Large Sets with Multiplicity

TL;DR

This paper explores the existence and construction of large sets of combinatorial designs, including Steiner quadruple systems and H-designs, using various combinatorial and algebraic methods.

Contribution

It provides new explicit constructions for large sets of combinatorial designs with specified multiplicities, expanding the known existence results.

Findings

Existence of large sets of combinatorial designs for many parameters.

Construction methods include orthogonal arrays, permutation sets, and graph factorizations.

Addresses open questions about Steiner quadruple systems and H-designs.

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 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.