonlinear codes more efficient than the linear Goppa codes
obtained from the same curves. These new codes now have the highest asymptotic
transmission rates known for certain ranges of alphabet size and error rate.
Both the theory and possible practical use of these new record codes require
the development of new tools. On the theoretical side, establishing the
transmission rate depends on an error estimate for a theorem of Schanuel
applied to the function field of an asymptotically optimal curve. On the
computational side, actual use of the codes will hinge on the solution of new
problems in the computational algebraic geometry of curves.
|
math/0104222
|
Decoding method for generalized algebraic geometry codes
|
math.NT cs.IT math.AG math.IT
|
We propose a decoding method for the generalized algebraic geometry codes
proposed by Xing et al. To show its practical usefulness, we give an example of
generalized algebraic geometry codes of length 567 over F_8 whose numbers of
correctable errors by the proposed method are larger than the shortened codes
of the primitive BCH codes of length 4095 in the most range of dimension.
|
math/0105235
|
Mathematics of learning
|
math.PR cs.LG math.CO math.DS
|
We study the convergence properties of a pair of learning algorithms
(learning with and without memory). This leads us to study the dominant
eigenvalue of a class of random matrices. This turns out to be related to the
roots of the derivative of random polynomials (generated by picking their roots
uniformly at random in the interval [0, 1], although our results extend to
other distributions). This, in turn, requires the study of the statistical
behavior of the harmonic mean of random variables as above, which leads us to
delicate question of the rate of convergence to stable laws and tail estimates
for stable laws. The reader can find the proofs of most of the results
announced here in the paper entitled "Harmonic mean, random polynomials, and
random matrices", by the same authors.
|
math/0105236
|
Harmonic mean, random polynomials and stochastic matrices
|
math.PR cs.LG math.CA math.CO math.DS
|
Motivated by a problem in learning theory, we are led to study the dominant
eigenvalue of a class of random matrices. This turns out to be related to the
roots of the derivative of random polynomials (generated by picking their roots
uniformly at random in the interval [0, 1], although our results extend to
other distributions). This, in turn, requires the study of the statistical
behavior of the harmonic mean of random variables as above, and that, in turn,
leads us to delicate question of the rate of convergence to stable laws and
tail estimates for stable laws.
|
math/0106089
|
The coset weight distributions of certain BCH codes and a family of
curves
|
math.AG cs.IT math.CO math.IT
|
We study the distribution of the number of rational points in a family of
curves over a finite field of characteristic 2. This distribution determines
the coset weight distribution of a certain BCH code.
|
math/0108096
|
Geometrically Uniform Frames
|
math.FA cs.IT math.GR math.IT
|
We introduce a new class of frames with strong symmetry properties called
geometrically uniform frames (GU), that are defined over an abelian group of
unitary matrices and are generated by a single generating vector. The notion of
GU frames is then extended to compound GU (CGU) frames which are generated by
an abelian group of unitary matrices using multiple generating vectors.
The dual frame vectors and canonical tight frame vectors associated with GU
frames are shown to be GU and therefore generated by a single generating
vector, which can be computed very efficiently using a Fourier transform
defined over the generating group of the frame. Similarly, the dual frame
vectors and canonical tight frame vectors associated with CGU frames are shown
to be CGU.
The impact of removing single or multiple elements from a GU frame is
considered. A systematic method for constructing optimal GU frames from a given
set of frame vectors that are not GU is also developed. Finally, the Euclidean
distance properties of GU frames are discussed and conditions are derived on
the abelian group of unitary matrices to yield GU frames with strictly positive
distance spectrum irrespective of the generating vector.
|
math/0110157
|
Some Applications of Algebraic Curves to Computational Vision
|
math.AG cs.IT math.IT
|
We introduce a new formalism and a number of new results in the context of
geometric computational vision. The classical scope of the research in
geometric computer vision is essentially limited to static configurations of
points and lines in $P^3$ . By using some well known material from algebraic
geometry, we open new branches to computational vision. We introduce algebraic
curves embedded in $P^3$ as the building blocks from which the tensor of a
couple of cameras (projections) can be computed. In the process we address
dimensional issues and as a result establish the minimal number of algebraic
curves required for the tensor variety to be discrete as a function of their
degree and genus. We then establish new results on the reconstruction of an
algebraic curves in $P^3$ from multiple projections on projective planes
embedded in $P^3$ . We address three different presentations of the curve: (i)
definition by a set of equations, for which we show that for a generic
configuration, two projections of a curve of degree d defines a curve in $P^3$
with two irreducible components, one of degree d and the other of degree $d(d -
1)$, (ii) the dual presentation in the dual space $P^{3*}$, for which we derive
a lower bound for the number of projections necessary for linear reconstruction
as a function of the degree and the genus, and (iii) the presentation as an
hypersurface of $P^5$, defined by the set of lines in $P^3$ meeting the curve,
for which we also derive lower bounds for the number of projections necessary
for linear reconstruction as a function of the degree (of the curve). Moreover
we show that the latter representation yields a new and efficient algorithm for
dealing with mixed configurations of static and moving points in $P^3$.
|
math/0110214
|
Coding Distributive Lattices with Edge Firing Games
|
math.CO cs.IT math-ph math.DS math.IT math.MP
|
In this note, we show that any distributive lattice is isomorphic to the set
of reachable configurations of an Edge Firing Game. Together with the result of
James Propp, saying that the set of reachable configurations of any Edge Firing
Game is always a distributive lattice, this shows that the two concepts are
equivalent.
|
math/0111159
|
Constructing elliptic curves with a known number of points over a prime
field
|
math.NT cs.IT math.AG math.IT
|
Elliptic curves with a known number of points over a given prime field with n
elements are often needed for use in cryptography. In the context of primality
proving, Atkin and Morain suggested the use of the theory of complex
multiplication to construct such curves. One of the steps in this method is the
calculation of a root modulo n of the Hilbert class polynomial H(X) for a
fundamental discriminant D. The usual way is to compute H(X) over the integers
and then to find the root modulo n. We present a modified version of the
Chinese remainder theorem (CRT) to compute H(X) modulo n directly from the
knowledge of H(X) modulo enough small primes. Our complexity analysis suggests
that asymptotically our algorithm is an improvement over previously known
methods.
|
math/0112216
|
Classification of Finite Dynamical Systems
|
math.DS cs.MA math.CO
|
This paper is motivated by the theory of sequential dynamical systems,
developed as a basis for a mathematical theory of computer simulation. It
contains a classification of finite dynamical systems on binary strings, which
are obtained by composing functions defined on the coordinates. The
classification is in terms of the dependency relations among the coordinate
functions. It suggests a natural notion of the linearization of a system.
