Generalising the scattered property of subspaces

Abstract

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 $h$. In 2000 Blokhuis and Lavrauw proved that ${\dim }_{{\mathbb{F}}_q}U\le rn/2$ 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.

We implemented the referees comments and changed the title. To appear in Combinatorica

References in corpus (3)

• On kernels and nuclei of rank metric codes
• MRD codes with maximum idealizers
• A Carlitz type result for linearized polynomials