Relative difference sets fixed by inversion and Cayley graphs

Using graph theoretical technique, we present a construction of a ( 30 , 2 , 29 , 14 ) -relative difference set fixed by inversion in the smallest finite simple group—the alternating group A 5 . To our knowledge this is the first example known of relative difference sets in the finite simple groups with a non-trivial forbidden subgroup. A connection is then established between some relative difference sets fixed by inversion and certain antipodal distance-regular Cayley graphs. With the connection, several families of antipodal distance-regular Cayley graphs which are coverings of complete graphs are presented.