Furthermore, it contains a sharp upper bound on the number of systems in terms
of the dependencies among the coordinate functions. This upper bound
generalizes an upper bound for sequential dynamical systems.
|
math/0202276
|
A numerical method for solution of ordinary differential equations of
fractional order
|
math.NA cs.CE physics.comp-ph
|
In this paper we propose an algorithm for the numerical solution of arbitrary
differential equations of fractional order. The algorithm is obtained by using
the following decomposition of the differential equation into a system of
differential equation of integer order connected with inverse forms of
Abel-integral equations. The algorithm is used for solution of the linear and
non-linear equations.
|
math/0203059
|
On linear programming bounds for spherical codes and designs
|
math.CO cs.IT math.IT math.OC
|
We investigate universal bounds on spherical codes and spherical designs that
could be obtained using Delsarte's linear programming methods. We give a lower
estimate for the LP upper bound on codes, and an upper estimate for the LP
lower bound on designs. Specifically, when the distance of the code is fixed
and the dimension goes to infinity, the LP upper bound on codes is at least as
large as the average of the best known upper and lower bounds. When the
dimension n of the design is fixed, and the strength k goes to infinity, the LP
bound on designs turns out, in conjunction with known lower bounds, to be
proportional to k^{n-1}.
|
math/0205218
|
A New Operation on Sequences: the Boustrouphedon Transform
|
math.CO cs.IT math.IT
|
A generalization of the Seidel-Entringer-Arnold method for calculating the
alternating permutation numbers (or secant-tangent numbers) leads to a new
operation on integer sequences, the Boustrophedon transform.
|
math/0205299
|
The Lattice of N-Run Orthogonal Arrays
|
math.CO cs.IT math.IT
|
If the number of runs in a (mixed-level) orthogonal array of strength 2 is
specified, what numbers of levels and factors are possible? The collection of
possible sets of parameters for orthogonal arrays with N runs has a natural
lattice structure, induced by the ``expansive replacement'' construction
method. In particular the dual atoms in this lattice are the most important
parameter sets, since any other parameter set for an N-run orthogonal array can
be constructed from them. To get a sense for the number of dual atoms, and to
begin to understand the lattice as a function of N, we investigate the height
and the size of the lattice. It is shown that the height is at most [c(N-1)],
where c= 1.4039... and that there is an infinite sequence of values of N for
which this bound is attained. On the other hand, the number of nodes in the
lattice is bounded above by a superpolynomial function of N (and
superpolynomial growth does occur for certain sequences of values of N). Using
a new construction based on ``mixed spreads'', all parameter sets with 64 runs
are determined. Four of these 64-run orthogonal arrays appear to be new.
|
math/0205301
|
Some Canonical Sequences of Integers
|
math.CO cs.IT math.IT
|
Extending earlier work of R. Donaghey and P. J. Cameron, we investigate some
canonical "eigen-sequences" associated with transformations of integer
sequences. Several known sequences appear in a new setting: for instance the
sequences (such as 1, 3, 11, 49, 257, 1531, ...) studied by T. Tsuzuku, H. O.
Foulkes and A. Kerber in connection with multiply transitive groups are
eigen-sequences for the binomial transform. Many interesting new sequences also
arise, such as 1, 1, 2, 26, 152, 1144, ..., which shifts one place left when
transformed by the Stirling numbers of the second kind, and whose exponential
generating function satisfies A'(x) = A(e^x -1) + 1.
|
math/0205303
|
On Asymmetric Coverings and Covering Numbers
|
math.CO cs.IT math.IT
|
An asymmetric covering D(n,R) is a collection of special subsets S of an
n-set such that every subset T of the n-set is contained in at least one
special S with |S| - |T| <= R. In this paper we compute the smallest size of
any D(n,1) for n <= 8. We also investigate ``continuous'' and ``banded''
versions of the problem. The latter involves the classical covering numbers
C(n,k,k-1), and we determine the following new values: C(10,5,4) = 51,
C(11,7,6,) =84, C(12,8,7) = 126, C(13,9,8)= 185 and C(14,10,9) = 259. We also
find the number of nonisomorphic minimal covering designs in several cases.
|
math/0207121
|
The Shannon-McMillan Theorem for Ergodic Quantum Lattice Systems
|
math.DS cs.DS cs.IT math-ph math.IT math.MP math.OA quant-ph
|
We formulate and prove a quantum Shannon-McMillan theorem. The theorem
demonstrates the significance of the von Neumann entropy for translation
invariant ergodic quantum spin systems on n-dimensional lattices: the entropy
gives the logarithm of the essential number of eigenvectors of the system on
large boxes. The one-dimensional case covers quantum information sources and is
basic for coding theorems.
|
math/0207146
|
A Zador-Like Formula for Quantizers Based on Periodic Tilings
|
math.CO cs.IT math.IT
|
We consider Zador's asymptotic formula for the distortion-rate function for a
variable-rate vector quantizer in the high-rate case. This formula involves the
differential entropy of the source, the rate of the quantizer in bits per
sample, and a coefficient G which depends on the geometry of the quantizer but
is independent of the source. We give an explicit formula for G in the case
when the quantizing regions form a periodic tiling of n-dimensional space, in
terms of the volumes and second moments of the Voronoi cells. As an application
we show, extending earlier work of Kashyap and Neuhoff, that even a
variable-rate three-dimensional quantizer based on the ``A15'' structure is
still inferior to a quantizer based on the body-centered cubic lattice. We also
determine the smallest covering radius of such a structure.
|
math/0207147
|
Quantizing Using Lattice Intersections
|
math.CO cs.IT math.IT
|
The usual quantizer based on an n-dimensional lattice L maps a point x in R^n
to a closest lattice point. Suppose L is the intersection of lattices L_1, ...,
L_r. Then one may instead combine the information obtained by simultaneously
quantizing x with respect to each of the L_i. This corresponds to decomposing
R^n into a honeycomb of cells which are the intersections of the Voronoi cells
for the L_i, and identifying the cell to which x belongs. This paper shows how
to write several standard lattices (the face-centered and body-centered cubic
lattices, the root lattices D_4, E_6*, E_8, the Coxeter-Todd, Barnes-Wall and
Leech lattices, etc.) in a canonical way as intersections of a small number of
simpler, decomposable, lattices. The cells of the honeycombs are given
explicitly and the mean squared quantizing error calculated in the cases when
the intersection lattice is the face-centered or body-centered cubic lattice or
the lattice D_4.
|
math/0207186
|
A Simple Construction for the Barnes-Wall Lattices
|
math.CO cs.IT math.IT
|
A certain family of orthogonal groups (called "Clifford groups" by G. E.
Wall) has arisen in a variety of different contexts in recent years. These
groups have a simple definition as the automorphism groups of certain
generalized Barnes-Wall lattices. This leads to an especially simple
construction for the usual Barnes-Wall lattices. This is based on the third
author's talk at the Forney-Fest, M.I.T., March 2000, which in turn is based on
our paper "The Invariants of the Clifford Groups", Designs, Codes, Crypt., 24
(2001), 99--121, to which the reader is referred for further details and
proofs.
|
math/0207197
|
On Single-Deletion-Correcting Codes
|
math.CO cs.IT math.IT
|
This paper gives a brief survey of binary single-deletion-correcting codes.
The Varshamov-Tenengolts codes appear to be optimal, but many interesting
unsolved problems remain. The connections with shift-register sequences also
remain somewhat mysterious.
|
math/0207208
|
The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes
|
math.CO cs.IT math.IT
|
Certain notorious nonlinear binary codes contain more codewords than any
known linear code. These include the codes constructed by Nordstrom-Robinson,
Kerdock, Preparata, Goethals, and Delsarte-Goethals. It is shown here that all
these codes can be very simply constructed as binary images under the Gray map
of linear codes over Z_4, the integers mod 4 (although this requires a slight
modification of the Preparata and Goethals codes). The construction implies
that all these binary codes are distance invariant. Duality in the Z_4 domain
implies that the binary images have dual weight distributions. The Kerdock and
"Preparata" codes are duals over Z_4 -- and the Nordstrom-Robinson code is
self-dual -- which explains why their weight distributions are dual to each
other. The Kerdock and "Preparata" codes are Z_4-analogues of first-order
Reed-Muller and extended Hamming codes, respectively. All these codes are
extended cyclic codes over Z_4, which greatly simplifies encoding and decoding.
An algebraic hard-decision decoding algorithm is given for the "Preparata" code
and a Hadamard-transform soft-decision decoding algorithm for the Kerdock code.
