On substructures of abelian difference sets with classical parameters

Abstract

AbstractWe 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 3 and 3. We show here that any abelian difference set with these parameters inherits a very regular intersection property with regard to subgroups. We show in 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. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 182–190, 2008