Short rank-metric codes and scattered subspaces

Abstract: By exploiting the connection between scattered $\mathbb{F}q$-subspaces of $\mathbb{F}{q^m}^3$ and minimal non degenerate $3$-dimensional rank metric codes of $\mathbb{F}{q^m}^{n}$, $n \geq m+2$, described in [2], we will exhibit a new class of codes with parameters $[m+2,3,m-2]{q^m/q}$ for infinite values of $q$ and $m \geq 5$ odd. Moreover, by studying the geometric structures of these scattered subspaces, we determine the rank weight distribution of the associated codes.