Binary first- and second-order Reed-Muller codes are also linear over Z_4, but
extended Hamming codes of length n >= 32 and the Golay code are not. Using
Z_4-linearity, a new family of distance regular graphs are constructed on the
cosets of the "Preparata" code.
|
math/0207209
|
Interleaver Design for Turbo Codes
|
math.CO cs.IT math.IT
|
The performance of a Turbo code with short block length depends critically on
the interleaver design. There are two major criteria in the design of an
interleaver: the distance spectrum of the code and the correlation between the
information input data and the soft output of each decoder corresponding to its
parity bits. This paper describes a new interleaver design for Turbo codes with
short block length based on these two criteria. A deterministic interleaver
suitable for Turbo codes is also described. Simulation results compare the new
interleaver design to different existing interleavers.
|
math/0207256
|
The Sphere-Packing Problem
|
math.CO cs.IT math.IT
|
A brief report on recent work on the sphere-packing problem.
|
math/0207291
|
On Kissing Numbers in Dimensions 32 to 128
|
math.CO cs.IT math.IT
|
An elementary construction using binary codes gives new record kissing
numbers in dimensions from 32 to 128.
|
math/0208001
|
Self-Dual Codes
|
math.CO cs.IT math.IT
|
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.
|
math/0208017
|
Packing Planes in Four Dimensions and Other Mysteries
|
math.CO cs.IT math.IT
|
How should you choose a good set of (say) 48 planes in four dimensions? More
generally, how do you find packings in Grassmannian spaces? In this article I
give a brief introduction to the work that I have been doing on this problem in
collaboration with A. R. Calderbank, J. H. Conway, R. H. Hardin, E. M. Rains
and P. W. Shor. We have found many nice examples of specific packings (70
4-spaces in 8-space, for instance), several general constructions, and an
embedding theorem which shows that a packing in Grassmannian space G(m,n) is a
subset of a sphere in R^D, where D = (m+2)(m-1)/2, and leads to a proof that
many of our packings are optimal. There are a number of interesting unsolved
problems.
|
math/0208155
|
Toric codes over finite fields
|
math.AG cs.IT math.CO math.IT
|
In this note, a class of error-correcting codes is associated to a toric
variety associated to a fan defined over a finite field $\fff_q$, analogous to
the class of Goppa codes associated to a curve. For such a ``toric code''
satisfying certain additional conditions, we present an efficient decoding
algorithm for the dual of a Goppa code. Many examples are given. For small $q$,
many of these codes have parameters beating the Gilbert-Varshamov bound. In
fact, using toric codes, we construct a $(n,k,d)=(49,11,28)$ code over
$\fff_8$, which is better than any other known code listed in Brouwer's on-line
tables for that $n$ and $k$.
|
math/0209407
|
Uniformly distributed sequences of p-adic integers, II
|
math.NT cs.IT math.DS math.IT
|
The paper describes ergodic (with respect to the Haar measure) functions in
the class of all functions, which are defined on (and take values in) the ring
of p-adic integers, and which satisfy (at least, locally) Lipschitz condition
with coefficient 1. Equiprobable (in particular, measure-preserving) functions
of this class are described also. In some cases (and especially for p=2) the
descriptions are given by explicit formulae. Some of the results may be viewed
as descriptions of ergodic isometric dynamical systems on p-adic unit disk. The
study was motivated by the problem of pseudorandom number generation for
computer simulation and cryptography. From this view the paper describes
nonlinear congruential pseudorandom generators modulo M which produce stricly
periodic uniformly distributed sequences modulo M with maximal possible period
length (i.e., exactly M). Both the state change function and the output
function of these generators could be, e.g., meromorphic functions (in
particular, polynomials with rational, but not necessarily integer
coefficients, or rational functions), or compositions of arithmetical
operations (like addition, multiplication, exponentiation, raising to integer
powers, including negative ones) with standard computer operations, such as
bitwise logical operations (XOR, OR, AND, etc.). The linear complexity of the
produced sequences is also studied.
|
math/0210018
|
Topological robotics: motion planning in projective spaces
|
math.AT cs.RO math.DG
|
We study an elementary problem of topological robotics: rotation of a line,
which is fixed by a revolving joint at a base point: one wants to bring the
line from its initial position to a final position by a continuous motion in
the space. The final goal is to construct an algorithm which will perform this
task once the initial and final positions are given.
Any such motion planning algorithm will have instabilities, which are caused
by topological reasons. A general approach to study instabilities of robot
motion was suggested recently by the first named author. With any
path-connected topological space X one associates a number TC(X), called the
topological complexity of X. This number is of fundamental importance for the
motion planning problem: TC(X) determines character of instabilities which have
all motion planning algorithms in X.
In the present paper we study the topological complexity of real projective
spaces. In particular we compute TC(RP^n) for all n<24. Our main result is that
(for n distinct from 1, 3, 7) the problem of calculating of TC(RP^n) is
equivalent to finding the smallest k such that RP^n can be immersed into the
Euclidean space R^{k-1}.
|
math/0210115
|
Topological Robotics: Subspace Arrangements and Collision Free Motion
Planning
|
math.AT cs.RO math.DG
|
We study an elementary problem of the topological robotics: collective motion
of a set of $n$ distinct particles which one has to move from an initial
configuration to a final configuration, with the requirement that no collisions
occur in the process of motion. The ultimate goal is to construct an algorithm
which will perform this task once the initial and the final configurations are
given. This reduces to a topological problem of finding the topological
complexity TC(C_n(\R^m)) of the configutation space C_n(\R^m) of $n$ distinct
ordered particles in \R^m. We solve this problem for m=2 (the planar case) and
for all odd m, including the case m=3 (particles in the three-dimensional
space). We also study a more general motion planning problem in Euclidean space
with a hyperplane arrangement as obstacle.
|
math/0210408
|
Representations of finite groups on Riemann-Roch spaces
|
math.AG cs.IT math.GR math.IT
|
We study the action of a finite group on the Riemann-Roch space of certain
divisors on a curve. If $G$ is a finite subgroup of the automorphism group of a
projective curve $X$ over an algebraically closed field and $D$ is a divisor on
$X$ left stable by $G$ then we show the irreducible constituents of the natural
representation of $G$ on the Riemann-Roch space $L(D)=L_X(D)$ are of dimension
$\leq d$, where $d$ is the size of the smallest $G$-orbit acting on $X$. We
give an example to show that this is, in general, sharp (i.e., that dimension
$d$ irreducible constituents can occur). Connections with coding theory, in
particular to permutation decoding of AG codes, are discussed in the last
section. Many examples are included.
|
math/0211040
|
On cyclic convolutional codes
|
math.RA cs.IT math.CO math.IT
|
We investigate the notion of cyclicity for convolutional codes as it has been
introduced by Piret and Roos in the seventies. Codes of this type are described
as submodules of the module of all vector polynomials in one variable with some
additional generalized cyclic structure but also as specific left ideals in a
skew polynomial ring. Extending a result of Piret, we show in a purely
algebraic setting that these ideals are always principal. This leads to the
notion of a generator polynomial just like for cyclic block codes. Similarly a
control polynomial can be introduced by considering the right annihilator
ideal. An algorithmic procedure is developed which produces unique reduced
generator and control polynomials. We also show how basic code properties and a
minimal generator matrix can be read off from these objects. A close link
between polynomial and vector description of the codes is provided by certain
generalized circulant matrices.
|
math/0211107
|
On Near-MDS Elliptic Codes
|
math.AG cs.IT math.CO math.IT
|
The main conjecture on maximum distance separable (MDS) codes states that,
execpt for some special cases, the maximum length of a q-ary linear MDS code is
q+1. This conjecture does not hold true for near maximum distance separable
codes because of the existence of q-ary near MDS elliptic codes having length
bigger than q+1. An interesting related question is whether a near MDS elliptic
code can be extended to a longer near MDS code. Our results are some
non-extendability results and an alternative and simpler construction for
certain known near MDS elliptic codes.
|
math/0211269
|
On ASGS framework: general requirements and an example of implementation
|
math.CO cs.CR cs.DM cs.IT math.IT
|
In the paper we propose general framework for Automatic Secret Generation and
Sharing (ASGS) that should be independent of underlying secret sharing scheme.
