On the List and Bounded Distance Decodability of Reed-Solomon Codes

For an error-correcting code and a distance bound, the list decoding problem is to compute all the codewords within a given distance to a received message. The bounded distance decoding problem is to find one codeword if there is at least one codeword within the given distance, or to output the empty set if there is not. Obviously the bounded distance decoding problem is not as hard as the list decoding problem. For a Reed-Solomon code [n, k]q, a simple counting argument shows that for any integer 0 < g < n, there exists at least one Hamming ball of radius n−g, which contains at least ( n g ) /qg−k many codewords. Let ĝ(n, k, q) be the smallest positive integer g such that ( n g ) /qg−k ≤ 1. One knows that k − 1 ≤ ĝ(n, k, q) ≤ √ n(k − 1) ≤ n. For the distance bound up to n− √ n(k − 1), it is known that both the list and bounded distance decoding can be solved efficiently. For the distance bound between n − √ n(k − 1) and n − ĝ(n, k, q), we do not know whether the Reed-Solomon code is list, or bounded distance decodable, nor do we know whether there are polynomially many codewords in all balls of the radius. It is generally believed that the answers to both questions are no. In this paper, we prove: (1) List decoding can not be done for radius n− ĝ(n, k, q) or larger, unless the discrete logarithm over Fqĝ(n,k,q)−k is easy. ∗This work is partially supported by NSF Career Award CCR-0237845. †Partially supported by NSF. (2) Let h and g be positive integers satisfying q ≥ max(g, (h− 1) ) and g ≥ ( 4 +2)(h+1) for a constant > 0. We show that the discrete logarithm problem over Fqh can be efficiently reduced by a randomized algorithm to the bounded distance decoding problem of the Reed-Solomon code [q, g − h]q with radius q − g. These results show that the decoding problems for the Reed-Solomon code are at least as hard as the discrete logarithm problem over certain finite fields. For the list decoding problem of Reed-Solomon codes, although the infeasible radius that we obtain is much larger than the radius which is known to be feasible, it is the first non-trivial bound. Our result on the bounded distance decodability of Reed-Solomon codes is also the first of its kind. The main tools to obtain these results are an interesting connection between the problem of list-decoding of Reed-Solomon code and the problem of discrete logarithm over finite fields, and a generalization of Katz’s theorem on representations of elements in an extension finite field by products of distinct linear factors.