Improved decoding of Reed-Solomon and algebraic-geometry codes

Given an error-correcting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding Reed-Solomon codes. The list decoding problem for Reed-Solomon codes reduces to the following “curve-fitting” problem over a field F : Given n points f(xi:yi)gni=1, xi; yi 2 F , and a degree parameter k and error parameter e, find all univariate polynomials p of degree at most k such that yi = p(xi) for all but at most e values of i 2 f1; : : : ; ng. We give an algorithm that solves this problem for e < n pkn, which improves over the previous best result [27], for every choice of k and n. Of particular interest is the case of k=n > 13 , where the result yields the first asymptotic improvement in four decades [21]. The algorithm generalizes to solve the list decoding problem for other algebraic codes, specifically alternant codes (a class of codes including BCH codes) and algebraic-geometry codes. In both cases, we obtain a list decoding algorithm that corrects up to n pn(n d0) errors, where n is the block length and d0 is the designed distance of the code. The improvement for the case of algebraic-geometry codes extends the methods of [24] and improves upon their bound for every choice of n and d0. We also present some other consequences of our algorithm including a solution to a weighted curve fitting problem, which may be of use in soft-decision decoding algorithms for Reed-Solomon codes.