Essential positive covers of the cube

Linial and Radhakrishnan introduced the following problem. A pair (a, c) with a ∈ R and c ∈ R defines the hyperplane {x : ∑ i aixi = c} ⊂ R. Say that a collection of hyperplanes (a, c), . . . , (a, c) is an essential cover of the cube {0, 1} if it is a cover, with no redundant hyperplanes, and every co-ordinate used: i.e., every point in {0, 1} is contained in some hyperplane; for every j ∈ [m] there exists x ∈ {0, 1} such that x is contained only in (a , c); and for every i ∈ [n] there exists j ∈ [m] with aji 6= 0. What is the minimum size of an essential cover? Linial and Radhakrishnan showed that the answer lies between ( √ 4n+ 1 + 1)/2 and dn/2e + 1. We give a best possible bound for the case where aji ≥ 0 for every i, j. Additionally, we reduce the original problem to a conjecture concerning permanents of matrices.