Improved upper bounds for partial spreads

TL;DR

This paper advances the understanding of partial spreads in finite projective spaces by providing new upper bounds, especially resolving the case where the dimension parameter satisfies a specific congruence condition for binary fields.

Contribution

It completely determines the maximum size of partial spreads in the case where n ≡ 2 mod k for q=2, filling a notable gap in the existing theory.

Findings

Resolved the case n ≡ 2 mod k for q=2.

Provided improved upper bounds for partial spreads.

Enhanced understanding of partial spreads in finite projective spaces.

Abstract

A partial $(k-1)$-spread in $\operatorname{PG}(n-1,q)$ is a collection of $(k-1)$-dimensional subspaces with trivial intersection, i.e., each point is covered at most once. So far the maximum size of a partial $(k-1)$-spread in $\operatorname{PG}(n-1,q)$ was known for the cases $n\equiv 0\pmod k$, $n\equiv 1\pmod k$ and $n\equiv 2\pmod k$ with the additional requirements $q=2$ and $k=3$. We completely resolve the case $n\equiv 2\pmod k$ for the binary case $q=2$.