A note on Tight Irreducible Affine Spreads

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 $\textit{affine vector space partition}$ 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 $\textit{completely tight}$ if for any pair $C,C'\in{\mathcal P}$ with $C=v+S$, $C'=v'+S'$, where $v,v'\in{\rm AG}(n,q)$ and $S\neq S'$ are linear subspaces of ${\mathbb F}_q^n$, we have $S\cap S'=\{{\bf 0}\}$. An affine partition ${\mathcal P}$ is said to be $\textit{irreducible}$ if there is no subset ${\mathcal P}'\subset {\mathcal P}$ such that $1<|{\mathcal P}'|<|{\mathcal P}|$ and the union of all subspaces in ${\mathcal P}'$ is a subspace of AG$(n,q)$. For all $d \geq 1$ 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.