Cyclic relative difference sets with classical parameters

We investigate the existence of cyclic relative difference sets with parame ters (q(d) - 1)/(q - 1), n, q(d-1), q(d-2)(q - 1)(n), q any prime power. On e can think of these as "liftings" or "extensions" of the complements of Si nger difference sets. When q is odd or d is even. we find that relative dif ference sets with these parameters exist if and only if n is a divisor of q - 1. In case q is even and d is odd, relative difference sets with these p arameters exist if and only if n is a divisor of 2(q - 1). (C) 2001 Academi c Press.