Most $q$-matroids are not representable
Combinatorics
2025-11-27 v2
Abstract
A -matroid is the analogue of a matroid which arises by replacing the finite ground set of a matroid with a finite-dimensional vector space over a finite field. These -matroids are motivated by coding theory as the representable -matroids are the ones that stem from rank-metric codes. In this note, we establish a -analogue of Nelson's theorem in matroid theory by proving that asymptotically almost all -matroids are not representable. This answers a question about representable -matroids by Jurrius and Pellikaan strongly in the negative.
Cite
@article{arxiv.2408.06795,
title = {Most $q$-matroids are not representable},
author = {Sebastian Degen and Lukas Kühne},
journal= {arXiv preprint arXiv:2408.06795},
year = {2025}
}
Comments
10 pages, To appear in Combinatorial Theory