Generalising the Scattered Property of Subspaces


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 \({\dim_{\mathbb{F}_{q}}}\) Urn/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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Corresponding author

Correspondence to Olga Polverino.

Additional information

The research was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA — INdAM). The first author was partially supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the National Research, Development and Innovation Office — NKFIH under the grants PD 132463 and K 124950. The last two authors were supported by the project “VALERE: VAnviteLli pEr la RicErca” of the University of Campania “Luigi Vanvitelli”.

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Csajbók, B., Marino, G., Polverino, O. et al. Generalising the Scattered Property of Subspaces. Combinatorica 41, 237–262 (2021).

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Mathematics Subject Classification (2010)

  • 05B25
  • 51E20
  • 33C20