On the one-dimensional extensions of $q$-matroids
Koji Imamura, Shinya Kawabuchi, Keisuke Shiromoto
TL;DR
This work develops a rigorous $q$-analogue of single-element matroid extensions by introducing one-dimensional extensions for $q$-matroids and a corresponding modular cut/selector framework. It proves a bijection between one-dimensional extensions and modular cut selectors and defines canonical representatives to enable enumeration of non-isomorphic $q$-matroids without exhaustive isomorphism testing. An explicit classification algorithm is proposed and implemented, yielding enumerations for ground fields $\,\mathbb{F}_2$ and $\,\mathbb{F}_3$ up to specified dimensions and highlighting connections to the $q$-Fano plane. The results advance systematic enumeration and classification of $q$-matroids and illuminate their connections to $q$-Steiner systems and potential $q$-Fano plane structures.
Abstract
In this paper we introduce a $q$-analogue of the single-element extensions of matroids for $q$-matroids, which we call one-dimensional extensions. To enumerate such extensions, we define a $q$-analogue of modular cuts and define a certain function which we call a modular cut selector. It assigns each newly appearing one-dimensional subspace to a modular cut. By using these notion, we prove the one-to-one correspondence between the one-dimensional extensions and the modular cut selectors. Furthermore, we define the canonnical representatives of the isomorphic class of the $q$-matroids, which enable us to enumerate non-isomorphic $q$-matroids without the paiwise isomorphism testing. As an application, we develop a classification algorithm for $q$-matroids, and classify all the $q$-matroids on ground spaces over $\mathbb{F}_2$ and $\mathbb{F}_3$ of dimension $4$ and $5$ respectively. We also determine some $5$-dimensional $q$-matroids related to the $q$-Fano plane, which is the $q$-analogue of the Fano plane, over $\mathbb{F}_2$.