Linear complementary dual codes and double circulant codes over a semi-local ring
Xiangdong Cheng, Xiwang Cao, Liqin Qian · Advances in Mathematics of Communications · 2022
Let \begin{document}$ q $\end{document} be an odd prime power and \begin{document}$ \mathbb{F}_q $\end{document} be the finite field with \begin{document}$ q $\end{document} elements. In this paper, suppose ring \begin{document}$ R = \mathbb{F}_{q}+ \mu \mathbb{F}_{q}+ u \mathbb{F}_{q}+ \mu u \mathbb{F}_{q} $\end{document} , where \begin{document}$ \mu u = u \mu, \mu^{2} = \mu, u^{2} = u. $\end{document} We first give a Gray map from \begin{document}$ R $\end{document} onto \begin{document}$ \mathbb{F}_q^{4} $\end{document} and consider a decomposition of the ring \begin{document}$ R $\end{document} . Additionally, we investigate linear complementary dual (LCD) codes over the ring \begin{document}$ R $\end{document} . Some conditions for such linear codes over \begin{document}$ R $\end{document} to be linear complementary dual are given. Furthermore, based on the Artin conjecture, we get a class of good codes by calculating the total number of LCD double circulant codes over \begin{document}$ R $\end{document} .