  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 to provide
constructions for sets on Steiner systems in which each block (quadruple or a
$k$-subset) is contained in exactly $\mu$ systems. The second question is to
provide constructions for large set of H-designs (mainly for quadruples, but
also for larger block size). 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.
