Scattered subspaces and related codes

Abstract: After a seminal paper by Shekeey (2016), a connection between maximum $h$-scattered $\mathbb{F}_q$-subspaces of $V(r,q^n)$ and maximum rank distance (MRD) codes has been established in the extremal cases $h=1$ and $h=r-1$. In this paper, we propose a connection for any $h\in{1,\ldots,r-1}$, extending and unifying all the previously known ones. As a consequence, we obtain examples of non-square MRD codes which are not equivalent to generalized Gabidulin or twisted Gabidulin codes. Up to equivalence, we classify MRD codes having the same parameters as the ones in our connection. Also, we determine the weight distribution of codes related to the geometric counterpart of maximum $h$-scattered subspaces.