Zeta functions for tensor codes

Abstract

In this work, we introduce a new class of optimal tensor codes related to the Ravagnani-type anticodes defined in 2023. We show that it extends the family of [Formula: see text]-maximum rank distance codes and contains the [Formula: see text]-binomial moment determined codes (with respect to the Ravagnani-type anticodes) as a proper subclass. We define and study the zeta function for tensor codes. We establish connections between this object and the weight enumerator of a tensor code with respect to the Ravagnani-type anticodes. We introduce a new refinement of the invariants of tensor codes exploiting the structure of product lattices of some classes of anticodes classified in 2023 and we derive the corresponding MacWilliams identities. In this framework, we also define a multivariate version of the tensor weight enumerator and we establish relations with the corresponding zeta function. As an application, we derive connections on the tensor weights.