Induced-bisecting families of bicolorings for hypergraphs

Two n-dimensional vectors A and B, A, B epsilon R-n, are said to be trivially orthogonal if in every coordinate i epsilon [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 epsilon (0, 1}(n) in the Hamming cube has some V epsilon 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 epsilon Omega(n(0.5+epsilon)) and d is even, for any epsilon, 0 < epsilon <= 0.5. (C) 2018 Elsevier B.V. All rights reserved