Mathieu groups as flag-transitive automorphism groups of block designs

Abstract

In this article, we study $2$-$(v,k,\lambda)$ designs $\mathcal{D}$ admitting a flag-transitive almost simple automorphism group $G$ with socle one of the Mathieu groups. In conclusion, we obtain all such $2$-designs with explicit constructions with appropriate references to known designs. To our knowledge, we obtain $101$ newly-constructed flag-transitive $2$-designs.