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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Book ChapterDOI
Linear-Time List Recovery of High-Rate Expander Codes
Brett Hemenway,Mary Wootters +1 more
TL;DR: It is shown that expander codes, when properly instantiated, are high-rate list recoverable codes with linear-time list recovery algorithms, which approach the optimal trade-off between the number of non-trivial lists provided and the rate of the code.
Journal ArticleDOI
Review of key-binding-based biometric data protection schemes
TL;DR: This study inspects the merging of multimodal biometrics with the fuzzy systems and discusses some open challenges in this domain, and investigates in details the core ideas behind the development of the fuzzy frameworks.
Book ChapterDOI
On Fast Interpolation Method for Guruswami-Sudan List Decoding of One-Point Algebraic-Geometry Codes
TL;DR: Fast interpolation methods for the original and improved versions of list decoding of one-point algebraic-geometry codes are presented, based on the Grobner basis theory and the BMS algorithm for multiple arrays.
Proceedings ArticleDOI
List Decoding Barnes-Wall Lattices
Elena Grigorescu,Chris Peikert +1 more
TL;DR: The results imply a polynomial-time listdecoding algorithm for any error radius bounded away from the minimum distance, thus beating a typical barrier for natural error-correcting codes posed by the Johnson radius.
Journal ArticleDOI
On deep holes of generalized Reed-Solomon codes
Shaofang Hong,Rongjun Wu +1 more
TL;DR: This work shows that the received word u is a deep hole of the standard Reed-Solomon codes [q-1, k] q if its Lagrange interpolation polynomial is the sum of monomial of degree q-2 and a polynometric of degree at most k-1.
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.