Large sets of t-designs from t-homogeneous groups

A direct method for constructing large sets of t-designs is based on the concept of assembling orbits of a permutation group G on ksubsets 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.