A Classification of the flag-transitive $2$-$(v,k,2)$ designs

TL;DR

This paper classifies flag-transitive 2-(v,k,2) designs with affine automorphism groups, constructs new design families, and completes the classification except for a specific semilinear case.

Contribution

Provides a complete classification of flag-transitive 2-(v,k,2) designs with affine automorphism groups, and introduces seven new design families including infinite ones.

Findings

Complete classification of such designs with affine automorphism groups.

Construction of seven new families of flag-transitive 2-designs.

Identification of remarkable objects involved in new designs.

Abstract

In this paper, we provide a complete classification of $2$-$(v,k,2)$ design admitting a flag-transitive automorphism group of affine type with the only exception of the semilinear $1$-dimensional group. Alongside this analysis we provide a construction of seven new families of such flag-transitive $2$-designs, two of them infinite, and some of them involve remarkable objects such as $t$-spreads, translation planes, quadrics and Segre varieties. Our result together with those Alavi et al. [1,2], Praeger et al. [15], Zhou and the first author [37,38] provides a complete classification of $2$-$(v,k,2)$ design admitting a flag-transitive automorphism group with the only exception of the semilinear $1$-dimensional case.