Abstract

AbstractLet denote an ‐dimensional vector space over , the finite field of elements. Then is also an ‐dimension vector space over . An ‐subspace of is ‐evasive if it meets the ‐dimensional ‐subspaces of in ‐subspaces of dimension at most . The ‐evasive subspaces are known as scattered and they have been intensively studied in finite geometry, their maximum size has been proved to be when is even or . We investigate the maximum size of ‐evasive subspaces, study two duality relations among them and provide various constructions. In particular, we present the first examples, for infinitely many values of , of maximum scattered subspaces when and . We obtain these examples in characteristics 2, 3 and 5.

Highlights

  • Let F be a set of subsets of a set A and let S ⊆ A

  • In their work A was taken to be the set of vectors of an r-dimensional vector space V over F2, the finite field of two elements, and F was the set of all d-dimensional affine subspaces of V for some positive integer d

  • The aim of this paper is to study these evasive sets

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Summary

Introduction

Let F be a set of subsets of a set A and let S ⊆ A. Uof V is called scattered with respect to a spread S if U meets each element of S in at most a one-dimensional Fq-subspace, i.e. when U is q-evasive for S. In [10] the authors generalized the Blokhuis–Lavrauw bound and proved that the dimension of an h-scattered subspace is at most rn/(h + 1) They introduced a relation, called Delsarte duality, on Fq-subspaces of V and proved that the Delsarte dual of an rn/(h + 1)dimensional h-scattered subspace is h -scattered with dimension r n/(h + 1) in some vector space V (r , qn).

General properties and existence results
Dualities on evasive subspaces
Delsarte dual of evasive subspaces
On the size of maximum evasive subspaces
Findings
Result

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