On the one-dimensional extensions of $q$-matroids

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 ${\mathbb{F}}_2$.