Critical pairs for the Product Singleton Bound

TL;DR

This paper characterizes pairs of linear codes called Product-MDS pairs, focusing on the case where the pair is (C,C), and identifies conditions under which such codes are generalized Reed-Solomon or direct sums of self-dual codes, with connections to additive combinatorics.

Contribution

It provides a characterization of Product-MDS pairs of codes, especially for (C,C), and establishes coding-theoretic analogues of classical additive combinatorics theorems.

Findings

If (C,C) is Product-MDS and the square of C has minimum distance at least 2, then C is either a generalized Reed-Solomon code or a direct sum of self-dual codes.

The paper proves new characterizations of Product-MDS pairs of codes.

Connections are made between coding theory and additive combinatorics.

Abstract

We characterize Product-MDS pairs of linear codes, i.e.\ pairs of codes $C,D$ whose product under coordinatewise multiplication has maximum possible minimum distance as a function of the code length and the dimensions $\dim C, \dim D$. We prove in particular, for $C=D$, that if the square of the code $C$ has minimum distance at least $2$, and $(C,C)$ is a Product-MDS pair, then either $C$ is a generalized Reed-Solomon code, or $C$ is a direct sum of self-dual codes. In passing we establish coding-theory analogues of classical theorems of additive combinatorics.