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 subdes igns. In terms of the corresponding difference sets, the set associated wit h 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 i n PG(m - 1, q) as blocks into sub-designs arising from PG(d - 1, q) wheneve r d divides m. Parametrically, these geometrical examples provide the only known examples of the situation we are studying. But there are many nonisom orphic examples with the same parameters, namely the complements of the cla ssical GMW designs and some generalizations. We also discuss the possibilit ies for obtaining new difference sets in this way and point out a connectio n to the recent constructions of Ionin for symmetric designs. (C) 1999 Acad emic Press.