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.