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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DissertationDOI
On the number field sieve: polynomial selection and smooth elements in number fields
TL;DR: The number field sieve is the asymptotically fastest known algorithm for factoring large integers that are free of small prime factors and two aspects of the algorithm are considered in this thesis: polynomial selection and smooth elements in number fields.
Book ChapterDOI
Near-Optimal collusion-secure fingerprinting codes for efficiently tracing illegal re-distribution
Xin-Wen Wu,Alan Wee-Chung Liew +1 more
TL;DR: A class of codes which have an efficient tracing algorithm and have length O(s2log(1/e)log(N)) are constructed, much shorter than those by Cortrina-Navau and Fernandez.
Proceedings ArticleDOI
Punctured Low-Bias Codes Behave Like Random Linear Codes
TL;DR: In this paper , the authors show that the worst-case achievability bounds for random linear codes can be extended to random puncturings of any low-bias (or large alphabet) "mother" code.
Posted Content
Algebraic Soft-Decision Decoding of Hermitian Codes
TL;DR: An algebraic soft-decision decoder for Hermitian codes is presented and an interpolation algorithm is presented to find the Q-polynomial that plays a key role in the decoding.
Book ChapterDOI
Fuzzy Image Authentication with Error Localization and Correction
Obaid Ur-Rehman,Nataša Živić +1 more
TL;DR: Algorithms for fuzzy image authentication, with error localization and correction capabilities are discussed and simulation results given in this chapter show how such algorithms react to bit (and/or block) errors and how the error correction and fuzzy authentication has little or no visual impact on the resultant image quality.
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.
Book
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.
Book
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.
Book
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.
Book
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.