Large sets with multiplicity

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