Generalising the Scattered Property of Subspaces

Let V be an r -dimensional $${\mathbb{F}_{{q^n}}}$$ F q n -vector space. We call an $${\mathbb{F}_{q}}$$ F q -subspace U of V h -scattered if U meets the h -dimensional $${\mathbb{F}_{{q^n}}}$$ F q n -subspaces of V in $${\mathbb{F}_{q}}$$ F q -subspaces of dimension at most h . In 2000 Blokhuis and Lavrauw proved that $${\dim_{\mathbb{F}_{q}}}$$ dim F q U ≤ rn /2 when U is 1-scattered. Sub-spaces 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.