Almost Affinely Disjoint Subspaces

TL;DR

This paper introduces the concept of almost affinely disjoint subspace families in finite vector spaces, explores their maximal sizes, constructs optimal examples for small dimensions, and discusses bounds and coding theory connections.

Contribution

It defines almost affinely disjoint subspace families, constructs optimal cases for certain dimensions, and provides bounds and coding theory links.

Findings

Optimal families constructed for k=1 and k=2.

Bounds on polynomial growth of maximal family size.

Connections established with coding theory problems.

Abstract

In this work, we introduce a natural notion concerning finite vector spaces. A family of $k$-dimensional subspaces of $\mathbb{F}_q^n$, which forms a partial spread, is called almost affinely disjoint if any $(k+1)$-dimensional subspace containing a subspace from the family non-trivially intersects with only a few subspaces from the family. The central question discussed in the paper is the polynomial growth (in $q$) of the maximal cardinality of these families given the parameters $k$ and $n$. For the cases $k=1$ and $k=2$, optimal families are constructed. For other settings, we find lower and upper bounds on the polynomial growth. Additionally, some connections with problems in coding theory are shown.