Critical pairs for the Product Singleton Bound

Diego Mirandola (1, 2, 3) , Gilles Zémor (1)

1 IMB - Institut de Mathématiques de Bordeaux
2 CWI - Centrum voor Wiskunde en Informatica
3 Universiteit Leiden = Leiden University

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.

Domaines: Théorie de l'information et codage [math.IT]

Diego Mirandola, Gilles Zémor. Critical pairs for the Product Singleton Bound. IEEE Transactions on Information Theory, 2015, 61 (9), pp.4928-4937. ⟨10.1109/TIT.2015.2450207⟩. ⟨hal-01261399⟩