An Interpolation Procedure for List Decoding Reed--Solomon codes Based on Generalized Key Equations
arXiv:1110.3898 · doi:10.1109/TIT.2011.2162160
Abstract
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 Reed-Solomon code, a multiplicity and a list size $\listl$, our algorithm has time complexity \ON{\listl s^4n^2}.
IEEE Transactions on Information Theory (2011)