<inline-formula><tex-math id="M1">\begin{document}$ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$\end{document}</tex-math></inline-formula>-additive cyclic codes

Tingting Wu; Jian Gao; Yun Gao; Fang-Wei Fu

doi:10.3934/amc.2018038

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November 2018, 12(4): 641-657. doi: 10.3934/amc.2018038

$ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes

Tingting Wu ^1, , Jian Gao ^2,, , Yun Gao ^3, and Fang-Wei Fu ^3,

1.	College of Science, Civil Aviation University of China, Tianjin 300300, China
2.	School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255091, China
3.	Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China

* Corresponding author: Jian Gao

Received April 2016 Published September 2018

Fund Project: This research is supported by the National Natural Science Foundation of China (Grant Nos. 11701336, 11626144, 11671235, 61571243 and 61171082), the Scientific Research Foundation of Civil Aviation University of China (Grant No. 2017QD22X).

This paper is concerned with ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes. These codes can be identified as submodules of the ring ${\mathbb{Z}}_{2}[x]/\langle x^r-1\rangle × {\mathbb{Z}}_{2}[x]/\langle x^s-1\rangle × {\mathbb{Z}}_{4}[x]/\langle x^t-1\rangle$. There are two major ingredients. First, we determine the generator polynomials and minimum generating sets of this kind of codes. Furthermore, we investigate their dual codes. We determine the structure of the dual of separable ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes completely. For the dual of non-separable ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes, we give their structural properties in a few special cases.

Keywords: $ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-cyclic codes, generator polynomials, minimum generating sets, dual code.

Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 11T71.

Citation: Tingting Wu, Jian Gao, Yun Gao, Fang-Wei Fu. $ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 641-657. doi: 10.3934/amc.2018038

References:

[1]	T. Abualrub, I. Siap and N. Aydin, $ \mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes, IEEE Trans. Inform. Theory, 60 (2014), 1508-1514. doi: 10.1109/TIT.2014.2299791.
[2]	I. Aydogdu, T. Abualrub and I. Siap, $ \mathbb{Z}_2\mathbb{Z}_2[u]$-additive codes, Int. J. Comput. Math., 92 (2015), 1806-1814. doi: 10.1080/00207160.2013.859854.
[3]	I. Aydogdu and I. