Efficient root-finding algorithm with application to list decoding of algebraic-geometric codes

Abstract

A list decoding for an error-correcting code is a decoding algorithm that generates a list of codewords within a Hamming distance t from the received vector, where t can be greater than the error-correction bound. In previous work by M. Shokrollahi and H. Wasserman (see ibid., vol.45, p.432-7, March 1999) a list-decoding procedure for Reed-Solomon codes was generalized to algebraic-geometric codes. Recent work by V. Guruswami and M. Sudan (see ibid., vol.45, p.1757-67, Sept. 1999) gives improved list decodings for Reed-Solomon codes and algebraic-geometric codes that work for all rates and have many applications. However, these list-decoding algorithms are rather complicated. R. Roth and G. Ruckenstein (see ibid., vol.46, p.246-57, Jan. 2000) proposed an efficient implementation of the list decoding of Reed-Solomon codes. In this correspondence, extending Roth and Ruckenstein's fast algorithm for finding roots of univariate polynomials over polynomial rings, i.e., the reconstruct algorithm, we present an efficient algorithm for finding the roots of univariate polynomials over function fields. Based on the extended algorithm, we give an efficient list-decoding algorithm for algebraic-geometric codes.