q-Polymatroids associated with restricted rank-metric codes

TL;DR

This paper investigates q-polymatroids derived from restricted rank-metric codes, characterizing their rank functions and highlighting differences from unrestricted matrix code polymatroids.

Contribution

It provides a characterization of the rank functions of q-polymatroids associated with restricted rank-metric codes and identifies conditions for their determination.

Findings

Characterization of rank functions for restricted rank-metric code polymatroids

Identification of conditions for full determination of these rank functions

Differences observed compared to unrestricted matrix code q-polymatroids

Abstract

In this article, we study polymatroids that are representable by means of linear restricted rank-metric codes, namely, by subspaces of the space of alternating, symmetric, or Hermitian square matrices endowed with the rank metric. More precisely, we characterize the rank function defining these polymatroids and establish sufficient conditions on the relevant parameters under which it is fully determined. We show that there are several differences in compared to the behaviour of $q$-polymatroids of unrestricted matrix codes.