Most \(q\)-matroids are not representable

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