Efficient decoding of Reed-Solomon codes beyond half the minimum distance

A list decoding algorithm is presented for [n,k] Reed-Solomon (RS) codes over GF(q), which is capable of correcting more than [(n-k)/2] errors. Based on a previous work of Sudan (see J. Compl., vol.13, p.180-93, 1997), an extended key equation (EKE) is derived for RS codes, which reduces to the classical key equation when the number of errors is limited to [(n-k)/2]. Generalizing Massey´s (1969) algorithm that finds the shortest recurrence that generates a given sequence, an algorithm is obtained for solving the EKE in time complexity O(l·(n-k)² ), where l is a design parameter, typically a small constant, which s an upper bound on the size of the list of decoded codewords. (The case l=1 corresponds to classical decoding of up to [(n-k)/2] errors where the decoding ends with at most one codeword.) This improves on the time complexity O(n³) needed for solving the equations of Sudan´s algorithm by a naive Gaussian elimination. The polynomials found by solving the EKE are then used for reconstructing the codewords in time complexity O((llog²l)k(n+llogq)) using root-finders of degree-l univariate polynomials