This paper studies flag-transitive 2- $\left(v,k,\lambda \right)$ designs with $\left(r,\lambda \right)=2$. We prove that if $\mathcal{D}$ is a flag-transitive nontrivial 2- $\left(v,k,\lambda \right)$ design with $\left(r,\lambda \right)=2$, then $G$ is of affine or almost simple type. Furthermore, we analyze the flag-transitive automorphism groups of almost simple type with $Soc\left(G\right)=A_n$ for $n\ge 5$, and obtain that, up to isomorphism, there are 11 designs.

中文翻译：

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减少标志传递 2- ( v , k , λ ) 设计，( r , λ ) = 2

本文研究了标志传递 2- $\left(v,克,\lambda \right)$ 设计与 $\left(r,\lambda \right)=2$. 我们证明如果$\mathcal{D}$ 是标志传递的非平凡 2- $\left(v,克,\lambda \right)$ 设计与 $\left(r,\lambda \right)=2$， 然后 $G$是仿射或几乎简单的类型。此外，我们分析了几乎简单类型的标志传递自同构群$秒○C\left(G\right)={\mathrm{一种}}_n$ 为了 $n\ge 5$，并得到，直到同构，有 11 个设计。