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
Waterfilling-Like Multiplicity Assignment Algorithm for Algebraic Soft-Decision Decoding of Reed-Solomon Codes
TL;DR: A simplified method for algebraic soft- decision decoding of Reed-Solomon codes, which is based on the Chernoff bound algorithm proposed by El-Khamy and McEliece, which achieves better performance by assigning multiplicity in the log probability domain.
Proceedings ArticleDOI
Using Ciphers for Failure-Recovery in ITS Systems
TL;DR: The presented techniques show that ciphering functions (as deterministic, non-linear bijective functions) can serve to achieve error correction enhancement and hence allow error recovery and scalable security trade-offs with or without additional ECC components.
Dissertation
On the Algorithms of Guruswami-Sudan List Decoding over Finite Rings
TL;DR: This thesis adapts the Guruswami-Sudan (GS) list decoding algorithm to generalized Reed-Solomon (GRS) codes over finite rings with identity, and explores more deeply a lifting technique for list decoding.
Dissertation
New error correcting codes from lifting
TL;DR: The “lifted Reed-Solomon code” is constructed, a supercode of the Reed-Muller code with the same error-correcting capabilities yet vastly greater rate, and is the first high-rate code known to be both locally decodable and locally testable.
Proceedings ArticleDOI
A Class of Traceability Codes with an Efficient Tracing Algorithm
TL;DR: This paper places emphasis upon the conditions under which list decoding algorithm can be applied successfully for Reed-Solomon codes and the maximum numbers of users and traceable traitors for particular codes.
References
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Alfred V. Aho,John E. Hopcroft +1 more
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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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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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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.