ASGS allows to prevent the dealer from knowing the secret or even to eliminate
him at all. Two situations are discussed. First concerns simultaneous
generation and sharing of the random, prior nonexistent secret. Such a secret
remains unknown until it is reconstructed. Next, we propose the framework for
automatic sharing of a known secret. In this case the dealer does not know the
secret and the secret owner does not know the shares. We present opportunities
for joining ASGS with other extended capabilities, with special emphasize on
PVSS and proactive secret sharing. Finally, we illustrate framework with
practical implementation.
Keywords: cryptography, secret sharing, data security, extended capabilities,
extended key verification protocol
|
math/0212038
|
A Goppa-like bound on the trellis state complexity of algebraic
geometric codes
|
math.AG cs.IT math.IT
|
For a linear code $\cC$ of length $n$ and dimension $k$, Wolf noticed that
the trellis state complexity $s(\cC)$ of $\cC$ is upper bounded by
$w(\cC):=\min(k,n-k)$. In this paper we point out some new lower bounds for
$s(\cC)$. In particular, if $\cC$ is an Algebraic Geometric code, then
$s(\cC)\geq w(\cC)-(g-a)$, where $g$ is the genus of the underlying curve and
$a$ is the abundance of the code.
|
math/0212212
|
Coverage control for mobile sensing networks
|
math.OC cs.IT math.IT
|
This paper presents control and coordination algorithms for groups of
vehicles. The focus is on autonomous vehicle networks performing distributed
sensing tasks where each vehicle plays the role of a mobile tunable sensor. The
paper proposes gradient descent algorithms for a class of utility functions
which encode optimal coverage and sensing policies. The resulting closed-loop
behavior is adaptive, distributed, asynchronous, and verifiably correct.
|
math/0301135
|
Grassmannian Frames with Applications to Coding and Communication
|
math.FA cs.IT math.IT
|
For a given class ${\cal F}$ of uniform frames of fixed redundancy we define
a Grassmannian frame as one that minimizes the maximal correlation $|< f_k,f_l
>|$ among all frames $\{f_k\}_{k \in {\cal I}} \in {\cal F}$. We first analyze
finite-dimensional Grassmannian frames. Using links to packings in Grassmannian
spaces and antipodal spherical codes we derive bounds on the minimal achievable
correlation for Grassmannian frames. These bounds yield a simple condition
under which Grassmannian frames coincide with uniform tight frames. We exploit
connections to graph theory, equiangular line sets, and coding theory in order
to derive explicit constructions of Grassmannian frames. Our findings extend
recent results on uniform tight frames. We then introduce infinite-dimensional
Grassmannian frames and analyze their connection to uniform tight frames for
frames which are generated by group-like unitary systems. We derive an example
of a Grassmannian Gabor frame by using connections to sphere packing theory.
Finally we discuss the application of Grassmannian frames to wireless
communication and to multiple description coding.
|
math/0301268
|
Improving Search Algorithms by Using Intelligent Coordinates
|
math.OC cond-mat.stat-mech cs.MA nlin.AO
|
We consider the problem of designing a set of computational agents so that as
they all pursue their self-interests a global function G of the collective
system is optimized. Three factors govern the quality of such design. The first
relates to conventional exploration-exploitation search algorithms for finding
the maxima of such a global function, e.g., simulated annealing. Game-theoretic
algorithms instead are related to the second of those factors, and the third is
related to techniques from the field of machine learning. Here we demonstrate
how to exploit all three factors by modifying the search algorithm's
exploration stage so that rather than by random sampling, each coordinate of
the underlying search space is controlled by an associated
machine-learning-based ``player'' engaged in a non-cooperative game.
Experiments demonstrate that this modification improves SA by up to an order of
magnitude for bin-packing and for a model of an economic process run over an
underlying network. These experiments also reveal novel small worlds phenomena.
|
math/0302043
|
Extended visual cryptography systems
|
math.CO cs.IT math.IT
|
Visual cryptography schemes have been introduced in 1994 by Naor and Shamir.
Their idea was to encode a secret image into $n$ shadow images and to give
exactly one such shadow image to each member of a group $P$ of $n$ persons.
Whereas most work in recent years has been done concerning the problem of
qualified and forbidden subsets of $P$ or the question of contrast optimizing,
in this paper we study extended visual cryptography schemes, i.e. shared secret
systems where any subset of $P$ shares its own secret.
|
math/0302132
|
Computing Symmetrized Weight Enumerators for Lifted Quadratic Residue
Codes
|
math.CO cs.IT math.IT
|
The paper describes a method to determine symmetrized weight enumerators of
$p^m$-linear codes based on the notion of a disjoint weight enumerator.
Symmetrized weight enumerators are given for the lifted quadratic residue codes
of length 24 modulo $2^m$ and modulo $3^m$, for any positive $m$.
|
math/0302154
|
Twisted Klein curves modulo 2
|
math.NT cs.IT math.AG math.IT
|
We give an explicit description of all 168 quartic curves over the field of
two elements that are isomorphic to the Klein curve over an algebraic
extension. Some of the curves have been known for their small class number,
others for attaining the maximal number of rational points.
|
math/0302172
|
Results on zeta functions for codes
|
math.CO cs.IT math.IT math.NT
|
We give a new and short proof of the Mallows-Sloane upper bound for self-dual
codes. We formulate a version of Greene's theorem for normalized weight
enumerators. We relate normalized rank-generating polynomials to two-variable
zeta functions. And we show that a self-dual code has the Clifford property,
but that the same property does not hold in general for formally self-dual
codes.
|
math/0303104
|
Bounding the trellis state complexity of algebraic geometric codes
|
math.AG cs.IT math.IT
|
Let C be an algebraic geometric code of dimension k and length n constructed
on a curve X over $F_q$. Let s(C) be the state complexity of C and set
w(C):=min{k,n-k}, the Wolf upper bound on s(C). We introduce a numerical
function R that depends on the gonality sequence of X and show that s(C)\geq
w(C)-R(2g-2), where g is the genus of X. As a matter of fact, R(2g-2)\leq
g-(\gamma_2-2) with \gamma_2 being the gonality over F_q of X, and thus in
particular we have that s(C)\geq w(C)-g+\gamma_2-2.
|
math/0303254
|
Strongly MDS Convolutional Codes
|
math.RA cs.IT math.IT math.OC
|
MDS convolutional codes have the property that their free distance is maximal
among all codes of the same rate and the same degree. In this paper we
introduce a class of MDS convolutional codes whose column distances reach the
generalized Singleton bound at the earliest possible instant. We call these
codes strongly MDS convolutional codes. It is shown that these codes can decode
a maximum number of errors per time interval when compared with other
convolutional codes of the same rate and degree. These codes have also a
maximum or near maximum distance profile. A code has a maximum distance profile
if and only if the dual code has this property.
|
math/0304192
|
On reconstructing n-point configurations from the distribution of
distances or areas
|
math.AC cs.CV cs.SC
|
One way to characterize configurations of points up to congruence is by
considering the distribution of all mutual distances between points. This paper
deals with the question if point configurations are uniquely determined by this
distribution. After giving some counterexamples, we prove that this is the case
for the vast majority of configurations. In the second part of the paper, the
distribution of areas of sub-triangles is used for characterizing point
configurations. Again it turns out that most configurations are reconstructible
from the distribution of areas, though there are counterexamples.
|
math/0304283
|
Whitehead method and Genetic Algorithms
|
math.GR cs.NE cs.SC
|
In this paper we discuss a genetic version (GWA) of the Whitehead's
algorithm, which is one of the basic algorithms in combinatorial group theory.
