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

Abstract: Let $\Bbb F_r$ be an extension of a finite field $\Bbb F_q$ with $r=q<sup>m$. Let each $g_i$ be of order $n_i$ in $\Bbb F_r<sup>*$ and $\gcd(n_i, n_j)=1$ for $1\leq i \neq j \leq u$. We define a cyclic code over $\Bbb F_q$ by $$\mathcal C_{(q, m, n_1,n_2, ..., n_u)}={c(a_1, a_2, ..., a_u) : a_1, a_2, ..., a_u \in \Bbb F_r},$$ where $$c(a_1, a_2, ..., a_u)=({Tr}<em>{r/q}(\sum</em>{i=1}^{ua_ig_i^0),}..., {Tr}<em>{r/q}(\sum</em>{i=1}<sup>ua_ig_i<sup>{n-1}))$$ and $n=n_1n_2... n_u$. In this paper, we present a method to compute the weights of $\mathcal C_{(q, m, n_1,n_2, ..., n_u)}$. Further, we determine the weight distributions of the cyclic codes $\mathcal C_{(q, m, n_1,n_2)}$ and $\mathcal C_{(q, m, n_1,n_2,1)}$.