In this work, we study a class of skew cyclic codes over the ring $R:=\mathbb{Z}_4+v\mathbb{Z}_4,$ where $v^2=v,$ with an automorphism $\theta$ and a derivation $\Delta_\theta,$ namely codes as modules over a skew polynomial ring $R[x;\theta,\Delta_{\theta}],$ whose multiplication is defined using an automorphism $\theta$ and a derivation $\Delta_{\theta}.$ We investigate the structures of a skew polynomial ring $R[x;\theta,\Delta_{\theta}].$ We define $\Delta_{\theta}$-cyclic codes as a generalization of the notion of cyclic codes. The properties of $\Delta_{\theta}$-cyclic codes as well as dual $\Delta_{\theta}$-cyclic codes are derived. As an application, some new linear codes over $\mathbb{Z}_4$ with good parameters are obtained by Plotkin sum construction, also via a Gray map as well as residue and torsion codes of these codes.
翻译：带有导数的$\mathbb{Z}_4+v\mathbb{Z}_4$数域的偏循环码：结构特性和计算结果
翻译后的摘要：
在本文中，我们研究了一类偏循环码，它们是作为具有自同构$\theta$和导数$\Delta_\theta$的偏多项式环$R[x;\theta,\Delta_{\theta}]$的模来定义的，这里的乘法是使用一个自同构$\theta$和导数$\Delta_{\theta}$定义的。我们研究了偏多项式环$R[x;\theta,\Delta_{\theta}]$的结构，定义了$\Delta_{\theta}$循环码，作为循环码概念的推广。推导了$\Delta_{\theta}$循环码及其对偶码的特性。作为应用，我们通过Plotkin和构造、灰度映射以及这些码的残余和扭曲码，获得了一些新的参数良好的$\mathbb{Z}_4$线性码。
相关内容
微信扫码咨询专知VIP会员