# Generalising the scattered property of subspaces

## TL;DR

This paper extends the theory of scattered subspaces in finite vector spaces by establishing a new upper bound for their dimension when h>1, and explores their properties, duality, and applications.

## Contribution

It generalizes the known bound for 1-scattered subspaces to h-scattered subspaces and provides constructions, duality relations, and equivalence analysis.

## Key findings

- Established the upper bound rn/(h+1) for h-scattered subspaces.
- Constructed examples achieving the new bound.
- Analyzed intersection properties and duality of these subspaces.

## Abstract

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 $\dim_{\mathbb{F}_q} U \leq rn/2$ 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.