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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Journal ArticleDOI
On the Use of Soft-Decision Error-Correction Codes in nand Flash Memory
TL;DR: A nonuniform memory sensing strategy is proposed to reduce the memory sensing precision and, thus, sensing latency while still maintaining good error-correction performance and carrying out extensive computer simulations to demonstrate the effectiveness and involved tradeoffs.
Journal ArticleDOI
Efficient decoding of Reed-Solomon codes beyond half the minimum distance
Ron M. Roth,G. Ruckenstein +1 more
TL;DR: A list decoding algorithm is presented for [n,k] Reed-Solomon (RS) codes over GF(q), which is capable of correcting more than [(n-k)/2] errors and improves on the time complexity O(n/sup 3/) needed for solving the equations of Sudan's algorithm by a naive Gaussian elimination.
Book ChapterDOI
Lower Bounds to Error Probability for Coding on Discrete Memoryless Channels. I
Neil J. A. Sloane,Aaron D. Wyner +1 more
TL;DR: The paper is presented in two parts: the first, appearing here, summarizes the major results and treats the case of high transmission rates in detail; the second, to appear in the subsequent issue, treats the cases of low transmission rates.
Proceedings ArticleDOI
Correcting errors beyond the Guruswami-Sudan radius in polynomial time
Farzad Parvaresh,Alexander Vardy +1 more
TL;DR: A new family of error-correcting codes that have a polynomial-time encoder and a poynomial- time list-decoder, correcting a fraction of adversarial errors up to /spl tau//sub M/ = 1 - /sup M+1//spl radic/(M/sup M/R/Sup M/) where R is the rate of the code and M /spl ges/ 1 is an arbitrary integer parameter.
Journal Article
Improved low-degree testing and its applications
Sanjeev Arora,Madhu Sudan +1 more
TL;DR: A new, and surprisingly strong, analysis is presented which shows that the preceding statement is true for arbitrarily small δ, provided the field size is polynomially larger than d/δ, and produces a self tester/corrector for any buggy program that computes a polynomial over a finite field.
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.
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.
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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.