Arc-transitive distance-regular coverings of cliques with $\lambda=\mu$

We study antipodal distance-regular graphs of diameter 3 such that their group of automorphisms acts transitively on the set of pairs $(a,b)$, where $\{a,b\}$ is an edge of the graph. Hence the group of automorphisms of the graph acts $2$-transitively on the set of antipodal classes, so the classification of $2$-transitive permutation groups can be used. We classify arc-transitive distance-regular graphs of diameter 3 in which any two vertices with distance at most two have exactly $\mu$ common neighbors.