Essential positive covers of the cube

• Authors: David Saxton
• Venue: Journal of Combinatorial Theory Series A
• Year: 2013

Linial and Radhakrishnan introduced the following problem. A pair (a,c) with a@?R^n and c@?R defines the hyperplane {x:@?"ia"ix"i=c}@?R^n. Say that a collection of hyperplanes (a^1,c^1),...,(a^m,c^m) is an essential cover of the cube {0,1}^n if it is a cover, with no redundant hyperplanes, and every co-ordinate used: i.e., every point in {0,1}^n is contained in some hyperplane; for every j@?[m] there exists x@?{0,1}^n such that x is contained only in (a^j,c^j); and for every i@?[n] there exists j@?[m] with a"i^j0. What is the minimum size of an essential cover? Linial and Radhakrishnan showed that the answer lies between (4n+1+1)/2 and @?n/2@?+1. We give a best possible bound for the case where a"i^j=0 for every i,j. Additionally, we reduce the original problem to a conjecture concerning permanents of matrices.