循环码关于两两互质序元素的重量分布
Weight distributions of cyclic codes with respect to pairwise coprime order elements
摘要：
设$\Bbb F_r$是有限域$\Bbb F_q$的扩张，其上有$r=q^m$ .令每个$g_i$为$\Bbb F_r^*$中的$n_i$阶，以及为$1\leq i \neq j \leq u$的$\gcd(n_i, n_j)=1$阶。我们通过$$\mathcal C_{(q, m, n_1,n_2, ..., n_u)}=\{c(a_1, a_2, ..., a_u) : a_1, a_2, ..., a_u \in \Bbb F_r\},$$定义$\Bbb F_q$上的循环码，其中$$c(a_1, a_2, ..., a_u)=({Tr}_{r/q}(\sum_{i=1}^ua_ig_i^0), ..., {Tr}_{r/q}(\sum_{i=1}^ua_ig_i^{n-1}))$$和$n=n_1n_2... n_u$。本文提出了一种计算$\mathcal C_{(q, m, n_1,n_2, ..., n_u)}$权值的方法。进一步，我们确定了循环码$\mathcal C_{(q, m, n_1,n_2)}$和$\mathcal C_{(q, m, n_1,n_2,1)}$的重量分布。
英文摘要：
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 $\mathcal C_{(q, m, n_1,n_2,1)}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
