Block-transitive and point-primitive $2$-$(v,k,2)$ designs with sporadic socle

TL;DR

This paper classifies certain symmetric block designs with a specific automorphism group structure, identifying a unique example involving the Higman-Sims group acting on a 2-(176,8,2) design.

Contribution

It provides a complete classification of block-transitive, point-primitive 2-(v,k,2) designs with sporadic socle, revealing a unique design associated with the Higman-Sims group.

Findings

Only one such design exists with parameters (176,8,2).

The automorphism group of this design is the Higman-Sims group.

The design is uniquely characterized by these properties.

Abstract

The purpose of this paper is to classify all pairs $(\mathcal{D}, G)$, where $\mathcal{D}$ is a non-trivial $2$-$(v, k, 2)$ design, and $G\leq Aut(\mathcal{D})$ acts transitively on the set of blocks of $\mathcal{D}$ and primitively on the set of points of $\mathcal{D}$ with sporadic socle. We prove that there exists only one such pair $(\mathcal{D}, G)$ in which $\mathcal{D}$ is a $2$-$(176,8,2)$ design and $G=HS$, the Higman-Sims simple group.