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

Let $\Bbb Fr$ be an extension of a finite field $\Bbb Fq$ with $r=q^m$. Let each $gi$ be of order $ni$ in $\Bbb Fr^*$ and $\gcd(ni, nj)=1$ for $1\leq i \neq j \leq u$. We define a cyclic code over $\Bbb Fq$ by $$\mathcal C{(q, m, n1,n2, ..., nu)}={c(a1, a2, ..., au) : a1, a2, ..., au \in \Bbb Fr},$$ where $$c(a1, a2, ..., au)=({Tr}{r/q}(\sum{i=1}^uaigi0),..., {Tr}{r/q}(\sum{i=1}^{uaigi{n-1}))$$} and $n=n1n2... nu$. In this paper, we present a method to compute the weights of $\mathcal C{(q, m, n1,n2, ..., nu)}$. Further, we determine the weight distributions of the cyclic codes $\mathcal C{(q, m, n1,n2)}$ and $\mathcal C{(q, m, n1,n_2,1)}$.