On defining generalized rank weights

TL;DR

This paper explores generalized rank weights for rank metric codes over Galois extensions, establishing equivalence with prior definitions and discussing degenerate codes, thus broadening the theoretical framework of rank metric coding.

Contribution

It introduces a unified definition of generalized rank weights over arbitrary characteristic fields and proves its equivalence to existing definitions.

Findings

Established equivalence with previous definitions of generalized rank weights.

Extended the concept to codes over arbitrary characteristic fields.

Discussed properties of degenerate codes with respect to the rank metric.

Abstract

This paper investigates the generalized rank weights, with a definition implied by the study of the generalized rank weight enumerator. We study rank metric codes over $L$, where $L$ is a finite Galois extension of a field $K$. This is a generalization of the case where $K = \mathbb{F}_q$ and $L = \mathbb{F}_{q^m}$ of Gabidulin codes to arbitrary characteristic. We show equivalence to previous definitions, in particular the ones by Kurihara-Matsumoto-Uyematsu, Oggier-Sboui and Ducoat. As an application of the notion of generalized rank weights, we discuss codes that are degenerate with respect to the rank metric.