Algebraic Soft-Decision Decoding of Reed-Solomon Codes

Abstract

A polynomial-time soft-decision decoding algorithm for Reed-Solomon codes is developed. This list-decoding algorithm is algebraic in nature and builds upon the interpolation procedure proposed by Guruswami and Sudan for hard-decision decoding. Algebraic soft-decision decoding is achieved by means of converting the probabilistic reliability information into a set of interpolation points, along with their multiplicities. The proposed conversion procedure is shown to be asymptotically optimal for a certain probabilistic model. The resulting soft-decoding algorithm significantly outperforms both the Guruswami-Sudan decoding and the generalized minimum distance (GMD) decoding of Reed-Solomon codes, while maintaining a complexity that is polynomial in the length of the code. Asymptotic analysis for a large number of interpolation points is presented, leading to a geometric characterization of the decoding regions of the proposed algorithm. It is then shown that the asymptotic performance can be approached as closely as desired with a list size that does not depend on the length of the code.