List Decodability of Symbol-Pair Codes

TL;DR

This paper studies the list decodability of symbol-pair codes, establishing bounds and algorithms that improve understanding of their decoding capabilities beyond traditional limits.

Contribution

It proves that list decodability cannot surpass the Gilbert-Varshamov bound for all symbol-pair codes, but random codes can achieve it with high probability, and introduces a list decoding algorithm for Reed-Solomon codes beyond the Johnson bound.

Findings

List decodability of symbol-pair codes is limited by the Gilbert-Varshamov bound.

Random symbol-pair codes can be list decoded up to the Gilbert-Varshamov bound with high probability.

A new list decoding algorithm for Reed-Solomon codes surpassing the Johnson bound.

Abstract

We investigate the list decodability of symbol-pair codes in the present paper. Firstly, we show that list decodability of every symbol-pair code does not exceed the Gilbert-Varshamov bound. On the other hand, we are able to prove that with high probability, a random symbol-pair code can be list decoded up to the Gilbert-Varshamov bound. Our second result of this paper is to derive the Johnson-type bound, i.e., a lower bound on list decoding radius in terms of minimum distance. Finally, we present a list decoding algorithm of Reed-Solomon codes beyond the Johnson-type bound.