Most $q$-matroids are not representable

TL;DR

This paper proves that, similar to classical matroids, most $q$-matroids are not representable, highlighting a fundamental difference in their structure and implications for coding theory.

Contribution

The paper establishes a $q$-analogue of Nelson's theorem, showing that asymptotically almost all $q$-matroids are non-representable, answering a key open question.

Findings

Most $q$-matroids are not representable

Asymptotic non-representability of $q$-matroids

Answers negatively to Jurrius and Pellikaan's question

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.