A note on constacyclic and skew constacyclic codes over the ring Zp[u, v]/hu 2 − u, v2 − v, uv − vui

Year 2019, , 163 - 172, 13.09.2019

Tushar Bag Habibul Islam , Om Prakash , Ashish K. Upadhyay

https://doi.org/10.13069/jacodesmath.617244

Cited By: 7

Abstract

For odd prime $p$, this paper studies $(1+(p-2)u)$-constacyclic codes over the ring $R= \mathbb{Z}_{p} [u,v]/\langle u^2-u,v^2-v,uv-vu\rangle$. We show that the Gray images of $(1+(p-2)u)$-constacyclic codes over $R$ are cyclic and permutation equivalent to a quasi cyclic code over $\mathbb{Z}_{p}$. We derive the generators for $(1+(p-2)u)$-constacyclic and principally generated $(1+(p-2)u)$-constacyclic codes over $R$. Among others, we extend our results for skew $(1+(p-2)u)$-constacyclic codes over $R$ and exhibit the relation between skew $(1+(p-2)u)$-constacyclic codes with the other linear codes. Finally, as an application of our study, we compute several non trivial linear codes by using the Gray images of $(1+(p-2)u)$-constacyclic codes over this ring $R$.

Keywords

Constacyclic codes, Skew constacyclic codes, Gray map, Quasi-cyclic codes

References

• [1] T. Abualrub, I. Siap, Constacyclic codes over $\mathbb{F}_{2} +u\mathbb{F}_{2}$, J. Franklin Inst. 346(5) (2009) 520–529.
• [2] M. Ashraf, G. Mohammed, $(1+2u)$-constacyclic codes over $\mathbb{Z}_{4} +u\mathbb{Z}_{4}$ (preprint) (2015).
• [3] N. Aydin, Y. Cengellenmis, A. Dertli, On some constacyclic codes over $\mathbb{Z}_{4}[u]/\langle u^{2}-1\rangle$, their $\mathbb{Z}_{4}$ images, and new codes, Des. Codes Cryptogr. 86(6) (2018) 1249–1255.
• [4] T. Bag, H. Islam, O. Prakash, A. K. Upadhyay, A study of constacyclic codes over the ring $\mathbb{Z}_{4}[u]/\langle u^{2}-3\rangle$, Discrete Math. Algorithms Appl. 10(4) (2018) 1850056.
• [5] W. Bosma, J. Cannon, Handbook of Magma Functions, Univ. of Sydney 1995.
• [6] H. Islam, O. Prakash, A study of cyclic and constacyclic codes over $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+v\mathbb{Z}_{4}$, Int. J. Inf. Coding Theory 5(2) (2018) 155–168.
• [7] H. Islam, T. Bag, O. Prakash, A class of constacyclic codes over $\mathbb{Z}_{4}[u]/\langle u^{k}\rangle$, J. Appl. Math. Comput. 60(1–2) (2019) 237–251.
• [8] H. Islam, O. Prakash, A note on skew constacyclic codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+v\mathbb{F}_{q}$, Discrete Math. Algorithms Appl. 11(03) (2019) 1950030.
• [9] S. Karadeniz, B. Yildiz, (1 + v)-constacyclic codes over $\mathbb{F}_{2}+u\mathbb{F}_{2}+v\mathbb{F}_{2}+uv\mathbb{F}_{2}$, J. Franklin Inst. 348(9) (2011) 2625–2632.
• [10] P. K. Kewat, B. Ghosh, S. Pattanayak, Cyclic codes over $\mathbb{Z}_{p}[u, v] /\left\langle u^{2}, v^{2}, u v-v u\right\rangle$, Finite Fields Appl. 34 (2015) 161-175.
• [11] M. Ozen, F. Z. Uzekmek, N. Aydin, N. T. Ozzaim, Cyclic and some constacyclic codes over the ring $\mathbb{Z}_{4}[u]/\langle u^2 -1\rangle$, Finite Fields Appl. 38 (2016) 27-39.
• [12] J. F. Qian, L. N. Zhang, S. X. Zhu, $(1+u)$ Constacyclic and cyclic codes over $\mathbb{F}_{2} +u\mathbb{F}_{2}$, Appl. Math. Lett. 19(8) (2006) 820-823.
• [13] M. Shi, L. Qian, L. Sok, N. Aydin, P. Sole, On constacyclic codes over $\mathbb{Z}_{4}[u]/\langle u^2 -1\rangle$ and their Gray images, Finite Fields Appl. 45 (2017) 86-95.
• [14] H. Yu, Y. Wang, M. Shi, $(1+u)$--Constacyclic codes over $\mathbb{Z}_{4} +u\mathbb{Z}_{4}$, Springer Plus 5 (2016) 1325(1-8).