Block-transitive $t$-($k^2,k,λ$) designs with $PSL(n,q)$ as socle
Guoqiang Xiong, Haiyan Guan
TL;DR
The paper classifies block-transitive $t$-designs with socle $X=PSL(n,q)$ for $n\ge3$, showing any such design must be a $t$-$(v,k,\lambda)$ with $v=k^2$ and $t=2$, and identifying the only possibilities for $(n,q,v,k)$ as $(3,3,144,12)$, $(4,7,400,20)$, and $(5,3,121,11)$. When $\lambda$ divides $k$, the analysis forces $X\cong PSL(3,3)$ and yields a $2$-$(144,12,\lambda)$ design with $\lambda\in\{3,6,12\}$, which the authors realize concretely: a unique $2$-$(144,12,3)$ design, a unique $2$-$(144,12,6)$ design up to isomorphism, and $96$ non-isomorphic $2$-$(144,12,12)$ designs (with at least one flag-transitive example). The approach combines Aschbacher’s subgroup classification, subdegree divisibility, and order bounds with computational constructions (GAP/MAGMA) to complete the classification and exhibit explicit designs.
Abstract
Let $\mathcal{D}=(\mathcal{P},\mathcal{B})$ be a non-trivial block-transitive $t$-$(k^2,k,λ)$ 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 $2$-$(144,12,λ)$ design with $λ\in\{3,6,12\}$ if $λ\mid k$.