Block-transitive $t$-($k^2,k,\lambda$) designs with $PSL(n,q)$ as socle

TL;DR

This paper classifies certain highly symmetric block designs with automorphism groups related to the projective special linear groups, identifying specific parameters and conditions under which these designs exist.

Contribution

It proves that such block-transitive $t$-designs must have $t=2$ and determines explicit parameter sets, advancing the classification of symmetric designs with these groups.

Findings

Identifies that $t=2$ for these designs.

Provides explicit parameter sets $(n,q,v,k)$ for the designs.

Shows the existence of a $2$-$(144,12, ext{lambda})$ design with specific $ ext{lambda}$ values.

Abstract

Let $\mathcal{D}=(\mathcal{P},\mathcal{B})$ be a non-trivial block-transitive $t$-$(k^2,k,\lambda)$ design with $G\leq \Aut(\mathcal{D})$ and $X\unlhd G\leq \Aut(X)$, where $X=PSL(n,q)(n\geq3).$ We prove that $t=2$ and the parameters $(n,q,v,k)$ is $(3,3,144,12),(4,7,400,20)$ or $(5,3,121,11).$ Moreover, $\mathcal{D}$ is a $2$-$(144,12,\lambda)$ design with $\lambda\in\{3,6,12\}$ if $\lambda\mid k$.