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
Lower bounds on almost-separating binary codes
TL;DR: It is shown how almost separating codes can be used to construct a family of fingerprinting codes and the lower bounds on the rate are greater than the currently known lower bounds for ordinary separating codes.
Improved Decoding of Partial Unit Memory Codes Using List Decoding of Reed--Solomon Codes
TL;DR: An existing bounded minimum distance decoding algorithm for Partial Unit Memory codes is improved by using list decoding of Reed–Solomon codes.
Patent
Error correction with iterations of belief propagation decoding and algebraic decoding
Makiko Kan,Toshiyuki Miyauchi +1 more
TL;DR: An ABP decoding apparatus (31) for decoding error correction codes such as RS, BCH or AG codes diagonalizes a parity check matrix and decodes the LLR values using a soft or hard algebraic decoding algorithm as mentioned in this paper.
Proceedings ArticleDOI
Application of Soft-Decision Decoders to Non Narrow-Sense Reed-Solomon Codes
TL;DR: This paper clarifies the relationships among the several definitions of RS codes, especially focusing on the connection between non-binary BCH codes, which are not necessarily narrow-sense, and Generalized Reed- Solomon (GRS) codes, and introduces code space conversion which converts objects of a GRS code into corresponding objects of another GRS codes with the same dimension.
Proceedings ArticleDOI
A link between Guruswami-Sudan's list-decoding and decoding of interleaved Reed-Solomon codes
Alexander Zeh,Christian Senger +1 more
TL;DR: It turns out that the decoding of Interleaved RS codes can be formulated as a modified Guruswami-Sudan problem with a specific multiplicity assignment and the new approach results in the same solution space as the Welch-Berlekamp scheme.
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.