Representability of $q$-matroids via rank-metric codes

TL;DR

This paper introduces a $q$-analogue of multilinear representability for $q$-matroids, exploring conditions and limitations for such representations and classifying multilinearity in small cases.

Contribution

It defines $m$-multilinear representability for $q$-matroids, establishes non-existence results for certain classes, and classifies multilinearity for small $q$-matroids.

Findings

Nontrivial uniform $q$-matroids admit no purely multilinear representations.

The non-Pappus $q$-matroid, if multilinearly representable, must have block size at least 9.

No rank-2 $q$-matroid on $ ext{F}_2^4$ admits a purely $m$-multilinear representation for $1<m<4$.

Abstract

Multilinear representability extends classical linear representability of matroids by assigning subspaces, rather than vectors, to ground elements. This notion is closely related to almost affine codes. In this paper, we introduce and study a $q$-analogue of multilinear representability for $q$-matroids, motivated by known connections between $q$-matroids, classical matroids, and rank-metric codes. We define $m$-multilinear representability in terms of almost affine matrix rank-metric codes satisfying a natural divisibility condition. We prove that nontrivial uniform $q$-matroids admit no purely multilinear representations, and we derive necessary conditions for multilinear representations of almost uniform $q$-matroids. We further show that the non-Pappus $q$-matroid, if multilinearly representable, must have block size at least $9$. Finally, we prove that no rank-$2$ $q$-matroid on $\mathbb{F}_2^4$ admits a purely $m$-multilinear representation for $1<m<4$, and we classify pure multilinearity for all $q$-matroids on $\mathbb{F}_2^3$ and $\mathbb{F}_2^4$ in the corresponding ranges. At present, no example is known of a purely multilinear $q$-matroid.