New Cyclic Relative Difference Sets Constructed From<tex>$d$</tex>-Homogeneous Functions With Difference-Balanced Property

Abstract

For a prime power q, we show that a cyclic relative difference set with parameters (q/sup n/-1/q-1,q-1,q/sup n-1/,q/sup n-2/) can be constructed from a d-homogeneous function from F/sub q//sup n//spl bsol/{0} onto F/sub q/ with difference-balanced property, where F/sub q//sup n/ is the finite field with q/sup n/ elements. This construction method enables us to construct several new cyclic relative difference sets with parameters (p/sup n/-1/p/sup l/-1,p/sup l/-1,p/sup n-l/,p/sup n-2l/) from p-ary sequences of period p/sup 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.