The cross-correlation values of a class of sequences with different lengths

Abstract

In this paper, let Fq be a finite field with q elements and m1,m2 two positive integers with d=gcd⁡(m1,m2). Let s(1)=(T1(α1ei))i=0n−1 and s(2)=(T2(α2i))i=0n−1 be two q-ary sequences of period qm1−1 and qm2−1, respectively, where Fqmi⁎=〈αi〉, Ti denote the trace function from Fqmi to Fq, i=1,2, n=(qm1−1)(qm2−1)/(qd−1), and gcd⁡(e,qm1−1)=1.If e≡1(modqd−1) and m2=m1+d, then two sequences have p values cross correlations and are asymptotically optimal with respect to the cross correlation bound. If e≡1(modqd−1) and m2=m1+2d, then the cross correlations have at most qd−1 values by closed-form expressions of Kloosterman sums over finite fields. If e≡−1(modqd−1), then the cross correlations also have at most qd−1 values. Furthermore, we support some examples by Magma program.