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 Fq^n-subspaces of V in Fq-subspaces of dimension at most h. In 2000 Blokhuis and Lavrauw proved that dim 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.