Weight distributions of cyclic codes with respect to pairwise coprime order elements

Let FrFr be an extension of a finite field FqFq with r=qmr=qm. Let each gigi be of order nini in Fr⁎ and gcd⁡(ni,nj)=1gcd⁡(ni,nj)=1 for 1⩽i≠j⩽u1⩽i≠j⩽u. We define a cyclic code over FqFq byC(q,m,n1,n2,…,nu)={C(a1,a2,…,au):a1,a2,…,au∈Fr},C(q,m,n1,n2,…,nu)={C(a1,a2,…,au):a1,a2,…,au∈Fr}, whereC(a1,a2,…,au)=(Trr/q(∑i=1uaigi0),…,Trr/q(∑i=1uaigin−1)) and n=n1n2⋯nun=n1n2⋯nu. In this paper, we present a method to compute the weights of C(q,m,n1,n2,…,nu)C(q,m,n1,n2,…,nu). Further, we determine the weight distributions of the cyclic codes C(q,m,n1,n2)C(q,m,n1,n2) and C(q,m,n1,n2,1)C(q,m,n1,n2,1).