On the rank weight hierarchy of $M$-codes

TL;DR

This paper investigates the rank weight hierarchy of linear codes stable under specific linear endomorphisms, providing conditions for rank weights and explicit formulas, especially for cyclic endomorphisms.

Contribution

It introduces new criteria for the first rank weight and formulas for the last rank weight of codes stable under cyclic endomorphisms.

Findings

Necessary and sufficient condition for first rank weight to be 1

Explicit formula for the last rank weight

Characterization of codes stable under cyclic endomorphisms

Abstract

We study the rank weight hierarchy of linear codes which are stable under a linear endomorphism defined over the base field, in particular when the endomorphism is cyclic. In this last case, we give a necessary and sufficient condition for such a code to have first rank weight equal to $1$ in terms of its generator polynomial, as well as an explicit formula for its last rank weight.