A special class of scattered subspaces

Let $V$ be an $r$-dimensional $\mathbb{F}_{q^n}$-vector space. We call an $\mathbb{F}_q$-subspace $U$ of $V$ $h$-scattered if $U$ meets the $h$-dimensional $\mathbb{F}_{q^n}$-subspaces of $V$ in $\mathbb{F}_q$-subspaces of dimension at most $h$. For $h=1$ and $\dim_q U=rn/2$ these subspaces were first studied by Blokhuis and Lavrauw in 2000 and are called maximum scattered. They have been investigated intensively because of their relations with two-weight codes and strongly regular graphs. In the recent years maximum scattered and $n$-dimensional $(r-1)$-scattered $\mathbb{F}_q$-subspaces have been linked also to MRD-codes with a maximum idealiser. In this paper we prove the upper bound $rn/(h+1)$ for the dimension of $h$-scattered subspaces, $h>1$, and construct examples with this dimension. We study their intersection numbers with hyperplanes, introduce a duality relation among them, and study the equivalence problem of the corresponding linear sets.