Improved list decoding of generalized Reed-Solomon and alternant codes over Galois rings

Abstract

We present a two-stage list decoder comprising an errors-only Guruswami-Sudan (GS) decoder and an errors-and-erasures GS decoder as component decoders in the first and second stage, respectively. The two stages are coupled via a post-processor which selects a codeword from the output list of the first component decoder, from which erasure locations are obtained for the second stage. When applied to a generalized Reed-Solomon (RS) code over a Galois ring R that maps into a generalized RS code of the same length n and minimum (Hamming) distance d over the corresponding residue field, the proposed decoder exploits the presence of zero divisors in R to correct s errors where w = ⌈ n - √n(n - d) - 1 ⌉ < s ≤ ⌈ n - √/(n - w)(n - d) - 1 ⌉ with a probability determined by s, w, and the ratio of the number of non-trivial zero divisors to the number of units in the code alphabet. Focusing primarily on alternant codes over ℤ2l, animportant class of subring subcodes of generalized RS codes over GR (2l, a), we demonstrate that the GS decoding radius w can be exceeded by a substantial margin with significant probability. © 2005 IEEE.