Induced-bisecting families of bicolorings for hypergraphs

Two n-dimensional vectors A and B, A,B∈Rn, are said to be trivially orthogonal if in every coordinate i∈[n], at least one of A(i) or B(i) is zero. Given the n-dimensional Hamming cube {0,1}n, we study the minimum cardinality of a set V of n-dimensional {−1,0,1} vectors, each containing exactly d non-zero entries, such that every 'possible' point A∈{0,1}n in the Hamming cube has some V∈V which is orthogonal, but not trivially orthogonal, to A. We give asymptotically tight lower and (constructive) upper bounds for such a set V except for the case where d∈Ω(n0.5+ϵ) and d is even, for any ϵ, 0<ϵ≤0.5.