On a hyperplane arrangement problem and tighter analysis of an error-tolerant pooling design

Abstract In this paper, we formulate and investigate the following problem: given integers d,k and r where k>r≥1,d≥2, and a prime power q, arrange d hyperplanes on $\mathbb{F}_{q}^{k}$ to maximize the number of r-dimensional subspaces of $\mathbb{F}_{q}^{k}$ each of which belongs to at least one of the hyperplanes. The problem is motivated by the need to give tighter bounds for an error-tolerant pooling design based on finite vector spaces.

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