A class of cyclic codes from two distinct finite fields

Let FqFq be a finite field with q   elements and m1m1, m2m2 two distinct positive integers such that gcd⁡(m1,m2)=dgcd⁡(m1,m2)=d. Suppose that α1α1 and α2α2 are two primitive elements of Fqm1Fqm1 and Fqm2Fqm2, respectively. Let n=(qm1−1)(qm2−1)/(qd−1)n=(qm1−1)(qm2−1)/(qd−1) and TiTi denote the trace function from FqmiFqmi to FqFq for i=1,2i=1,2. We define a cyclic codeC(q,m1,m2)={c(a,b):a∈Fqm1,b∈Fqm2},C(q,m1,m2)={c(a,b):a∈Fqm1,b∈Fqm2}, wherec(a,b)=(T1(aα10)+T2(bα20),T1(aα11)+T2(bα21),…,T1(aα1n−1)+T2(bα2n−1)). In this paper, we use Gauss sums to investigate the weight distribution of C(q,m1,m2)C(q,m1,m2) and prove that it has at most four nonzero weights if d=2d=2 and gcd⁡(m1−m22,q−1)=1. Furthermore, we get a class of three-weight cyclic codes if |m1−m2|=2|m1−m2|=2. Some optimal or nearly optimal cyclic codes are presented.