Cyclic Relative Difference Sets and their p-Ranks

By modifying the constructions in Helleseth et al. [10] and No [15], we construct a family of cyclic ((q^3k–1)/(q–1), q–1, q^3k–1, q^3k–2) relative difference sets, where q=3^e. These relative difference sets are liftings of the difference sets constructed in Helleseth et al. [10] and No [15]. In order to demonstrate that these relative difference sets are in general new, we compute p-ranks of the classical relative difference sets and 3-ranks of the newly constructed relative difference sets when q=3. By rank comparison, we show that the newly constructed relative difference sets are never equivalent to the classical relative difference sets, and are in general inequivalent to the affine GMW difference sets.