New List Decoding Algorithms for Reed-Solomon and BCH Codes

In this paper we formulate the list decoding of (generalized) Reed-Solomon codes as a rational curve-fitting problem, utilizing the polynomials constructed by the Berlekamp-Massey algorithm. We present a novel list decoding algorithm that ldr corrects up to 1-radic1-D errors for (generalized) Reed-Solomon codes, identical to that of the Guruswami-Sudan algorithm which is built upon the Berlekamp-Welch algorithm, where D denote the normalized minimum distance, ldr with appropriate modifications, corrects up to 1/2(1-radic1-2D) errors for binary BCH codes, which is the best known bound under polynomial complexity, ldr exhibits polynomial complexity in nature, in particular, requires O(n⁶(1-radic1-D)⁷) field operations for Reed-Solomon codes in achieving its maximum list error correction capability (n denotes code length), whereas the Guruswami-Sudan algorithm has complexity O(n¹⁰(1-D)⁴).