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

Authors

MEHMET ÖZEN
NAZMİYE TUĞBA ÖZZAİM
NUH AYDIN

DOI

10.3906/mat-1602-35

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 $\mathbb{Z}_4$

First Page

1235

Last Page

1247

Recommended Citation

ÖZEN, MEHMET; ÖZZAİM, NAZMİYE TUĞBA; and AYDIN, NUH (2017) "Cyclic codes over $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}$," Turkish Journal of Mathematics: Vol. 41: No. 5, Article 14. https://doi.org/10.3906/mat-1602-35
Available at: https://journals.tubitak.gov.tr/math/vol41/iss5/14