Cyclic codes over $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}$

Authors: MEHMET ÖZEN, NAZMİYE TUĞBA ÖZZAİM, NUH AYDIN

Abstract: In this paper, we study cyclic codes over the ring $R=\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}$,where $u^{3}=0$. We investigate Galois extensions of this ring and the ideal structure of these extensions.The results are then used to obtain facts about cyclic codes over $R$. We also determine the general form of the generator of a cyclic code and find its minimal spanning sets. Finally, we obtain many new linear codes over $\mathbb{Z}_4$ by considering Gray images of cyclic codes over $R$.

Keywords: Cyclic codes, Galois extensions, codes over rings, codes over $\mathbb{Z}_4$,

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