On substructures of abelian difference sets with classical parameters

We constrain the structure of difference sets with classical parameters in abelian groups. These include the classical Singer [7] and Gordon et al. [4] constructions and also more recent constructions due to Helleseth et al. [5], [6] arising from the study of sequences with ideal autocorrelation properties. A unified overview of the known families is given in [2] and [3].We show here that any abelian difference set with these parameters inherits a very regular intersection property with regard to subgroups. We showin particular that a planar difference set can always be found embedded in an abelian difference set of odd order whose parameters are those of a 5-dimensional projective geometry.