Generalized Singleton Bound and List-Decoding Reed–Solomon Codes Beyond the Johnson Radius

Abstract

In this paper we take a combinatorial approach to the problem of list-decoding, which allows us to determine the precise relation (up to the exact constant) between the decoding radius, list size, and code rate. We prove a generalized Singleton bound for a given list size, and conjecture that the bound is tight for most Reed–Solomon (RS) codes over large enough finite fields. We also show that the conjecture holds true for list sizes 2 and 3, and as a by product show that most RS codes with a rate of at least 1/9 are list-decodable beyond the Johnson radius. Last, we give the first explicit construction in the literature of such RS codes. The main tools used in the proof are a new type of linear dependency between codewords of a code that are contained in a small Hamming ball, and a surprising connection between list-decoding and the notion of cycle space in graph theory. Both of them are new, and may be of independent interest.