Analysing 2-$(v,k,2)$ designs admitting a flag-transitive almost simple automorphism group with socle $PSL(2,q)$ by means of conics and hyperovals of $PG(2,q)$

TL;DR

This paper classifies certain 2-designs with specific symmetry properties using geometric objects like conics and hyperovals in projective planes, completing previous open cases in the field.

Contribution

It provides a complete classification of 2-(v,k,2) designs with flag-transitive automorphism groups having socle PSL(2,q), utilizing geometric methods.

Findings

Complete classification of the designs in question.

Identification of geometric structures corresponding to these designs.

Resolution of two previously open cases.

Abstract

The classification of the $2$-designs with $\lambda=2$ admitting a flag-transitive automorphism groups with socle $PSL(2,q)$ is completed by settling the two open cases in \cite{ABDT}. The result is achieved by using conics and hyperovals of $PG(2,q)$.