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.