Work

Most \(q\)-matroids are not representable

Year: 2026

Type: article

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. Mathematics Subject Classifications: 05B35, 05A30 Keywords: \(q\)-matroids, representability, rank-metric codes

Source: Combinatorial Theory

DOI: 10.5070/c6.68695

Authors: Sebastian Degen, Lukas Kühne

Language: English

Topic: Advanced Combinatorial Mathematics

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: Deutsche Forschungsgemeinschaft

Awards: Simplizialität in Arrangements und Matroiden, G5893294622, G6601682123, TRR 358: Ganzzahlige Strukturen in Geometrie und Darstellungstheorie, G8667429248

OpenAlex ID: W4402427161

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