On the rank weight hierarchy of $M$-codes
G. Berhuy, J. Molina
TL;DR
The paper addresses the rank-weight hierarchy of linear codes stable under a base-field endomorphism defined by $M$, with a focus on $M$-cyclic codes when $M$ is cyclic. It develops a generator-polynomial framework tied to the minimal polynomial of $M$, derives general upper bounds for the rank-weight hierarchy via decompositions into cyclic subspaces, and provides exact formulas for the first and last generalized rank weights in the $M$-cyclic setting. A key contribution is a necessary condition for the existence of nontrivial MRD $M$-codes, plus a probabilistic counting result for $M$-cyclic codes having first rank weight equal to $1$, and explicit expressions for the last rank distance of $M$-cyclic codes and their duals. Overall, the work unifies cyclic, quasi-cyclic, and polynomial codes under the $M$-code framework and offers tools for rank-metric code design with potential network-coding and cryptographic applications.
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.