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
Design of Guruswami-Sudan List Decoding for Elliptic Codes
Yunqi Wan,Li Chen,Fangguo Zhang +2 more
TL;DR: The Guruswami-Sudan (GS) list decoding algorithm for elliptic codes is proposed, demonstrating their advantage over RS codes and an efficient interpolation algorithm is further presented.
Proceedings ArticleDOI
On optimal weight assignmnents for multivariate interpolation list-decoding
TL;DR: In this article, the authors considered the weight assignment problem for multivariate interpolation-based decoding with the assumption that the codes are not too short (the integrality gap for our relaxations approaches zero as the code length grows) and the optimal weight assignment scheme was given for the criterion of maximizing an allowable additive cost (e.g., Hamming distance or log probability) of an error event.
Proceedings ArticleDOI
A high-rate fingerprinting code
TL;DR: This work proposes the construction of a random-like fingerprinting code, intended to operate at rates close to fingerprinting capacity, and shows that at rates ranging from 30% to 50% of capacity, it still has a low error probability.
Patent
Decoding of product codes
TL;DR: In this article, a method for decoding data includes iteratively decoding all first subsets of a set of data two or more times in each half iteration using two-or more C1-decoding methods when a first subset is not decoded successfully using a first C1 decoding.
Book ChapterDOI
A key sharing fuzzy vault scheme
TL;DR: In this fuzzy vault scheme, the two users cooperatively build their shared fuzzy vault with a shared key hidden in it using their own biometric features, and they can respectively use theirBiometric features to unlock the fuzzy vault to get their shared key without running the risk of disclosure of their biometric Features later.
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.