q-Analogues of Matroids and Codes

In classical combinatorics, matroids generalize the notion of linear independence of vectors over a field. In this talk, we will introduce the concept of ${\mathbb{F}}_{q^m}$-independence of ${\mathbb{F}}_q$-spaces and we show that $q$-matroids generalize this notion. As a consequence, the independent spaces of a representable $q$-matroid will be defined as the ${\mathbb{F}}_{q^m}$-independent subspaces of the $q$-system associated to an ${\mathbb{F}}_{q^m}$-linear rank-metric code. Moreover, we will further investigate the link between codes and matroids.