Complexity of Decoding Positive-Rate Reed-Solomon Codes

TL;DR

This paper proves that decoding Reed-Solomon codes with positive rate is as hard as discrete logarithm, resolving an open problem and implying no polynomial-time decoding algorithm exists under cryptographic assumptions.

Contribution

It removes the rate restriction in the complexity of Reed-Solomon decoding and provides explicit constructions of Hamming balls with many codewords for any positive rate.

Findings

Decoding complexity matches discrete logarithm hardness for all positive rates.

Constructs Hamming balls with exponentially many codewords for any positive rate.

Provides subexponentially many codewords in Hamming balls for rates approaching one.

Abstract

The complexity of maximal likelihood decoding of the Reed-Solomon codes $[q-1, k]_q$ is a well known open problem. The only known result in this direction 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, we remove the rate restriction and prove that the same complexity result holds for any positive information rate. In particular, this resolves an open problem left in [4], and rules out the possibility of a polynomial time algorithm for maximal likelihood decoding problem of Reed-Solomon codes of any rate under a well known cryptographical hardness assumption. As a side result, we give an explicit construction of Hamming balls of radius bounded away from the minimum distance, which contain exponentially many codewords for Reed-Solomon code of any positive rate less than one. The previous constructions only apply to Reed-Solomon codes of diminishing rates. We also give an explicit construction of Hamming balls of relative radius less than 1 which contain subexponentially many codewords for Reed-Solomon code of rate approaching one.