Distance-regular Cayley graphs over ${\mathbb{Z}}_{p^s}\oplus {\mathbb{Z}}_p$

In [Distrance-regular Cayley graphs on dihedral groups, J. Combin. Theory Ser B 97 (2007) 14–33], Miklavič and Potočnik proposed the problem of characterizing distance-regular Cayley graphs, which can be viewed as an extension of the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, all distance-regular Cayley graphs over ${\mathbb{Z}}_{p^s}\oplus {\mathbb{Z}}_p$ with $p$ being an odd prime are determined. It is shown that every such graph is isomorphic to a complete graph, a complete multipartite graph, or the line graph of a transversal design $TD(r,p)$ with $2\le r\le p-1$.