Flag-transitive non-symmetric 2-designs with $\lambda$ prime and exceptional groups of Lie type

This paper contributes to the classification of flag-transitive 2- $(v,k,\lambda )$ designs. Let $\mathcal{D}$ be a non-trivial and non-symmetric $2$- $(v,k,\lambda )$ design with $\lambda$ prime and $G$ be a flag-transitive point-primitive automorphism group of $\mathcal{D}$. A recent work by the first author and Chen has proven that the socle of $G$ is either a nonabelian simple group or an elementary abelian $p$-group for some prime $p$. In this paper, we focus on the case where the socle of $G$ is an exceptional group of Lie type and give all possible parameters of such 2-designs.