The weight distribution of a class of cyclic codes containing a subclass with optimal parameters

Let α be a primitive element of a finite field Fr, where r=qm1m2 and gcd⁡(m1,m2)=d, so α1=αr−1qm1−1 and α2=αr−1qm2−1 are primitive elements of Fqm1 and Fqm2, respectively. Let e be a positive integer such that gcd⁡(e,qm2−1qd−1)=1, Fqm2=Fq(α2e), and α1 and α2e are not conjugates over Fq. We define a cyclic codeC={c(a,b):a∈Fqm1,b∈Fqm2},c(a,b)=(T1(aα1i)+T2(bα2ei))i=0n−1, where Ti denotes the trace function from Fqmi to Fq for i=1,2. In this paper, we use Gauss sums to investigate the weight distribution of C, which generalizes the results of C. Li and Q. Yue in [13,14]. Furthermore, we explicitly determine the weight distribution of C if d=1,2. Moreover, we prove it is optimal three-weight achieving the Griesmer bound if d=1 and gcd⁡(m2−em1,q−1)=1.