Evasive subspaces

TL;DR

This paper explores the properties and maximum sizes of evasive subspaces in finite vector spaces, introduces duality relations, and provides new constructions, including the first examples of maximum scattered subspaces for specific parameters.

Contribution

It introduces the concept of $(h,k)_q$-evasive subspaces, studies their duality, and constructs new maximum scattered subspaces for various parameters and characteristics.

Findings

Maximum size of $(h,k)_q$-evasive subspaces determined.

Duality relations among evasive subspaces established.

First examples of maximum scattered subspaces for certain parameters provided.

Abstract

Let $V$ denote an $r$-dimensional vector space over $\mathbb{F}_{q^n}$, the finite field of $q^n$ elements. Then $V$ is also an $rn$-dimension vector space over $\mathbb{F}_q$. An $\mathbb{F}_q$-subspace $U$ of $V$ is $(h,k)_q$-evasive if it meets the $h$-dimensional $\mathbb{F}_{q^n}$-subspaces of $V$ in $\mathbb{F}_q$-subspaces of dimension at most $k$. The $(1,1)_q$-evasive subspaces are known as scattered and they have been intensively studied in finite geometry, their maximum size has been proved to be $\lfloor rn/2 \rfloor$ when $rn$ is even or $n=3$. We investigate the maximum size of $(h,k)_q$-evasive subspaces, study two duality relations among them and provide various constructions. In particular, we present the first examples, for infinitely many values of $q$, of maximum scattered subspaces when $r=3$ and $n=5$. We obtain these examples in characteristics $2$, $3$ and $5$.