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

Abstract

Let ${\mathbb{F}}_r$ be an extension of a finite field ${\mathbb{F}}_q$ with $r=q^m$. Let each $g_i$ be of order $n_i$ in ${\mathbb{F}}_r^{\ast }$ and $\gcd (n_i,n_j)=1$ for $1\le i\ne j\le u$. We define a cyclic code over ${\mathbb{F}}_q$ by $${\mathcal{C}}_{(q,m,n_1,n_2,...,n_u)}=\{c(a_1,a_2,...,a_u):a_1,a_2,...,a_u\in {\mathbb{F}}_r\},$$ where $$c(a_1,a_2,...,a_u)=({Tr}_{r/q}(\sum _{i=1}^u{a}_i{g}_i^0),...,{Tr}_{r/q}(\sum _{i=1}^u{a}_i{g}_i^{n-1}))$$ and $n=n_1{n}_2...n_u$. 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 ${\mathcal{C}}_{(q,m,n_1,n_2,1)}$.

18 pages. arXiv admin note: substantial text overlap with arXiv:1306.5277

References in corpus (3)

• The Weight Distribution of a Class of Cyclic Codes Related to Hermitian Forms Graphs
• Weight distribution of two classes of cyclic codes with respect to two distinct order elements
• A Class of Three-Weight Cyclic Codes