Block-transitive 2-designs with a chain of imprimitive point-partitions

More than 30 years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive 2-design, with blocks of size k, could leave invariant a nontrivial point-partition, but only if the number of points was bounded in terms of k. Since then examples have been found where there are two nontrivial point partitions, either forming a chain of partitions, or forming a grid structure on the point set. We show, by construction of infinite families of designs, that there is no limit on the length of a chain of invariant point partitions for a block-transitive 2-design. We introduce the notion of an ‘array’ of a set of points which describes how the set interacts with parts of the various partitions, and we obtain necessary and sufficient conditions in terms of the ‘array’ of a point set, relative to a partition chain, for it to be a block of such a design.