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 dimFqU ≤ 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.

Generalising the Scattered Property of Subspaces

Marino G.;Polverino O.
;
Zullo F.
2021

Abstract

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 dimFqU ≤ 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11591/443714
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