Generalising the scattered property of subspaces

Let V be an r-dimensional Fqⁿ-vector space. We call an Fq-subspace U of V h-scattered if U meets the h-dimensional Fqⁿ-subspaces of V in Fq-subspaces of dimension at most h. In 2000 Blokhuis and Lavrauw proved that Fq U ≤ rn/2 when U is $1$-scattered. Subspaces attaining this bound have been investigated intensively because of their relations with projective two-weight codes and strongly regular graphs. MRD-codes with a maximum idealiser have also been linked to $rn/2$-dimensional $1$-scattered subspaces and to n-dimensional $(r-1)$-scattered subspaces. 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.