Pairwise transitive 2-designs

TL;DR

This paper classifies pairwise transitive 2-designs, identifying their structure and examples, including symmetric and quasisymmetric types, from various geometric and sporadic sources.

Contribution

It provides a complete classification of pairwise transitive 2-designs, detailing their geometric origins and specific examples, expanding understanding of their automorphism groups.

Findings

Classified pairwise transitive 2-designs into symmetric and quasisymmetric types.

Identified examples from projective, affine, and symplectic geometries.

Included sporadic examples and known designs like the Higman-Sims design.

Abstract

We classify the pairwise transitive 2-designs, that is, 2-designs such that a group of automorphisms is transitive on the following five sets of ordered pairs: point-pairs, incident point-block pairs, non-incident point-block pairs, intersecting block-pairs and non-intersecting block-pairs. These 2-designs fall into two classes: the symmetric ones and the quasisymmetric ones. The symmetric examples include the symmetric designs from projective geometry, the 11-point biplane, the Higman-Sims design, and designs of points and quadratic forms on symplectic spaces. The quasisymmetric examples arise from affine geometry and the point-line geometry of projective spaces, as well as several sporadic examples.