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