Edge-symmetric distance-regular coverings of cliques: The affine case

Let $\Gamma$ be an edge-symmetric distance-regular covering of a clique. Then the group $G=\mathrm{Aut}(\Gamma)$ acts twice transitively on the set $\Sigma$ of antipodal classes. We propose a classification for the graphs based on the description of twice transitive permutation groups. This program is realized for $a_1=c_2$. In this article we classify graphs in the case when the action of $G$ on $\Sigma$ is affine.