Generalising the scattered property of subspaces

Csajbók, Bence and Marino, Giuseppe and Polverino, Olga and Zullo, Ferdinando (2019) Generalising the scattered property of subspaces. COMBINATORICA. ISSN 0209-9683 (In Press)

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Abstract

Let V be an r-dimensional GF(q^n)-vector space. We call a GF(q)-subspace U of V h-scattered if U meets the h-dimensional GF(q^n)-subspaces of V in GF(q)-subspaces of dimension at most h. In 2000 Blokhuis and Lavrauw proved that dim(U) over GF(q) is at most 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.

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