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

Abstract

For a prime power q, we show that a cyclic relative difference set with parameters (q(n)-1/q-1, q(n-1), q(n-2)) can be constructed from a d-homogeneous function from F, \ fO I onto Fq with difference-balanced property, where F-qn is the finite field with q(n) elements. This construction method enables us to construct several new cyclic relative difference sets with parameters (p(n)-1/p(l)-1, p(l)-1, p(n-l) , p(n-2l)) from p-ary sequences of period p(n)-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.