On Bipartite Distance-Regular Cayley Graphs with Small Diameter

We study bipartite distance-regular Cayley graphs with diameter three or four. give sufficient conditions under which a graph can be constructed on the semidirect product of group — part this contains identity element and $\mathbb{Z}_{2}$. apply to case three, consider cases where are not satisfied for some specific groups such as dihedral group.We also extend result by Miklavič Potočnik that relates difference sets This new involves certain partial geometric in antipodal relative sets.