It turns out that GWA is surprisingly fast and outperforms the standard
Whitehead's algorithm in free groups of rank >= 5. Experimenting with GWA we
collected an interesting numerical data that clarifies the time-complexity of
the Whitehead's Problem in general. These experiments led us to several
mathematical conjectures. If confirmed they will shed light on hidden
mechanisms of Whitehead Method and geometry of automorphic orbits in free
groups.
|
math/0304292
|
The Ubiquity of Order Domains for the Construction of Error Control
Codes
|
math.AC cs.IT math.AG math.IT math.RA
|
The order domains are a class of commutative rings introduced by H{\o}holdt,
van Lint, and Pellikaan to simplify the theory of error control codes using
ideas from algebraic geometry. The definition is largely motivated by the
structures utilized in the Berlekamp-Massey-Sakata (BMS) decoding algorithm,
with Feng-Rao majority voting for unknown syndromes, applied to one-point
geometric Goppa codes constructed from curves. However, order domains are much
more general, and O'Sullivan has shown that the BMS algorithm can be applied to
decode all codes constructed from order domains by a suitable generalization of
Goppa's procedure for curves. In this article we will first discuss the
connection between order domains and valuations on function fields over a
finite field. Under some mild conditions, we will see that a general projective
variety over a finite field has projective models which can be used to
construct order domains and Goppa-type codes for which the BMS algorithm is
applicable. We will then give a slightly different interpretation of Geil and
Pellikaan's extrinsic characterization of order domains via the theory of
Gr\"obner bases, and show that their results are related to the existence of
toric deformations of varieties. To illustrate the potential usefulness of
these observations, we present a series of new explicit examples of order
domains associated to varieties with many rational points over finite fields:
Hermitian hypersurfaces, Grassmannians, and flag varieties.
|
math/0304306
|
Genetic algorithms and the Andrews-Curtis conjecture
|
math.GR cs.NE cs.SC
|
The Andrews-Curtis conjecture claims that every balanced presentation of the
trivial group can be transformed into the trivial presentation by a finite
sequence of "elementary transformations" which are Nielsen transformations
together with an arbitrary conjugation of a relator. It is believed that the
Andrews-Curtis conjecture is false; however, not so many possible
counterexamples are known. It is not a trivial matter to verify whether the
conjecture holds for a given balanced presentation or not. The purpose of this
paper is to describe some non-deterministic methods, called Genetic Algorithms,
designed to test the validity of the Andrews-Curtis conjecture. Using such
algorithm we have been able to prove that all known (to us) balanced
presentations of the trivial group where the total length of the relators is at
most 12 satisfy the conjecture. In particular, the Andrews-Curtis conjecture
holds for the presentation <x,y|x y x = y x y, x^2 = y^3> which was one of the
well known potential counterexamples.
|
math/0305121
|
Robust Estimators under the Imprecise Dirichlet Model
|
math.PR cs.IT cs.LG math.IT math.ST stat.TH
|
Walley's Imprecise Dirichlet Model (IDM) for categorical data overcomes
several fundamental problems which other approaches to uncertainty suffer from.
Yet, to be useful in practice, one needs efficient ways for computing the
imprecise=robust sets or intervals. The main objective of this work is to
derive exact, conservative, and approximate, robust and credible interval
estimates under the IDM for a large class of statistical estimators, including
the entropy and mutual information.
|
math/0305135
|
Distance bounds for convolutional codes and some optimal codes
|
math.RA cs.IT math.IT math.OC
|
After a discussion of the Griesmer and Heller bound for the distance of a
convolutional code we present several codes with various parameters, over
various fields, and meeting the given distance bounds. Moreover, the Griesmer
bound is used for deriving a lower bound for the field size of an MDS
convolutional code and examples are presented showing that, in most cases, the
lower bound is tight. Most of the examples in this paper are cyclic
convolutional codes in a generalized sense as it has been introduced in the
seventies. A brief introduction to this promising type of cyclicity is given at
the end of the paper in order to make the examples more transparent.
|
math/0305308
|
Numerical Analogues of Aronson's Sequence
|
math.NT cs.IT math.IT
|
Aronson's sequence 1, 4, 11, 16, ... is defined by the English sentence ``t
is the first, fourth, eleventh, sixteenth, ... letter of this sentence.'' This
paper introduces some numerical analogues, such as: a(n) is taken to be the
smallest positive integer greater than a(n-1) which is consistent with the
condition ``n is a member of the sequence if and only if a(n) is odd.'' This
sequence can also be characterized by its ``square'', the sequence a^(2)(n) =
a(a(n)), which equals 2n+3 for n >= 1. There are many generalizations of this
sequence, some of which are new, while others throw new light on previously
known sequences.
|
math/0306354
|
Coding and tiling of Julia sets for subhyperbolic rational maps
|
math.DS cs.IT math.IT
|
Let $f:\hat{C}\to\hat{C}$ be a subhyperbolic rational map of degree $d$. We
construct a set of coding maps $Cod(f)=\{\pi_r:\Sigma\to J\}_r$ of the Julia
set $J$ by geometric coding trees, where the parameter $r$ ranges over mappings
from a certain tree to the Riemann sphere. Using the universal covering space
$\phi:\tilde S\to S$ for the corresponding orbifold, we lift the inverse of $f$
to an iterated function system $I=(g_i)_{i=1,2,...,d}$. For the purpose of
studying the structure of $Cod(f)$, we generalize Kenyon and Lagarias-Wang's
results : If the attractor $K$ of $I$ has positive measure, then $K$ tiles
$\phi^{-1}(J)$, and the multiplicity of $\pi_r$ is well-defined. Moreover, we
see that the equivalence relation induced by $\pi_r$ is described by a finite
directed graph, and give a necessary and sufficient condition for two coding
maps $\pi_r$ and $\pi_{r'}$ to be equal.
|
math/0306395
|
Sur la non-linearite des fonctions booleennes
|
math.NT cs.IT math.IT
|
Boolean functions on the space $F_{2}^m$ are not only important in the theory
of error-correcting codes, but also in cryptography, where they occur in
private key systems. In these two cases, the nonlinearity of these function is
a main concept. In this article, I show that the spectral amplitude of boolean
functions, which is linked to their nonlinearity, is of the order of
$2^{m/2}\sqrt{m}$ in mean, whereas its range is bounded by $2^{m/2}$ and $2^m$.
Moreover I examine a conjecture of Patterson and Wiedemann saying that the
minimum of this spectral amplitude is as close as desired to $2^{m/2}$. I also
study a weaker conjecture about the moments of order 4 of their Fourier
transform. This article is inspired by works of Salem, Zygmund, Kahane and
others about the related problem of real polynomials with random coefficients.
|
math/0307064
|
The Number of Hierarchical Orderings
|
math.CO cs.IT math.IT
|
An ordered set-partition (or preferential arrangement) of n labeled elements
represents a single ``hierarchy''; these are enumerated by the ordered Bell
numbers. In this note we determine the number of ``hierarchical orderings'' or
``societies'', where the n elements are first partitioned into m <= n subsets
and a hierarchy is specified for each subset. We also consider the unlabeled
case, where the ordered Bell numbers are replaced by the composition numbers.
If there is only a single hierarchy, we show that the average rank of an
element is asymptotic to n/(4 log 2) in the labeled case and to n/4 in the
unlabeled case.
|
math/0307196
|
Convolutional Codes with Maximum Distance Profile
|
math.OC cs.IT math.IT math.RA
|
Maximum distance profile codes are characterized by the property that two
trajectories which start at the same state and proceed to a different state
will have the maximum possible distance from each other relative to any other
convolutional code of the same rate and degree.
