Sporadic finite simple groups and block designs

Abstract

The purpose of this paper is to classify all pairs $(\mathcal{D}, G)$, where $\mathcal{D}$ is a non-trivial $2$-$(v, k, \lambda)$ design with $\lambda\leq10$, and $G\leq \mathrm{Aut}(\mathcal{D})$ acts transitively on the set of blocks of $\mathcal{D}$ and primitively on the set of points of $\mathcal{D}$ with sporadic socle. We prove that there are exactly 15 such pairs $(\mathcal{D}, G)$.