On the existence of a (2,3)-spread in V(7,2)

TL;DR

This paper investigates the existence of (2,3)-spreads in V(7,2), introducing the concept of α-points and proving that every 6-dimensional subspace contains a point not an α-point, thus advancing understanding of spread structures.

Contribution

The paper introduces the concept of α-points for (2,3)-spreads in V(7,2) and proves a new property about the distribution of these points within subspaces.

Findings

Not all points in V(7,2) are α-points to a (2,3)-spread.

Every 6-dimensional subspace contains at least one point that is not an α-point.

The results strengthen previous theorems about the structure of spreads in finite vector spaces.

Abstract

An $(s,t)$-spread in a finite vector space $V=V(n,q)$ is a collection $\mathcal F$ of $t$-dimensional subspaces of $V$ with the property that every $s$-dimensional subspace of $V$ is contained in exactly one member of $\mathcal F$. It is remarkable that no $(s,t)$-spreads has been found yet, except in the case $s=1$. In this note, the concept $\alpha$-point to a $(2,3)$-spread $\mathcal F$ in {$V=V(7,2)$} is introduced. A classical result of Thomas, applied to the vector space $V$, states that all points of $V$ cannot be $\alpha$-points to a given $(2,3)$-spread $\mathcal F$ in $V$. {In this note, we strengthened this result by proving that} every 6-dimensional subspace of $V$ must contain at least one point that is not an $\alpha$-point to a given $(2,3)$-spread of $V$.