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The purpose of this paper is to classify all pairs $(\mathcal{D}, G)$, where$\mathcal{D}$ is a non-trivial $2$-$(v, k, 2)$ design, and $G\leqAut(\mathcal{D})$ acts transitively on the set of blocks of $\mathcal{D}$ andprimitively on the set of points of $\mathcal{D}$ with sporadic socle. We provethat there exists only one such pair $(\mathcal{D}, G)$ in which $\mathcal{D}$is a $2$-$(176,8,2)$ design and $G=HS$, the Higman-Sims simple group.

Block-transitive and point-primitive $2$-$(v,k,2)$ designs with sporadic socle