A note on Tight Irreducible Affine Spreads

Abstract

Let ${\mathbb{F}}_q^n$ denote the vector space of dimension $n$ over ${\mathbb{F}}_q$ and AG$(n,q)$ denote the corresponding affine space. An $\text{affine vector space partition}$ of AG$(n,q)$ is a collection $\mathcal{P}$ of affine subspaces that partition the points of AG$(n,q)$. If all subspaces in $\mathcal{P}$ have the same dimension $d$, then $\mathcal{P}$ is called an $\textit{affine $d$-spread}$. We say that an affine partition $\mathcal{P}$ is $\text{completely tight}$ if for any pair $C,C^{\prime }\in \mathcal{P}$ with $C=v+S$, $C^{\prime }=v^{\prime }+S^{\prime }$, where $v,v^{\prime }\in \mathrm{AG}(n,q)$ and $S\ne S^{\prime }$ are linear subspaces of ${\mathbb{F}}_q^n$, we have $S\cap S^{\prime }=\{0\}$. An affine partition $\mathcal{P}$ is said to be $\text{irreducible}$ if there is no subset ${\mathcal{P}}^{\prime }\subset \mathcal{P}$ such that $1<|{\mathcal{P}}^{\prime }|<|\mathcal{P}|$ and the union of all subspaces in ${\mathcal{P}}^{\prime }$ is a subspace of AG$(n,q)$. For all $d\ge 1$ and $n>2d$, we construct a completely tight irreducible affine $d$-spread of AG$(n,q)$. This also settles a recent conjecture of Bamberg et al. on the existence of tight irreducible affine $d$-spreads.

5 pages