Zeta Functions for Tensor Codes

TL;DR

This paper introduces a new class of optimal tensor codes called $j$-tensor maximum rank distance codes, explores their zeta functions, and establishes connections with weight enumerators and invariants, advancing tensor code theory.

Contribution

It defines and studies the generalized zeta function for tensor codes, extending the family of $j$-maximum rank distance codes and deriving new MacWilliams identities.

Findings

Introduces $j$-tensor maximum rank distance codes.

Establishes the relation between zeta functions and weight enumerators.

Derives new connections between tensor weights and anticodes.

Abstract

In this work we introduce a new class of optimal tensor codes related to the Ravagnani-type anticodes, namely the $j$-tensor maximum rank distance codes. We show that it extends the family of $j$-maximum rank distance codes and contains the $j$-tensor binomial moment determined codes (with respect to the Ravagnani-type anticodes) as a proper subclass. We define and study the generalized zeta function for tensor codes. We establish connections between this object and the weight enumerator of a 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 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 generalized tensor weights related to the Delsarte and Ravagnani-type anticodes.