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arXiv · 2609.37830

Near-uniform $q$-matroids

Giovanni Longobardi·Rocco Trombetti·Lei Xu
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Abstract

We study the $q$-matroids associated with nondegenerate $\F_{q^m}$-linear near-MRD codes, which we call near-uniform $q$-matroids. Using the correspondence between rank-metric codes and their associated $q$-systems, we determine explicitly their rank functions and characterize their cyclic flats. We show that near-uniform $q$-matroids form a class of representable paving $q$-matroids. and prove an upper bound on the number of their cyclic flats, which is shown to be sharp for certain parameters. Finally, when $n>m$, exploiting the rank distribution of near-MRD codes, we obtain an exact count of the nontrivial cyclic flats of the associated $q$-matroids.

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Giovanni Longobardi, Rocco Trombetti, Lei Xu. 2026-09-29. Near-uniform $q$-matroids. https://arxiv.org/abs/2609.37830

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