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Near-uniform q-matroids
Abstract
We study the q-matroids associated with nondegenerate Fqm-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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BibTeX
@misc{longobardi2026nearuniform,
title = {{Near-uniform $q$-matroids}},
author = {Giovanni Longobardi and Rocco Trombetti and Lei Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37830},
eprint = {2609.37830},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37830},
}
RIS
TY - UNPB
TI - Near-uniform q-matroids
AU - Giovanni Longobardi
AU - Rocco Trombetti
AU - Lei Xu
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37830
UR - https://arxiv.org/abs/2609.37830
AB - 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.
ER -