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
Correlated Algebraic-Geometric Codes: Improved List Decoding over Bounded Alphabets
TL;DR: A new family of error-correcting codes based on algebraic curves over finite fields is defined, and efficient list decoding algorithms for them are developed, showing that the PV framework applies to fairly general settings by elucidating the key algebraic concepts underlying it.
Proceedings ArticleDOI
On Transformed Folded Shortened Reed-Solomon Codes for the Correction of Phased Bursts
Z. Jianwen,Marc A. Armand +1 more
TL;DR: A shortened Reed-Solomon (RS) code may be folded into a two-dimensional array and used in a burst error channel model to reduce the probability of erroneous reception.
Posted Content
Efficient list decoding of punctured Reed-Muller codes.
TL;DR: A puncturing for which the RM code is a subcode of a certain algebraic-geometric code (which is known to be efficiently list decodable) is obtained by substituting for the variables functions with carefully chosen pole orders from an algebraic function field.
Proceedings ArticleDOI
Linear-time Erasure List-decoding of Expander Codes
TL;DR: This is the first linear-time algorithm that can list-decode expander codes from erasures beyond their (designed) distance of approximately δ2nd and shows how to improve the dependence of the running time on these parameters.
Journal ArticleDOI
Iterative Soft-Decision Decoding of Hermitian Codes
TL;DR: Performance analysis shows the proposed iterative decoding algorithm outperforms both the existing decoding approaches for Hermitian codes and the ABP-KV decoding of Reed-Solomon (RS) codes.
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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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.