Query complexity of sampling and small geometric partitions

Abstract: In this paper we study the following problem: Discrete partitioning problem (DPP): Let $\mathbb{F}q P^n$ denote the $n$-dimensional finite projective space over $\mathbb{F}_q$. For positive integer $k \leq n$, let ${ A^i}{i=1}^N$ be a partition of $(\mathbb{F}q P^n)^k$ such that (1) for all $i \leq N$, $A^i = \prod{j=1}^k A^i_j$ (partition into product sets), (2) for all $i \leq N$, there is a $(k-1)$-dimensional subspace $L^i \subseteq \mathbb{F}_q P^n$ such that $A^i \subseteq (L^i)^k$. What is the minimum value of $N$ as a function of $q,n,k$? We will be mainly interested in the case $k=n$.