Complexity of Decoding Positive-Rate Reed-Solomon Codes

Abstract

Abstract-It has been proved that the maximum likelihood decoding problem of Reed-Solomon codes is NP-hard. However, the length of the code in the proof is at most polylogarithmic in the size of the alphabet. For the complexity of maximum likelihood decoding of the primitive Reed-Solomon code, whose length is one less than the size of alphabet, the only known result states that it is at least as hard as the discrete logarithm in some cases where the information rate unfortunately goes to zero. In this paper, it is proved under a well known cryptography hardness assumption that 1) There does not exist a randomized polynomial time maximum likelihood decoder for the Reed-Solomon code family2) There does not exist a randomized polynomial time bounded-distance decoder for primitive Reed-Solomon codes at distance 2 3 + of the minimum distance for any constant 0 < < 1 3 . In particular, this rules out the possibility of a polynomial time algorithm for maximum likelihood decoding problem of primitive Reed-Solomon codes of any rate under the assumption.

Computing Error Distance of Reed-Solomon Codes

“…It would be very interesting to see if the square root condition d < c √ q in our main theorem can be improved to a linear condition d < cq for some positive constant c. A similar problem in different contexts occurs in [5] and [15].…”

On Reed-Solomon codes

“…For the standard Reed-Solomon code C, the complexity of decoding is unknown and much more subtle. It was shown in [2,4] to be at least as hard as the discrete logarithm in a large extension F q h , where h can be as large as √ q. If deg u(x) = k, then u is a deep hole.…”

A new sieve for distinct coordinate counting

“…In the special case when q is a square, this type of theorem was first proved in [4,5] by using the Brun sieve, Weil's bound and a dual argument. The general q case was raised as an open problem.…”

“…We are interested in when N m (k, b) > 0 and when there is a good asymptotic formula. This problem arises from several applications in coding theory [1,3,4,10]. In graph theory, it reduces to the study of the girth of Chung's graph [6], which has been studied extensively in the literature.…”

“…This is something that cannot be avoided if m is very close to k − 2. In coding theory applications [4,5], one would like to have the situation that q is linear in k (corresponding to the condition that the information rate is positive) and m can be somewhat smaller but not too much smaller. In this case, k is necessarily large and we can try to apply our new sieving formula.…”