q-Polymatroids associated with restricted rank-metric codes

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.

💡 Research Summary

This paper investigates q‑polymatroids that can be represented by linear restricted rank‑metric codes, i.e., subspaces of the space of alternating, symmetric, or Hermitian n × n matrices equipped with the rank metric. After recalling the basic notions of rank‑metric codes, Delsarte duality, and the Singleton‑like bound, the authors focus on the three families of restricted matrix spaces, denoted Altₙ,q, Symₙ,q, and Herₙ,q. Each of these spaces can be identified with a subspace of the algebra of q‑linearized polynomials 𝔽_{qⁿ}