id stringlengths 9 16 | title stringlengths 4 278 | categories stringlengths 5 104 | abstract stringlengths 6 4.09k |
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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
producin... |
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 fo... |
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 ... |
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 adva... |
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 specu... |
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{... |
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 o... |
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 ... |
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 pr... |
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... |
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 dis... |
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 abelia... |
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 ... |
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
equiv... |
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 ... |
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... |
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 eq... |
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 d... |
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 parti... |
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. ... |
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 proble... |
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 on... |
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 inde... |
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 i... |
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 Barn... |
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... |
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 b... |
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 formul... |
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.... |
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... |
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 thi... |
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 ... |
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 wi... |
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
rep... |
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
sk... |
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 th... |
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 a... |
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
$s(\cC)\geq w(\cC)-(g-a)$, wh... |
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 op... |
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
spa... |
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 ma... |
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 fo... |
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 ... |
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 ... |
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 ... |
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 th... |
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 interesti... |
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 ... |
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... |
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, cons... |
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... |
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 ``... |
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:\... |
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... |
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 hi... |
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 system... |
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... |
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 compres... |
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
or... |
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=... |
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 fro... |
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
guarante... |
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 ... |
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 o... |
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 lin... |
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 t... |
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 s... |
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 weake... |
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... |
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... |
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 co... |
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... |
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) signa... |
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 approxim... |
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
mod... |
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 ... |
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 f... |
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... |
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.... |
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... |
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 ... |
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", whic... |
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 almo... |
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 parsimon... |
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 o... |
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. W... |
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
vert... |
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 t... |
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