On Skew Polycyclic Codes over $\mathbb{Z}_4[u]/\langle u^2-2\rangle$
Received:April 01, 2022  Revised:October 04, 2022
Key Words: skew polycyclic code   polycyclic code   cyclic code   generator polynomial   Gray map
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.12201361).
 Author Name Affiliation Wei QI School of Mathematics and Statistics, Shandong University of Technology, Shandong 255000, P. R. China Xiaolei ZHANG School of Mathematics and Statistics, Shandong University of Technology, Shandong 255000, P. R. China
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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.