Cyclic and constacyclic codes over the ring Z4[u]/⟨u3 − u2⟩ and their Gray images

Abstract

In this article, the structure of generator polynomial of the cyclic codes with odd length is formed over the ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}$ where $u^3=u^2$. With the isomorphism we have defined, the generator polynomial of constacyclic codes with odd length over this ring is created from the generator of the cyclic codes. Additionally, necessary and sufficient conditions for a linear code in this ring to be a self dual code and a LCD code are mentioned. Furthermore, for all units over this ring, $\mathbb{Z}_{4}$-images of $\lambda$-constacyclic codes and also $\mathbb{Z}_{4}$-images of cyclic codes are examined by using related ones from defined three new Gray maps. Moreover, several new and optimal codes are constructed in terms of the Lee, Euclidean and Hamming weight in reference to the database.