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
Improved decoding of algebraic-geometric codes with respect to the Lee metric
TL;DR: An improved decoding algorithm of algebraic-geometric codes with respect to the Lee metric is presented and an upper bound is given for the Lee-error correcting performance of this decoding algorithm.
Proceedings ArticleDOI
Improved erasure list decoding locally repairable codes using alphabet-dependent list recovery
TL;DR: New optimal constructions of locally repairable codes over small fields and their polynomial-time erasure list decoding are considered and give optimal binary codes with locality r = 2; 3.
Proceedings ArticleDOI
New Traceability Codes and Identification Algorithm for Tracing Pirates
TL;DR: In this article, the authors proposed a new family of traceability codes that is much larger than the traditional TA codes by using existing decoding algorithms with respect to the Lee distance, and they showed that the identification algorithm of generalized TA codes can find more redistributors than those of traditional TA code.
Posted Content
On the Doubly Sparse Compressed Sensing Problem
TL;DR: A new variant of the Compressed Sensing problem is investigated when the number of measurements corrupted by errors is upper bounded by some value l but there are no more restrictions on errors and it is proved that in this case it is enough to make $$2t+l$$ measurements.
Journal ArticleDOI
The combination of sparse learning and list decoding of subspace codes for error correction in random network coding
Guangzhi Zhang,Shao-bin Cai,Shao-bin Cai,Dongqiu Zhang,Wanneng Shu,Chunhua Ma,Mu Niu,Xiangnan Ding +7 more
TL;DR: In this proposal, the original packets are coded with John Wright’s coding matrix, and then, the coded message is coded again with subspace codes, and the unique solution is achieved even though there are dense propagated errors in random network coding.
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.