Cyclic Relative Difference Sets and their p -Ranks

Abstract

By modifying the constructions in Helleseth et al. [10] and No [15], we construct a family of cyclic ((q3k−1)/(q−1), q−1, q3k−1, q3k−2) relative difference sets, where qe3e. 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 qe3. 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.