On some vertex-transitive distance-regular antipodal covers of complete graphs

In the present paper, we classify abelian antipodal distance-regular graphs $\Gamma$ of diameter 3 with the following property: $(*)$ $\Gamma$ has a transitive group of automorphisms $\widetilde{G}$ that induces a primitive almost simple permutation group $\widetilde{G}^{\Sigma}$ on the set ${\Sigma}$ of its antipodal classes. There are several infinite families of (arc-transitive) examples in the case when the permutation rank ${\rm rk}(\widetilde{G}^{\Sigma})$ of $\widetilde{G}^{\Sigma}$ equals $2$; moreover, all such graphs are now known. Here we focus on the case ${\rm rk}(\widetilde{G}^{\Sigma})=3$. Under this condition the socle of $\widetilde{G}^{\Sigma}$ turns out to be either a sporadic simple group, or an alternating group, or a simple group of exceptional Lie type, or a classical simple group. Earlier, it was shown that the family of non-bipartite graphs $\Gamma$ with the property $(*)$ such that $\mathrm{rk}(\widetilde{G}^{\Sigma})=3$ and the socle of $\widetilde{G}^{\Sigma}$ is a sporadic or an alternating group is finite and limited to a small number of potential examples. The present paper is aimed to study the case of classical simple socle for $\widetilde{G}^{\Sigma}$. We follow a classification scheme that is based on a reduction to minimal quotients of $\Gamma$ that inherit the property $(*)$. For each given group $\widetilde{G}^{\Sigma}$ with simple classical socle of degree $|{\Sigma}|\le 2500$, we determine potential minimal quotients of $\Gamma$, applying some previously developed techniques for bounding their spectrum and parameters in combination with the classification of primitive rank $3$ groups of the corresponding type and associated rank $3$ graphs. This allows us to essentially restrict the sets of feasible parameters of $\Gamma$ in the case of classical socle for $\widetilde{G}^{\Sigma}$ under condition $|{\Sigma}|\le 2500$.