Orientably-regular embeddings of a class of multipartite graphs

The orientably-regular embeddings of complete multipartite graphs have been determined by the contributions of several papers. After that, a natural question can be asked: How about the regular embeddings of the multipartite graphs with m parts, while each part contains n vertices (not necessarily complete multipartite). In this paper, we classify all the orientably-regular embeddings of these graphs when m is a prime q and n is a prime power pe.