Journal ArticleDOI
Improved decoding of Reed-Solomon and algebraic-geometry codes
Venkatesan Guruswami,Madhu Sudan +1 more
TLDR
An improved list decoding algorithm for decoding Reed-Solomon codes and alternant codes and algebraic-geometry codes is presented and a solution to a weighted curve-fitting problem is presented, which may be of use in soft-decision decoding algorithms for Reed- Solomon codes.Abstract:
Given an error-correcting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding Reed-Solomon codes. The list decoding problem for Reed-Solomon codes reduces to the following "curve-fitting" problem over a field F: given n points ((x/sub i//spl middot/y/sub i/))/sub i=1//sup n/, x/sub i/, y/sub i//spl isin/F, and a degree parameter k and error parameter e, find all univariate polynomials p of degree at most k such that y/sub i/=p(x/sub i/) for all but at most e values of i/spl isin/(1,...,n). We give an algorithm that solves this problem for e 1/3, where the result yields the first asymptotic improvement in four decades. The algorithm generalizes to solve the list decoding problem for other algebraic codes, specifically alternant codes (a class of codes including BCH codes) and algebraic-geometry codes. In both cases, we obtain a list decoding algorithm that corrects up to n-/spl radic/(n(n-d')) errors, where n is the block length and d' is the designed distance of the code. The improvement for the case of algebraic-geometry codes extends the methods of Shokrollahi and Wasserman (see in Proc. 29th Annu. ACM Symp. Theory of Computing, p.241-48, 1998) and improves upon their bound for every choice of n and d'. We also present some other consequences of our algorithm including a solution to a weighted curve-fitting problem, which may be of use in soft-decision decoding algorithms for Reed-Solomon codes.read more
Citations
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Proceedings ArticleDOI
Factors of low individual degree polynomials
TL;DR: This work generalizes the main factorization theorem from Dvir et al.
Journal ArticleDOI
Guest column: from randomness extraction to rotating needles
TL;DR: The finite field Kakeya problem deals with the way lines in different directions can overlap in a vector space over a finite field and several of its applications.
Quasilinear time list-decodable codes for space bounded channels.
TL;DR: In this paper, it was shown that for every constant 0 ≤ p < ½, and every sufficiently small constant ε>0, there are codes with rate R ≥ 1-H(p)-ε, list size poly(1/ε), and furthermore, their codes can handle channels with space s=n^Ω(1), which is much larger than O(log n) achieved by previous work.
Journal ArticleDOI
List-decoding methods for inferring polynomials in finite dynamical gene network models
Janis Dingel,Olgica Milenkovic +1 more
TL;DR: A novel algebraic modeling framework, termed stochastic polynomial dynamical systems (SPDSs) that can capture the dynamics of regulatory networks based on microarray expression data and shows that SPDSs constructed via list-decoders significantly outperform other algebraic reverse engineering methods, and that they also provide good guidelines for estimating the influence of genes on the Dynamics of the network.
DissertationDOI
Decoding Hermitian codes - an engineering approach
TL;DR: This thesis introduces and discusses a new algorithm for solving the key equation for Hermitian codes, that belong to the class of algebraic-geometric (AG) codes, and describes the idea of virtual extension to an interleaved code, which works only for codes with low rates and therefore the rate bound is given.
References
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The Theory of Error-Correcting Codes
TL;DR: This book presents an introduction to BCH Codes and Finite Fields, and methods for Combining Codes, and discusses self-dual Codes and Invariant Theory, as well as nonlinear Codes, Hadamard Matrices, Designs and the Golay Code.
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The Design and Analysis of Computer Algorithms
Alfred V. Aho,John E. Hopcroft +1 more
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
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Algebraic Coding Theory
TL;DR: This is the revised edition of Berlekamp's famous book, "Algebraic Coding Theory," originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering practice in this field.
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A Course in Computational Algebraic Number Theory
TL;DR: The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.
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Algebraic Function Fields and Codes
TL;DR: This new edition, published in the series Graduate Texts in Mathematics, has been considerably expanded and contains numerous exercises that help the reader to understand the basic material.