Large sets with multiplicity

Abstract

Large sets of combinatorial designs has always been a fascinating topic in design theory. These designs form a partition of the whole space into combinatorial designs with the same parameters. In particular, a large set of block designs, whose blocks are of size k taken from an n-set, is a partition of all the k-subsets of the n-set into disjoint copies of block designs, defined on the n-set, and with the same parameters. The current most intriguing question in this direction is whether large sets of Steiner quadruple systems exist and to provide explicit constructions for those parameters for which they exist. In view of its difficulty no one ever presented an explicit construction even for one nontrivial order. Hence, we seek for related generalizations. As generalizations, to the existence question of large sets, we consider two related questions. The first one is to provide constructions for sets on Steiner systems in which each block (quadruple or a k-subset) is contained in exactly $$\mu $$ systems. The constructions of such systems also yield secure protocols for the generalized Russian cards problem. The second question is to provide constructions for large set of H-designs (mainly for quadruples, but also for larger block size), which have applications in threshold schemes and in quantum jump codes. We prove the existence of such systems for many parameters using orthogonal arrays, perpendicular arrays, ordered designs, sets of permutations, and one-factorizations of the complete graph.