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

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3 Author(s)
Alexander Zeh ; Institute of Telecommunications and Applied Information Theory, University of Ulm, Germany and INRIA¿Saclay-Île-de-France and École Polytechnique, France ; Christian Gentner ; Daniel Augot

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(ls4n2).

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IEEE Transactions on Information Theory  (Volume:57 ,  Issue: 9 )