In this paper we use methods from systems theory to characterize maximum
distance profile codes algebraically. Tha main result shows that maximum
distance profile codes form a generic set inside the variety which parametrizes
the set of convolutional codes of a fixed rate and a fixed degree.
|
math/0308046
|
Still better nonlinear codes from modular curves
|
math.NT cs.IT math.AG math.IT
|
We give a new construction of nonlinear error-correcting codes over suitable
finite fields k from the geometry of modular curves with many rational points
over k, combining two recent improvements on Goppa's construction. The
resulting codes are asymptotically the best currently known.
|
math/0308110
|
Sphere packing bounds in the Grassmann and Stiefel manifolds
|
math.MG cs.IT math.IT
|
Applying the Riemann geometric machinery of volume estimates in terms of
curvature, bounds for the minimal distance of packings/codes in the Grassmann
and Stiefel manifolds will be derived and analyzed. In the context of
space-time block codes this leads to a monotonically increasing minimal
distance lower bound as a function of the block length. This advocates large
block lengths for the code design.
|
math/0308153
|
Mathematics and Logic as Information Compression by Multiple Alignment,
Unification and Search
|
math.GM cs.AI math.LO
|
This article introduces the conjecture that "mathematics, logic and related
disciplines may usefully be understood as information compression (IC) by
'multiple alignment', 'unification' and 'search' (ICMAUS)".
As a preparation for the two main sections of the article, concepts of
information and information compression are reviewed. Related areas of research
are also described including IC in brains and nervous systems, and IC in
relation to inductive inference, Minimum Length Encoding and probabilistic
reasoning. The ICMAUS concepts and a computer model in which they are embodied
are briefly described.
The first of the two main sections describes how many of the commonly-used
forms and structures in mathematics, logic and related disciplines (such as
theoretical linguistics and computer programming) may be seen as devices for
IC. In some cases, these forms and structures may be interpreted in terms of
the ICMAUS framework.
The second main section describes a selection of examples where processes of
calculation and inference in mathematics, logic and related disciplines may be
understood as IC. In many cases, these examples may be understood more
specifically in terms of the ICMAUS concepts.
|
math/0309081
|
Asymmetric binary covering codes
|
math.CO cs.IT math.IT
|
An asymmetric binary covering code of length n and radius R is a subset C of
the n-cube Q_n such that every vector x in Q_n can be obtained from some vector
c in C by changing at most R 1's of c to 0's, where R is as small as possible.
K^+(n,R) is defined as the smallest size of such a code. We show K^+(n,R) is of
order 2^n/n^R for constant R, using an asymmetric sphere-covering bound and
probabilistic methods. We show K^+(n,n-R')=R'+1 for constant coradius R' iff
n>=R'(R'+1)/2. These two results are extended to near-constant R and R',
respectively. Various bounds on K^+ are given in terms of the total number of
0's or 1's in a minimal code. The dimension of a minimal asymmetric linear
binary code ([n,R]^+ code) is determined to be min(0,n-R). We conclude by
discussing open problems and techniques to compute explicit values for K^+,
giving a table of best known bounds.
|
math/0309120
|
An invariant of finitary codes with finite expected square root coding
length
|
math.PR cs.IT math.IT
|
Let $p$ and $q$ be probability vectors with the same entropy $h$. Denote by
$B(p)$ the Bernoulli shift indexed by $\Z$ with marginal distribution $p$.
Suppose that $\phi$ is a measure preserving homomorphism from $B(p)$ to $B(q)$.
We prove that if the coding length of $\phi$ has a finite 1/2 moment, then
$\sigma_p^2=\sigma_q^2$, where $\sigma_p^2=\sum_i p_i(-\log p_i-h)^2$ is the
{\dof informational variance} of $p$. In this result, which sharpens a theorem
of Parry (1979), the 1/2 moment cannot be replaced by a lower moment. On the
other hand, for any $\theta<1$, we exhibit probability vectors $p$ and $q$ that
are not permutations of each other, such that there exists a finitary
isomorphism $\Phi$ from $B(p)$ to $B(q)$ where the coding lengths of $\Phi$ and
of its inverse have a finite $\theta$ moment. We also present an extension to
ergodic Markov chains.
|
math/0309123
|
Error Correcting Codes on Algebraic Surfaces
|
math.NT cs.IT math.AG math.IT
|
Error correcting codes are defined and important parameters for a code are
explained. Parameters of new codes constructed on algebraic surfaces are
studied. In particular, codes resulting from blowing up points in $\proj^2$ are
briefly studied, then codes resulting from ruled surfaces are covered. Codes
resulting from ruled surfaces over curves of genus 0 are completely analyzed,
and some codes are discovered that are better than direct product Reed Solomon
codes of similar length. Ruled surfaces over genus 1 curves are also studied,
but not all classes are completely analyzed. However, in this case a family of
codes are found that are comparable in performance to the direct product code
of a Reed Solomon code and a Goppa code. Some further work is done on surfaces
from higher genus curves, but there remains much work to be done in this
direction to understand fully the resulting codes. Codes resulting from blowing
points on surfaces are also studied, obtaining necessary parameters for
constructing infinite families of such codes.
Also included is a paper giving explicit formulas for curves with more
\field{q}-rational points than were previously known for certain combinations
of field size and genus. Some upper bounds are now known to be optimal from
these examples.
|
math/0309285
|
An Algorithm for Optimal Partitioning of Data on an Interval
|
math.NA astro-ph cs.CE cs.DS cs.IT math.CO math.IT
|
Many signal processing problems can be solved by maximizing the fitness of a
segmented model over all possible partitions of the data interval. This letter
describes a simple but powerful algorithm that searches the exponentially large
space of partitions of $N$ data points in time $O(N^2)$. The algorithm is
guaranteed to find the exact global optimum, automatically determines the model
order (the number of segments), has a convenient real-time mode, can be
extended to higher dimensional data spaces, and solves a surprising variety of
problems in signal detection and characterization, density estimation, cluster
analysis and classification.
|
math/0309389
|
Approximate Squaring
|
math.NT cs.IT math.IT
|
We study the ``approximate squaring'' map f(x) := x ceiling(x) and its
behavior when iterated. We conjecture that if f is repeatedly applied to a
rational number r = l/d > 1 then eventually an integer will be reached. We
prove this when d=2, and provide evidence that it is true in general by giving
an upper bound on the density of the ``exceptional set'' of numbers which fail
to reach an integer. We give similar results for a p-adic analogue of f, when
the exceptional set is nonempty, and for iterating the ``approximate
multiplication'' map f_r(x) := r ceiling(x) where r is a fixed rational number.
|
math/0309425
|
Algebraic Aspects of Multiple Zeta Values
|
math.QA cs.IT math.IT math.NT
|
Multiple zeta values have been studied by a wide variety of methods. In this
article we summarize some of the results about them that can be obtained by an
algebraic approach. This involves "coding" the multiple zeta values by
monomials in two noncommuting variables x and y. Multiple zeta values can then
be thought of as defining a map \zeta: H^0 -> R, where H^0 is the graded
rational vector space generated by the "admissible words" of the noncommutative
polynomial algebra Q<x,y>. Now H^0 admits two (commutative) products making
\zeta a homomorphism: the shuffle product and the "harmonic" product. The
latter makes H^0 a subalgebra of the algebra QSym of quasi-symmetric functions.
