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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Posted Content
Algebraic codes for Slepian-Wolf code design
Shizheng Li,Aditya Ramamoorthy +1 more
TL;DR: In this paper, the authors proposed the use of Reed-Solomon (RS) codes for the asymmetric version of the Slepian-Wolf problem, and showed that algebraic softdecision decoding of RS codes can be used effectively under certain correlation structures.
SIGACT News Complexity Theory Column 25
Lane A. Hemaspaandra,Madhu Sudan +1 more
TL;DR: The notion of list-decoding for error-correcting codes was introduced in the context of coding theory and complexity theory as discussed by the authors, and it has been used in a variety of applications within complexity theory.
Behavioral approach to list decoding
TL;DR: This work constructs the Most Powerful Unfalsified Model, a bivariate polynomial derived from a specific representation of the MPUM, which interpolates a given set of pairs taken from a finite field.
Proceedings ArticleDOI
A rate-distortion perspective on multiple decoding attempts for Reed-Solomon codes
TL;DR: A rate-distortion (R-D) approach is used to understand the asymptotic performance-versus-complexity trade-off of multiple error-and-erasure decoding of RS codes and is extended to analyze multiple algebraic soft-decision (ASD) decoding ofRS codes.
DissertationDOI
Decoding evaluation codes and their interleaving
TL;DR: In this paper, the authors propose a method to solve the problem of "uniformity" and "uncertainty" in the context of education.V.VV.
References
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Book
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.