Cyclic codes over non-Frobenius ring $\mathbb{Z}_4+u\mathbb{Z}_4+v\mathbb{Z}_4$

Articles in Press

Document Type : Original paper

Authors

1 Faculty of Mathematical Sciences and Statistics, Malayer University, Malayer, Iran

2 Department of Mathematics and Statistics, Kenyon College, 43022 Gambier, OH, USA

Abstract

In this study, cyclic codes of odd length over the ring $R=\mathbb Z_4+u\mathbb Z_4+v\mathbb Z_4$ are investigated. The structure of cyclic codes over this ring is characterized using  the work Samie and Alimoradi (Comp. Appl. Math. 2018). The  ranks and duals of such codes are determined. It is also shown that the  equality $|C||C^\bot|=|R|^n$  holds for a cyclic code $C$ of length $n$ over this ring if and only if $C$ is a cyclic code over the Frobenius subring $\mathbb{Z}_4+u\mathbb{Z}_4$ of the non-Frobenius ring $R.$ Moreover, we correct a false claim about this ring that appeared in the literature. Finally, several new linear codes over $\mathbb{Z}_4$ are obtained from cyclic codes over $R$ via a Gray map.

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References

[1] Database of $\mathbb Z_4$ codes, http://www.z4-codes.info, Accessed: 03 September 2016.
[2] T. Abualrub and I. Siap, Reversible cyclic codes over $\mathbb Z_4$, Australas. J. Combin. 38 (2007), 195–205.
[3] T.A. Abualrub and R.H. Oehmke, Cyclic codes of length $2^e$ over $\mathbb Z_4$, Discrete Appl. Math. 128 (2003), no. 1, 3–9. https://doi.org/10.1016/S0166-218X(02)00432-8
[4] N. Aydin and T. Asamov, A database of $\mathbb Z_4$-codes, J. Combin. Info. System Sci. 34 (2009), no. 1-4, 1–12.
[5] N. Aydin, Y. Lu, and V. Onta, An updated database of $\mathbb Z_4$ codes and an open problem about quasi-cyclic Codes, Codes, Cryptology and Information Security (C2SI 2023) (S. El-Hajji, S. Mesnager, and E.M. Souidi, eds.), Lecture Notes in Computer Science, vol. 13874, Springer, 2023.
[6] B. Çalışkan, N. Aydin, and P. Liu, Skew cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4+v\mathbb{Z}_4$,  Cryptogr. Commun. 15 (2023), no. 4, 845–858. https://doi.org/10.1007/s12095-023-00645-3
[7] H.S. Choi and B. Kim, Various structure of cyclic codes over the non-frobenius ring $F_p[u,v]/\langle u^2,v^2,uv,vu\rangle$, Adv. Math. Commun. 18 (2024), no. 2, 304–327.  https://doi.org/10.3934/amc.2023030
[8] S.T. Dougherty, J. Kim, H. Kulosman, and H. Liu, Self-dual codes over commutative Frobenius rings, Finite Fields Appl. 16 (2010), no. 1, 14–26. https://doi.org/10.1016/j.ffa.2009.11.004
[9] S.T. Dougherty and H. Liu, Independence of vectors in codes over rings, Des. Codes Cryptogr. 51 (2009), 55–68.
https://doi.org/10.1007/s10623-008-9243-1
[10] M. Greferath and S.E. Schmidt, Finite-ring combinatorics and Macwilliams equivalence theorem, J. Combin. Theory Ser. A 92 (2000), no. 1, 17–28. https://doi.org/10.1006/jcta.1999.3033
[11] A.R. Hammons, P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, and P. Solé, The $\mathbb Z_4$-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inf. Theory 40 (1994), no. 2, 301–319. https://doi.org/10.1109/18.312154
[12] W.C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, New York, 2003.
[13] H. Islam and 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 (2018), no. 2, 155–168. https://doi.org/10.1504/IJICOT.2018.095017
[14] B.R. McDonald, Finite Rings with Identity, Marcel Dekker, New York, 1974.
[15] M. Özen, N.T. Özzaim, and N. Aydin,  Cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$, Turkish J. Math. 41 (2017), 1235–1247. https://doi.org/10.3906/mat-1602-35
[16] K. Samie and M.R. Alimoradi, Cyclic codes over the ring $\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2$, Comput. Appl. Math. 37 (2018), no. 3, 2489–2502.
[17] M. Shi, L. Qian, L. Sok, N. Aydin, and P. Solé, On constacyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$ and their gray images, Finite Fields Appl. 45 (2017), 86–95. https://doi.org/10.1016/j.ffa.2016.11.016
[18] J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (1999), no. 3, 555–575.
[19] B. Yildiz and S. Karadeniz, Linear codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, Macwilliams identities, projections and formally self-dual codes, Finite Fields Appl. 27 (2014), 24–40.  https://doi.org/10.1016/j.ffa.2013.12.007
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 05 September 2026

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