2-Cancellative Hypergraphs and Codes

A family of sets F (and the corresponding family of 0-1 vectors) is called t-cancellative if, for all distinct t + 2 members A(1), ... , A(t) and B, C is an element of F, A(1) boolean OR ... boolean OR A(t) boolean OR B not equal A(1) boolean OR ... boolean OR A(t) boolean OR C. Let c(t)(n) be the size of the largest t-cancellative family on n elements, and let c(t)(n, r) denote the largest r-uniform family. We improve the previous upper bounds, e. g., we show c(2)(n) <= 2(0.322n) (for n > n(0)). Using an algebraic construction we show that c(2)(n, 2k) = Theta(n(k)) for each k when n -> infinity.