Some necessary conditions for vector space partitions

TL;DR

This paper derives new necessary conditions for vector space partitions, applies them to determine the maximum number of certain subspaces, and explores their relation to partial spreads, providing bounds that are shown to be tight.

Contribution

It introduces novel necessary conditions for vector space partitions and connects these to maximal partial t-spreads, advancing understanding of their structure and bounds.

Findings

Derived new necessary conditions for existence of vector space partitions

Established a tight lower bound for the number of spaces in a partition

Linked the problem to maximal partial t-spreads in vector spaces

Abstract

Some new necessary conditions for the existence of vector space partitions are derived. They are applied to the problem of finding the maximum number of spaces of dimension t in a vector space partition of V(2t,q) that contains m_d spaces of dimension d, where t/2<d<t, and also spaces of other dimensions. It is also discussed how this problem is related to maximal partial t-spreads in V(2t,q). We also give a lower bound for the number of spaces in a vector space partition and verify that this bound is tight.