Large Sets of $t$-Designs from $t$-Homogeneous Groups

Abstract

A direct method for constructing large sets of $t$-designs is based on the concept of assembling orbits of a permutation group $G$ on $k$-subsets of a $v$-set into block sets of $t$-designs so that these designs form a large set. If $G$ is $t$-homogeneous, then any orbit is a $t$-design and therefore we obtain a large set by partitioning the set of orbits into parts consisting of the same number of $k$-subsets. In general, it is hard to find such partitions. We solve this problem when orbit sizes are limited to two values. We then use its corollaries to obtain some results in a special case in which a simple divisibility condition holds and no knowledge about orbit sizes is assumed.