A study on constacyclic codes over the ring $\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$

Document Type : Research Paper


1 Department of Mathematics, Manipur University, Imphal, Manipur-795003, India.

2 Department of Mathematics, D. M. College of Science, Imphal, Manipur-795001, India.


This paper studies $\lambda$-constacyclic codes and skew $\lambda$-constacyclic codes over the finite commutative non-chain ring $R=\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$ with $u^3=0$ for $\lambda= (1+2u+2u^2)$ and $(3+2u+2u^2)$. We introduce distinct Gray maps and show that the Gray images of $\lambda$-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over $\mathbb{Z}_4$. It is also shown that the Gray images of skew $\lambda$-constacyclic codes are quasi-cyclic codes of length $2n$ and index 2 over $\mathbb{Z}_4$. Moreover, the structure of $\lambda$-constacyclic codes of odd length $n$ over the ring $R$ is determined and give some suitable examples.


[1] T. Abualrub and R. Oehmke, Cyclic codes over Z4 of length 2e, Discrete Appl. Math., 128 (2003) 3-9.
[2] T. Abualrub and I. Siap, Cyclic codes over the ring Z2 +uZ2 and Z2 +uZ2 +u2Z2, Des. Codes Cryptogr., 42 No. 3 (2007) 273-287.
[3] N. Aydin, Y. Cengellenmis and A. Dertli, On some constacyclic codes over Z4[u]/u2 1, their Z4 images and new codes, Des. Codes Cryptogr., 86 (2018) 1249-1255.
[4] T. Bag, A. Dertli, Y. Cengellenmis and A. K. Upadhyay, Application of constacyclic codes over the semilocal ring Fpm + vFpm, Indian J. Pure Appl. Math., 51 No. 1 (2020) 265-275.
[5] A. Bayram and I. Siap, Structure of codes over the ring Z3[v]/v3 − v, Appl. Algebra Engrg. Comm. Comput., 24 (2013) 369-386.
[6] Y. Cengellenmis, A. Dertli and N. Aydin, Some constacyclic codes over Z4[u]/u2, new Gray maps and new quaternary codes, Algebra Colloq., 25 No. 3 (2018) 369-376.
[7] A. Dertli and Y. Cengellenmis, On the codes over the ring Z4 +uZ4 +vZ4 cyclic, constacyclic, quasi-cyclic codes, their skew codes, cyclic DNA and skew cyclic DNA codes, Prespacetime Journal, 10 No. 2 (2019) 196-213.
[8] J. Gao, Some results on linear codes over Fp + uFp + u2Fp, J. Appl. Math. Comput., 47 (2015) 473-485.
[9] A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The Z4-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994) 301-319.
[10] H. Islam, T. Bag and O. Prakash, A class of constacyclic codes over Z4[u]/uk, J. Appl. Math. Comput., 60 No. 1-2 (2019) 237-251.
[11] H. Islam and O. Prakash, A study of cyclic and constacyclic codes over Z4 +uZ4 +vZ4, Int. J. Inf. Coding Theory, 5 No. 2 (2018) 155-168.
[12] H. Islam and O. Prakash, A class of constacyclic codes over the ring Z4[u, v]/u2, v2, uv − vu and their Gray images, Filomat, 33 No. 8 (2019) 2237-2248.
[13] M. Özen, N. T. Özzaim and N. Aydin, Cyclic codes over Z4 + uZ4 + u2Z4, Turkish J. Math., 42 (2016) 1235-1247.
[14] M. Özen, F. Z. Uzekmek, N. Aydin and N. T. Özzaim, Cyclic and some constacyclic codes over the ring Z4[u]/u2 1, Finite Fields Appl., 38 (2016) 27-39.
[15] V. S. Pless and Z. Qian, Cyclic codes and quadratic residue codes over Z4, IEEE Trans. Inform. Theory, 41 No. 5 (1996) 1594-1600.
[16] J. F. Qian, L. N. Zhang and S. X. Zhu, (1 + u)-Constacyclic and cyclic codes over F2 + uF2, Appl. Math. Lett., 19 (2006) 820-823.
[17] M. Shi, A. Alahmadi and P. Solé, Codes and Rings: Theory and Practice, First Edition, Academic Press, 2017.
[18] M. Shi, L. Qian, L. Sok, N. Aydin and P. Solé, On constayclic codes over Z4[u]/u2 1 and their Gray images, Finite Fields Appl., 45 (2017) 86-95.
[19] A. K. Singh and P. K. Kewat, On cyclic codes over the ring Zp[u]/uk, Des. Codes Cryptogr., 74 (2015) 1-13.
[20] Z. X. Wan, Quaternary Codes, World Scientific Publishing Company, Singapore, 1997.
[21] B. Yildiz and N. Aydin, On cyclic codes over Z4 + uZ4 and their Z4-images, Int. J. Inf. Coding Theory, 2 (2014) 226-237.
[22] H. Yu, Y. Wang and M. Shi, (1 + u)-Constacyclic codes over Z4 + uZ4, Springer Plus, 2016.