Divisible linear rank metric codes

TL;DR

This paper investigates whether all matrix subspaces with ranks divisible by a certain number originate from larger field matrices, providing conditions for when this is true and presenting counterexamples.

Contribution

It characterizes when divisible rank metric codes can be derived from larger fields and constructs counterexamples where this is not possible.

Findings

Identifies cases where divisible codes originate from larger fields

Constructs counterexamples of divisible codes not arising from larger fields

Provides criteria for the existence of such codes

Abstract

A subspace of matrices over $\mathbb{F}_{q^e}^{m\times n}$ can be naturally embedded as a subspace of matrices in $\mathbb{F}_q^{em\times en}$ with the property that the rank of any of its matrix is a multiple of $e$. It is quite natural to ask whether or not all subspaces of matrices with such a property arise from a subspace of matrices over a larger field. In this paper we explore this question, which corresponds to studying divisible codes in the rank metric. We determine some cases for which this question holds true, and describe counterexamples by constructing subspaces with this property which do not arise from a subspace of matrices over a larger field.