Critical Pairs for the Product Singleton Bound.

We characterize product-maximum distance separable (PMDS) pairs of linear codes, i.e., pairs of codes $C$ and $D$ whose product under coordinatewise multiplication has maximum possible minimum distance as a function of the code length and the dimensions $\dim C$ and $\dim D$ . 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 PMDS 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. [ABSTRACT FROM AUTHOR]