Affine vector space partitions

TL;DR

This paper studies affine vector space partitions, determining minimal sizes, classifying small cases, and providing constructions for any field size, advancing understanding of geometric partitions.

Contribution

It introduces new minimal size bounds, classifies small cases, and offers parametric constructions for affine vector space partitions across different field sizes.

Findings

Minimum sizes of affine vector space partitions determined

Enumeration of equivalence classes for small parameters

Parametric constructions provided for arbitrary field sizes

Abstract

An affine vector space partition of $\operatorname{AG}(n,q)$ is a set of proper affine subspaces that partitions the set of points. Here we determine minimum sizes and enumerate equivalence classes of affine vector space partitions for small parameters. We also give parametric constructions for arbitrary field sizes.