Vector space partitions of $\operatorname{GF}(2)^8$

TL;DR

This paper classifies all possible configurations of vector space partitions in the projective space PG(7,2), providing a comprehensive understanding of their structure and types.

Contribution

It determines all feasible types of vector space partitions in PG(7,2), advancing the classification of such partitions in finite projective spaces.

Findings

All possible partition types in PG(7,2) are characterized.

The classification aids in understanding the structure of vector space partitions.

Results contribute to the broader theory of finite projective space partitions.

Abstract

A vector space partition $\mathcal{P}$ of the projective space $\operatorname{PG}(v-1,q)$ is a set of subspaces in $\operatorname{PG}(v-1,q)$ which partitions the set of points. We say that a vector space partition $\mathcal{P}$ has type $(v-1)^{m_{v-1}} \dots 2^{m_2}1^{m_1}$ if precisely $m_i$ of its elements have dimension $i$, where $1\le i\le v-1$. Here we determine all possible types of vector space partitions in $\operatorname{PG}(7,2)$.