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

TL;DR

This paper investigates Delandtsheer--Doyen parameters in block-transitive, point-imprimitive 2-designs, establishing bounds on permutation ranks, exploring extremal cases, and proposing new constructions and open questions.

Contribution

It introduces bounds on permutation ranks using Delandtsheer--Doyen parameters and provides new design constructions achieving these bounds.

Findings

Delandtsheer--Doyen parameters bound permutation ranks.

Extremal cases where bounds are attained are characterized.

New design families achieving the bounds are constructed.

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 and 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.