q-Polymatroids associated with restricted rank-metric codes
Eimear Byrne, Giovanni Longobardi, and Rocco Trombetti
TL;DR
This work studies q-polymatroids representable by restricted rank-metric codes, focusing on alternating, symmetric, and Hermitian subspaces and establishing how their rank functions can be characterized. By leveraging linearized polynomial representations, dualities, and shortening/puncturing, the authors derive explicit rank-function descriptions for the associated q-polymatroids and identify parameter regimes where these functions are fully determined. A key finding is that, unlike the unrestricted case, the column and row q-polymatroids coincide for alternating and symmetric codes but generally differ for Hermitian codes, with several extremal code families having fully determined or tightly constrained rank functions. The results illuminate invariant-type properties of restricted rank-metric codes, clarify duality relations in restricted settings, and provide a basis for comparing restricted and unrestricted polymatroid behaviour with potential applications to code invariants and structure.
Abstract
In this article, we study polymatroids that are representable by means of linear {\it 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.