关于环$Z_4[u]/\langle u2-2\rangle$上的斜多元循环码

On Skew Polycyclic Codes over $\mathbb{Z}_4[u]/\langle u^2-2\rangle$

中文关键词:  斜多元循环码  多元循环码  循环码  生成多项式  Gray映射

英文关键词:skew polycyclic code  polycyclic code  cyclic code  generator polynomial  Gray map

中文摘要:

      本文探索了环$R=Z_4[u]/\langle u2-2\rangle$ 上的几类斜多元循环码和多元循环码. 首先得到了环$R$上$(1,2u)$-多元循环码的生成多项式. 其次由定义的Gray映射得到了环$R$上$(1,2u)$- 多元循环码的Gray像是$Z_4$上的循环码或指数为2的逆循环码. 最后, 通过环$R$上$(1,2u)$- 多元循环码的一些例子来展示本文的主要结果.

英文摘要:

      In this paper, we investigate some classes of skew polycyclic codes and polycyclic codes over $R=\mathbb{Z}_4[u]/\langle u^2-2\rangle$. We first obtain the generator polynomials of all $(1,2u)$-polycyclic codes over $R$. Then, by defining some Gray maps, we show that the images of (skew) $(1,2u)$-polycyclic codes over $R$ are cyclic or quasi-cyclic with index 2 over $\mathbb{Z}_4$. Finally, an example of some $(1,2u)$-polycyclic codes over $R$ is given to exhibit the main results of the paper.