Constacyclic and Negacyclic Codes over $\mathbb{F}_{2}+u\mathbb{F}_{2}+v\mathbb{F}_{2}$ and their Equivalents over $\mathbb{F}_{2}$

Year 2022, Volume: 5 Issue: 4, 228 - 233, 01.12.2022

Mustafa Özkan Berk Yenice Ayşe Tuğba Güroğlu

https://doi.org/10.33401/fujma.1124502

Abstract

In this work, we consider the finite ring $\mathbb{F}_{2}+u\mathbb{F}_{2}+v\mathbb{F}_{2}$, $u^{2}=1, v^{2}=0$, $u\cdot v=v\cdot u=0$ which is not Frobenius and chain ring. We studied constacyclic and negacyclic codes in $\mathbb{F}_{2}+u\mathbb{F}_{2}+v\mathbb{F}_{2}$ with odd length. These codes are compared with codes that had priorly been obtained on the finite field $\mathbb{F}_{2}$. Moreover, we indicate that the Gray image of a constacyclic and negacyclic code over $\mathbb{F}_{2}+u\mathbb{F}_{2}+v\mathbb{F}_{2}$ with odd length $n$ is a quasicyclic code of index $4$ with length $4n$ in $\mathbb{F}_{2}$. In particular, the Gray images are applied to two different rings $S_{1}=\mathbb{F}_{2}+v\mathbb{F}_{2}$, $v^{2}=0$ and $S_{2}=\mathbb{F}_{2}+u\mathbb{F}_{2}$, $u^{2}=1$ and negacyclic and constacyclic images of these rings are also discussed.

Keywords

Constacyclic code, Codes over rings, Negacyclic code

References

• [1] S. Roman, Coding and Information Theory, Springer-Verlag, 1992.
• [2] J. Wolfman, Negacyclic and cyclic codes over Z4, IEEE Trans. Inform. Theory, 45(7), (1999), 2527-2532.
• [3] J. F. Qian, L. N. Zhang, S. X. Zhu, (1+u)-constacyclic and cyclic codes over F2 +uF2, Appl. Math. Letters, 19(8), (2006), 820-823.
• [4] J. F. Qian, L. N. Zhang, S. X. Zhu, Constacyclic and cyclic codes over F2 +uF2 +u2F2, IEICE Trans. Fund. Electron., Commun. and Comput. Sci., 89(6), (2006), 1863-1865.
• [5] X. Xiaofang, (1+v)-constacyclic codes over F2 +uF2 +vF2, Computer Eng. and Appl., 49(12), (2013), 77-79.
• [6] M. Ozkan, A. Dertli, Y. Cengellenmis, On Gray images of constacyclic codes over the finite ring F2 +u1F2 +u2F2, TWMS J. Appl. Eng. Math., 9(4), (2019), 876-881.