Turkish Journal of Mathematics
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 $R$.
DOI
10.3906/mat-1602-35
Keywords
Cyclic codes, Galois extensions, codes over rings, codes over $\mathbb{Z}_4$
First Page
1235
Last Page
1247
Recommended Citation
ÖZEN, M, ÖZZAİM, N. T, & AYDIN, N (2017). Cyclic codes over $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}$. Turkish Journal of Mathematics 41 (5): 1235-1247. https://doi.org/10.3906/mat-1602-35
