Heden's bound on the tail of a vector space partition

TL;DR

This paper improves the lower bound on the smallest subspace size in vector space partitions over finite fields by introducing $q^r$-divisible sets and using geometric non-existence results.

Contribution

It introduces the concept of $q^r$-divisible sets of subspaces and applies geometric arguments to refine bounds on vector space partitions.

Findings

Improved lower bound on the smallest subspace size in partitions

Introduction of $q^r$-divisible sets concept

Non-existence results for certain divisible sets

Abstract

A vector space partition of $\mathbb{F}_q^v$ is a collection of subspaces such that every non-zero vector is contained in a unique element. We improve a lower bound of Heden, in a subcase, on the number of elements of the smallest occurring dimension in a vector space partition. To this end, we introduce the notion of $q^r$-divisible sets of $k$-subspaces in $\mathbb{F}_q^v$. By geometric arguments we obtain non-existence results for these objects, which then imply the improved result of Heden.