Delandtsheer--Doyen parameters for block-transitive point-imprimitive 2-designs

Abstract

Delandtsheer and Doyen bounded, in terms of the block size, the number of points of a point-imprimitive, block-transitive 2-design. To do this they introduced two integer parameters m, n, now called Delandtsheer-Doyen parameters, linking the block size with the parameters of an associated imprimitivity system on points. We show that the Delandtsheer-Doyen parameters provide upper bounds on the permutation ranks of the groups induced on the imprimitivity system and on a class of the system. We explore extreme cases where these bounds are attained, give a new construction for a family of designs achieving these bounds, and pose several open questions concerning the Delandtsheer-Doyen parameters.