# Rank-Metric Codes and Zeta Functions

## TL;DR

This paper introduces a new zeta function for rank-metric codes, linking it to weight enumerators, functional equations, and code invariants, thus extending the analogy with classical Hamming code theory.

## Contribution

It defines the rank-metric zeta function, establishes its properties, and explores its implications for code invariants and zero distribution, providing new tools for rank-metric code analysis.

## Key findings

- Zeta function generates weight enumerators for rank-metric codes.
- Functional equation relates the zeta function to code parameters.
- Upper bound on minimum distance derived from roots of the zeta function.

## Abstract

We define the rank-metric zeta function of a code as a generating function of its normalized $q$-binomial moments. We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank-metric codes. We further prove a functional equation and derive an upper bound for the minimum distance in terms of the reciprocal roots of the zeta function. Finally, we show invariance under suitable puncturing and shortening operators and study the distribution of zeroes of the zeta function for a family of codes.