Combinatorial list-decoding of Reed-Solomon codes beyond the Johnson radius

TL;DR

This paper advances the understanding of Reed-Solomon code list-decoding beyond the Johnson radius by establishing a combinatorial framework, proving bounds, and providing explicit constructions for such codes.

Contribution

It introduces a combinatorial approach to determine the relation between decoding radius and list size, proving a generalized Singleton bound and constructing explicit codes beyond the Johnson radius.

Findings

Most RS codes over large fields are list-decodable beyond the Johnson radius.

The paper proves the conjecture for list sizes 2 and 3.

Most RS codes with rate at least 1/9 can be list-decoded beyond the Johnson radius.

Abstract

List-decoding of Reed-Solomon (RS) codes beyond the so called Johnson radius has been one of the main open questions since the work of Guruswami and Sudan. It is now known by the work of Rudra and Wootters, using techniques from high dimensional probability, that over large enough alphabets most RS codes are indeed list-decodable beyond this radius. In this paper we take a more combinatorial approach which allows us to determine the precise relation (up to the exact constant) between the decoding radius and the list size. We prove a generalized Singleton bound for a given list size, and conjecture that the bound is tight for most RS codes over large enough finite fields. We also show that the conjecture holds true for list sizes $2 \text{ and }3$, 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. Lastly, we give the first explicit construction 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 the notion of cycle space from Graph Theory. Both of them have not been used before in the context of list-decoding.