This paper contributes to the classification of flag-transitive 2-designs with block size 5. In a recent paper, the flag-transitive automorphism groups of such designs are reduced to point-primitive groups of affine type and almost simple type, and a classification is given of such automorphism groups with sporadic socle. In the present paper, we classify such designs admitting a flag-transitive automorphism group whose socle is an alternating group. We prove that there are precisely six such designs and determine the corresponding automorphism groups.

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.