New cyclic relative difference sets constructed from d-homogeneous functions with difference-balanced property

For a prime power q, we show that a cyclic relative difference set with parameters (qⁿ-1/q-1,q-1,q^n-1,q^n-2) can be constructed from a d-homogeneous function from F_qⁿ/{0} onto F_q with difference-balanced property, where F_qⁿ is the finite field with qⁿ elements. This construction method enables us to construct several new cyclic relative difference sets with parameters (pⁿ-1/p^l-1,p^l-1,p^n-l,p^n-2l) from p-ary sequences of period pⁿ-1 with ideal autocorrelation property introduced by Helleseth and Gong. Using a lifting idea, other new cyclic relative difference sets can be constructed from the Helleseth-Gong (HG) sequences. Also, the 3-ranks and the trace representation of the characteristic sequences of cyclic relative difference sets from a specific class of ternary HG sequences and ternary Lin sequences are derived.