We also discuss some results about multiple zeta values that can be stated in
terms of derivations and cyclic derivations of Q<x,y>, and define an action of
the Hopf algebra QSym on Q<x,y> that appears useful. Finally, we apply the
algebraic approach to finite partial sums of multiple zeta value series.
|
math/0310148
|
Convolutional Codes of Goppa Type
|
math.OC cs.IT math.AG math.IT
|
A new kind of Convolutional Codes generalizing Goppa Codes is proposed. This
provides a systematic method for constructing convolutional codes with prefixed
properties. In particular, examples of Maximum-Distance Separable (MDS)
convolutional codes are obtained.
|
math/0310149
|
Convolutional Goppa Codes
|
math.OC cs.IT math.AG math.IT
|
We define Convolutional Goppa Codes over algebraic curves and construct their
corresponding dual codes. Examples over the projective line and over elliptic
curves are described, obtaining in particular some Maximum-Distance Separable
(MDS) convolutional codes.
|
math/0310385
|
De Bruijn Cycles for Covering Codes
|
math.CO cs.IT math.IT
|
A de Bruijn covering code is a q-ary string S so that every q-ary string is
at most R symbol changes from some n-word appearing consecutively in S. We
introduce these codes and prove that they can have length close to the smallest
possible covering code. The proof employs tools from field theory, probability,
and linear algebra. We also prove a number of ``spectral'' results on de Bruijn
covering codes. Included is a table of the best known bounds on the lengths of
small binary de Bruijn covering codes, up to R=11 and n=13, followed by several
open questions in this area.
|
math/0311004
|
Which Point Configurations are Determined by the Distribution of their
Pairwise Distances?
|
math.MG cs.CV math.AC math.AG
|
In a previous paper we showed that, for any $n \ge m+2$, most sets of $n$
points in $\RR^m$ are determined (up to rotations, reflections, translations
and relabeling of the points) by the distribution of their pairwise distances.
But there are some exceptional point configurations which are not
reconstructible from the distribution of distances in the above sense. In this
paper, we present a reconstructibility test with running time $O(n^{11})$. The
cases of orientation preserving rigid motions (rotations and translations) and
scalings are also discussed.
|
math/0311046
|
Codes and Invariant Theory
|
math.NT cs.IT math.IT
|
The main theorem in this paper is a far-reaching generalization of Gleason's
theorem on the weight enumerators of codes which applies to arbitrary-genus
weight enumerators of self-dual codes defined over a large class of finite
rings and modules. The proof of the theorem uses a categorical approach, and
will be the subject of a forthcoming book. However, the theorem can be stated
and applied without using category theory, and we illustrate it here by
applying it to generalized doubly-even codes over fields of characteristic 2,
doubly-even codes over the integers modulo a power of 2, and self-dual codes
over the noncommutative ring $\F_q + \F_q u$, where $u^2 = 0$..
|
math/0311129
|
Cayley-Bacharach and evaluation codes on complete intersections
|
math.AG cs.IT math.AC math.IT
|
In recent work, J. Hansen uses cohomological methods to find a lower bound
for the minimum distance of an evaluation code determined by a reduced complete
intersection in the projective plane. In this paper, we generalize Hansen's
results from P^2 to P^m; we also show that the hypotheses in Hansen's work may
be weakened. The proof is succinct and follows by combining the
Cayley-Bacharach theorem and bounds on evaluation codes obtained from reduced
zero-schemes.
|
math/0311289
|
Complete Weight Enumerators of Generalized Doubly-Even Self-Dual Codes
|
math.NT cs.IT math.IT
|
For any q which is a power of 2 we describe a finite subgroup of the group of
invertible complex q by q matrices under which the complete weight enumerators
of generalized doubly-even self-dual codes over the field with q elements are
invariant.
An explicit description of the invariant ring and some applications to
extremality of such codes are obtained in the case q=4.
|
math/0311319
|
Modular and p-adic cyclic codes
|
math.CO cs.IT math.IT
|
This paper presents some basic theorems giving the structure of cyclic codes
of length n over the ring of integers modulo p^a and over the p-adic numbers,
where p is a prime not dividing n. An especially interesting example is the
2-adic cyclic code of length 7 with generator polynomial X^3 + lambda X^2 +
(lambda - 1) X - 1, where lambda satisfies lambda^2 - lambda + 2 =0. This is
the 2-adic generalization of both the binary Hamming code and the quaternary
octacode (the latter being equivalent to the Nordstrom-Robinson code). Other
examples include the 2-adic Golay code of length 24 and the 3-adic Golay code
of length 12.
|
math/0312092
|
On the Parameters of Convolutional Codes with Cyclic Structure
|
math.RA cs.IT math.CO math.IT
|
In this paper convolutional codes with cyclic structure will be investigated.
These codes can be understood as left principal ideals in a suitable
skew-polynomial ring. It has been shown in [3] that only certain combinations
of the parameters (field size, length, dimension, and Forney indices) can occur
for cyclic codes. We will investigate whether all these combinations can indeed
be realized by a suitable cyclic code and, if so, how to construct such a code.
A complete characterization and construction will be given for minimal cyclic
codes. It is derived from a detailed investigation of the units in the
skew-polynomial ring.
|
math/0401045
|
Unitary Space Time Constellation Analysis: An Upper Bound for the
Diversity
|
math.CO cs.IT math.IT
|
The diversity product and the diversity sum are two very important parameters
for a good-performing unitary space time constellation. A basic question is
what the maximal diversity product (or sum) is. In this paper we are going to
derive general upper bounds on the diversity sum and the diversity product for
unitary constellations of any dimension $n$ and any size $m$ using packing
techniques on the compact Lie group U(n).
|
math/0401157
|
Generalized PSK in Space Time Coding
|
math.CO cs.IT math.IT math.OC
|
A wireless communication system using multiple antennas promises reliable
transmission under Rayleigh flat fading assumptions. Design criteria and
practical schemes have been presented for both coherent and non-coherent
communication channels. In this paper we generalize one dimensional phase shift
keying (PSK) signals and introduce space time constellations from generalized
phase shift keying (GPSK) signals based on the complex and real orthogonal
designs. The resulting space time constellations reallocate the energy for each
transmitting antenna and feature good diversity products, consequently their
performances are better than some of the existing comparable codes. Moreover
since the maximum likelihood (ML) decoding of our proposed codes can be
decomposed to one dimensional PSK signal demodulation, the ML decoding of our
codes can be implemented in a very efficient way.
|
math/0401279
|
Backward Optimized Orthogonal Matching Pursuit
|
math.GM cs.IT math.IT
|
A recursive approach for shrinking coefficients of an atomic decomposition is
proposed. The corresponding algorithm evolves so as to provide at each
iteration a) the orthogonal projection of a signal onto a reduced subspace and
b) the index of the coefficient to be disregarded in order to construct a
coarser approximation minimizing the norm of the residual error.
|
math/0402346
|
Applications of Lefschetz numbers in control theory
|
math.OC cs.SY math.AT
|
We develop some applications of techniques of the Lefschetz coincidence
theory in control theory. The topics are existence of equilibria and their
robustness, controllability and its robustness.
|
math/0403548
|
Remarks on codes from modular curves: MAGMA applications
|
math.NT cs.IT math.AG math.IT
|
Expository paper discussing AG or Goppa codes arising from curves, first from
an abstract general perspective then turning to concrete examples associated to
modular curves. We will try to explain these extremely technical ideas using a
special case at a level to a typical graduate student with some background in
modular forms, number theory, group theory, and algebraic geometry. Many
examples using MAGMA are included.
|
math/0404325
|
Asymptotic Improvement of the Gilbert-Varshamov Bound on the Size of
Binary Codes
|
math.CO cs.IT math.AC math.IT
|
Given positive integers $n$ and $d$, let $A_2(n,d)$ denote the maximum size
of a binary code of length $n$ and minimum distance $d$. The well-known
Gilbert-Varshamov bound asserts that $A_2(n,d) \geq 2^n/V(n,d-1)$, where
$V(n,d) = \sum_{i=0}^{d} {n \choose i}$ is the volume of a Hamming sphere of
radius $d$. We show that, in fact, there exists a positive constant $c$ such
that $$ A_2(n,d) \geq c \frac{2^n}{V(n,d-1)} \log_2 V(n,d-1) $$ whenever $d/n
\le 0.499$. The result follows by recasting the Gilbert- Varshamov bound into a
graph-theoretic framework and using the fact that the corresponding graph is
locally sparse. Generalizations and extensions of this result are briefly
discussed.
|
math/0405082
|
On the List and Bounded Distance Decodibility of the Reed-Solomon Codes
|
math.NT cs.IT math.IT
|
In this paper show that the list and bounded-distance decoding problems of
certain bounds for the Reed-Solomon code are at least as hard as the discrete
logarithm problem over finite fields.
|
math/0406077
|
A tutorial introduction to the minimum description length principle
|
math.ST cs.IT cs.LG math.IT stat.TH
|
This tutorial provides an overview of and introduction to Rissanen's Minimum
Description Length (MDL) Principle. The first chapter provides a conceptual,
entirely non-technical introduction to the subject. It serves as a basis for
the technical introduction given in the second chapter, in which all the ideas
of the first chapter are made mathematically precise. The main ideas are
discussed in great conceptual and technical detail. This tutorial is an
extended version of the first two chapters of the collection "Advances in
Minimum Description Length: Theory and Application" (edited by P.Grunwald, I.J.
