Decompositions of Difference Sets

We characterize those symmetric designs with a Singer group G which admit a quasi-regular G-invariant partition into strongly induced symmetric subdesigns. In terms of the corresponding difference sets, the set associated with the larger design can be decomposed into a difference set describing the small designs and a suitable relative difference set. This generalizes the decomposition of the classical design with the complements of hyperplanes in PG(m − 1, q) as blocks into sub-designs arising from PG(d − 1, q) whenever d divides m. Parametrically, these geometrical examples provide the only known examples of the situation we are studying. But there are many nonisomorphic examples with the same parameters, namely the complements of the classical GMW designs and some generalizations. We also discuss the possibilities for obtaining new difference sets in this way and point out a connection to the recent constructions of Ionin for symmetric designs.