Séminaire Lotharingien de Combinatoire, 93B.135 (2025), 12 pp.
Sebastian Degen and Lukas Kühne
Most q-Matroids are Not Representable
Abstract.
A q-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 q-matroids are motivated by coding theory as the representable q-matroids are the ones that stem from rank-metric codes. In this note, we establish a q-analogue of Nelson's theorem in matroid theory by proving that asymptotically almost all q-matroids are not representable. This answers a question about representable q-matroids by Jurrius and Pellikaan strongly in the negative.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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