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 ((x_i·y_i))_i=1 ⁿ, x_i, y_i∈F, and a degree parameter k and error parameter e, find all univariate polynomials p of degree at most k such that y_i=p(x_i) for all but at most e values of i∈(1,...,n). We give an algorithm that solves this problem for e<n-√(kn), which improves over the previous best result, for every choice of k and n. Of particular interest is the case of k/n>1/3, where the result yields the first asymptotic improvement in four decades. 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-√(n(n-d´)) errors, where n is the block length and d´ is the designed distance of the code. The improvement for the case of algebraic-geometry codes extends the methods of Shokrollahi and Wasserman (see in Proc. 29th Annu. ACM Symp. Theory of Computing, p.241-48, 1998) and improves upon their bound for every choice of n and d´. 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