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For positive integer $k \\leq n$, let $\\{ A^i\\}_{i=1}^N$ be a partition of $(\\mathbb{F}_q P^n)^k$ such that\n  (1) for all $i \\leq N$, $A^i = \\prod_{j=1}^k A^i_j$ (partition into product sets),\n  (2) for all $i \\leq N$, there is a $(k-1)$-dimensional subspace $L^i \\subseteq \\mathbb{F}_q P^n$ such that $A^i \\subseteq (L^i)^k$.\n  What is the minimum value of $N$ as a function of $q,n,k$? 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