Weight distributions of cyclic codes with respect to pairwise coprime order elements (1306.5809v2)

Abstract: Let $\Bbb F_r$ be an extension of a finite field $\Bbb F_q$ with $r=q^m$. Let each $g_i$ be of order $n_i$ in $\Bbb F_r^*$ 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 $$c(a_1, a_2, ..., a_u)=({Tr}{r/q}(\sum{i=1}^ua_ig_i^0),..., {Tr}{r/q}(\sum{i=1}^ua_ig_i^{n-1}))$$ 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 $\mathcal C_{(q, m, n_1,n_2,1)}$.