Computing Error Distance of Reed-Solomon Codes

Under polynomial time reduction, the maximum likelihood decoding of a linear code is equivalent to computing the error distance of a received word. It is known that the decoding complexity of standard Reed-Solomon codes at certain radius is at least as hard as the discrete logarithm problem over certain large finite fields. This implies that computing the error distance is hard for standard Reed-Solomon codes. Using the Weil bound and a new sieve for distinct coordinates counting, we are able to compute the error distance for a large class of received words. This significantly improves previous results in this direction. As a corollary, we also improve the existing results on the Cheng-Murray conjecture about the complete classification of deep holes for standard Reed-Solomon codes.