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
Spatially coupled ensembles universally achieve capacity under belief propagation
TL;DR: The key technical result is a proof that, under belief-propagation decoding, spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble.
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Algebraic Geometric Codes: Basic Notions
TL;DR: The theory of algebraic geometric codes, a subject formed on the border of several domains of mathematics, is studied in this paper, where the authors constantly look for interpretations that connect coding theory to algebraic geometry and number theory.
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Scalable, transparent, and post-quantum secure computational integrity.
TL;DR: The first realization of a transparent ZK system (ZK-STARK) in which verification scales exponentially faster than database size is reported, and this exponential speedup in verification is observed concretely for meaningful and sequential computations, described next.
Journal ArticleDOI
Unbalanced expanders and randomness extractors from Parvaresh--Vardy codes
TL;DR: A new, self-contained construction of randomness extractors that is optimal up to constant factors, while being much simpler than the previous construction of Lu et al.
Book
List Decoding of Error-Correcting Codes
Venkatesan Guruswami,Madhu Sudan +1 more
TL;DR: This thesis presents a detailed investigation of list decoding, and proves its potential, feasibility, and importance as a combinatorial and algorithmic concept and presents the first polynomial time algorithm to decode Reed-Solomon codes beyond d/2 errors for every value of the rate.
References
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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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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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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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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.