An Interpolation Procedure for List Decoding Reed-Solomon Codes Based on Generalized Key Equations

The key step of syndrome-based decoding of Reed-Solomon codes up to half the minimum distance is to solve the so-called Key Equation. List decoding algorithms, capable of decoding beyond half the minimum distance, are based on interpolation and factorization of multivariate polynomials. This article...

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Bibliographic Details
Published in:	IEEE transactions on information theory 2011-09, Vol.57 (9), p.5946-5959
Main Authors:	Zeh, A., Gentner, C., Augot, D.
Format:	Article
Language:	eng
Subjects:	Algorithms Applied sciences Block-Hankel matrix Coding theory Coding, codes Complexity theory Computer Science Decoding Exact sciences and technology fundamental iterative algorithm (FIA) Guruswami-Sudan interpolation Indexes Information Theory Information, signal and communications theory Interpolation Iterative decoding key equation list decoding Mathematics Polynomials Reed-Solomon codes Signal and communications theory Telecommunications and information theory
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Summary:	The key step of syndrome-based decoding of Reed-Solomon codes up to half the minimum distance is to solve the so-called Key Equation. List decoding algorithms, capable of decoding beyond half the minimum distance, are based on interpolation and factorization of multivariate polynomials. This article provides a link between syndrome-based decoding approaches based on Key Equations and the interpolation-based list decoding algorithms of Guruswami and Sudan for Reed-Solomon codes. The original interpolation conditions of Guruswami and Sudan for Reed-Solomon codes are reformulated in terms of a set of Key Equations. These equations provide a structured homogeneous linear system of equations of Block-Hankel form, that can be solved by an adaption of the Fundamental Iterative Algorithm. For an ( n , k ) Reed-Solomon code, a multiplicity s and a list size l , our algorithm has time complexity O ( ls 4 n 2 ).
ISSN:	0018-9448 1557-9654