Myung and M. Pitt, to be published by the MIT Press, Spring 2005).
|
math/0406221
|
Suboptimal behaviour of Bayes and MDL in classification under
misspecification
|
math.ST cs.IT cs.LG math.IT stat.TH
|
We show that forms of Bayesian and MDL inference that are often applied to
classification problems can be *inconsistent*. This means there exists a
learning problem such that for all amounts of data the generalization errors of
the MDL classifier and the Bayes classifier relative to the Bayesian posterior
both remain bounded away from the smallest achievable generalization error.
|
math/0408146
|
Learning a Machine for the Decision in a Partially Observable Markov
Universe
|
math.GM cs.AI cs.LG
|
In this paper, we are interested in optimal decisions in a partially
observable Markov universe. Our viewpoint departs from the dynamic programming
viewpoint: we are directly approximating an optimal strategic tree depending on
the observation. This approximation is made by means of a parameterized
probabilistic law. In this paper, a particular family of hidden Markov models,
with input and output, is considered as a learning framework. A method for
optimizing the parameters of these HMMs is proposed and applied. This
optimization method is based on the cross-entropic principle.
|
math/0409548
|
On mutual information, likelihood-ratios and estimation error for the
additive Gaussian channel
|
math.PR cs.IT math.IT math.ST stat.TH
|
This paper considers the model of an arbitrary distributed signal x observed
through an added independent white Gaussian noise w, y=x+w. New relations
between the minimal mean square error of the non-causal estimator and the
likelihood ratio between y and \omega are derived. This is followed by an
extended version of a recently derived relation between the mutual information
I(x;y) and the minimal mean square error. These results are applied to derive
infinite dimensional versions of the Fisher information and the de Bruijn
identity. The derivation of the results is based on the Malliavin calculus.
|
math/0410317
|
On doubly-cyclic convolutional codes
|
math.RA cs.IT math.IT
|
Cyclicity of a convolutional code (CC) is relying on a nontrivial
automorphism of the algebra F[x]/(x^n-1), where F is a finite field. If this
automorphism itself has certain specific cyclicity properties one is lead to
the class of doubly-cyclic CC's. Within this large class Reed-Solomon and BCH
convolutional codes can be defined. After constructing doubly-cyclic CC's,
basic properties are derived on the basis of which distance properties of
Reed-Solomon convolutional codes are investigated.This shows that some of them
are optimal or near optimal with respect to distance and performance.
|
math/0411515
|
Fast Non-Parametric Bayesian Inference on Infinite Trees
|
math.ST cs.LG math.PR stat.TH
|
Given i.i.d. data from an unknown distribution, we consider the problem of
predicting future items. An adaptive way to estimate the probability density is
to recursively subdivide the domain to an appropriate data-dependent
granularity. A Bayesian would assign a data-independent prior probability to
"subdivide", which leads to a prior over infinite(ly many) trees. We derive an
exact, fast, and simple inference algorithm for such a prior, for the data
evidence, the predictive distribution, the effective model dimension, and other
quantities.
|
math/0502315
|
Strong Asymptotic Assertions for Discrete MDL in Regression and
Classification
|
math.ST cs.AI cs.IT cs.LG math.IT math.PR stat.TH
|
We study the properties of the MDL (or maximum penalized complexity)
estimator for Regression and Classification, where the underlying model class
is countable. We show in particular a finite bound on the Hellinger losses
under the only assumption that there is a "true" model contained in the class.
This implies almost sure convergence of the predictive distribution to the true
one at a fast rate. It corresponds to Solomonoff's central theorem of universal
induction, however with a bound that is exponentially larger.
|
math/0504378
|
A Short Proof that Phylogenetic Tree Reconstruction by Maximum
Likelihood is Hard
|
math.PR cs.CC cs.CE math.ST q-bio.PE stat.TH
|
Maximum likelihood is one of the most widely used techniques to infer
evolutionary histories. Although it is thought to be intractable, a proof of
its hardness has been lacking. Here, we give a short proof that computing the
maximum likelihood tree is NP-hard by exploiting a connection between
likelihood and parsimony observed by Tuffley and Steel.
|
math/0504522
|
On the Classification of All Self-Dual Additive Codes over GF(4) of
Length up to 12
|
math.CO cs.IT math.IT
|
We consider additive codes over GF(4) that are self-dual with respect to the
Hermitian trace inner product. Such codes have a well-known interpretation as
quantum codes and correspond to isotropic systems. It has also been shown that
these codes can be represented as graphs, and that two codes are equivalent if
and only if the corresponding graphs are equivalent with respect to local
complementation and graph isomorphism. We use these facts to classify all codes
of length up to 12, where previously only all codes of length up to 9 were
known. We also classify all extremal Type II codes of length 14. Finally, we
find that the smallest Type I and Type II codes with trivial automorphism group
have length 9 and 12, respectively.
|
math/0507235
|
Analyticity of Entropy Rate of Hidden Markov Chains
|
math.PR cs.IT math.IT
|
We prove that under mild positivity assumptions the entropy rate of a hidden
Markov chain varies analytically as a function of the underlying Markov chain
parameters. A general principle to determine the domain of analyticity is
stated. An example is given to estimate the radius of convergence for the
entropy rate. We then show that the positivity assumptions can be relaxed, and
examples are given for the relaxed conditions. We study a special class of
hidden Markov chains in more detail: binary hidden Markov chains with an
unambiguous symbol, and we give necessary and sufficient conditions for
analyticity of the entropy rate for this case. Finally, we show that under the
positivity assumptions the hidden Markov chain {\em itself} varies
analytically, in a strong sense, as a function of the underlying Markov chain
parameters.
|
math/0508171
|
Matrices of Forests and the Analysis of Digraphs
|
math.CO cs.CV cs.NI
|
The matrices of spanning rooted forests are studied as a tool for analysing
the structure of digraphs and measuring their characteristics. The problems of
revealing the basis bicomponents, measuring vertex proximity, and ranking from
preference relations / sports competitions are considered. It is shown that the
vertex accessibility measure based on spanning forests has a number of
desirable properties. An interpretation for the normalized matrix of
out-forests in terms of information dissemination is given.
Keywords: Laplacian matrix, spanning forest, matrix-forest theorem, proximity
measure, bicomponent, ranking, incomplete tournament, paired comparisons
|
math/0508319
|
Combinations and Mixtures of Optimal Policies in Unichain Markov
Decision Processes are Optimal
|
math.CO cs.DM cs.LG math.OC math.PR
|
We show that combinations of optimal (stationary) policies in unichain Markov
decision processes are optimal. That is, let M be a unichain Markov decision
process with state space S, action space A and policies \pi_j^*: S -> A (1\leq
j\leq n) with optimal average infinite horizon reward. Then any combination \pi
of these policies, where for each state i in S there is a j such that
\pi(i)=\pi_j^*(i), is optimal as well. Furthermore, we prove that any mixture
of optimal policies, where at each visit in a state i an arbitrary action
\pi_j^*(i) of an optimal policy is chosen, yields optimal average reward, too.
|
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