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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 distict t+2 members A_1,... A_t and B,C from F the union of A_1,..., A_t and B differs from the union of A_1, ..., A_t and C. Let c(n,t) be the size of the largest t-cancellative family on n elements, and let c_k(n,t) denote the largest k-uniform family. We significantly improve the previous upper bounds, e.g., we show c(n,2)< 2^0.322n (for n> n_0). Using an algebraic construction we show that the order of magnitude of c_{2k}(n,2) is n^k for each k (when n goes to infinity).