Cyclic Relative Difference Sets with Classical Parameters

We investigate the existence of cyclic relative difference sets with parameters ((qd−1)/(q−1), n, qd−1, qd−2(q−1)/n), q any prime power. One can think of these as “liftings” or “extensions” of the complements of Singer difference sets. When q is odd or d is even, we find that relative difference 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 parameters exist if and only if n is a divisor of 2(q−1).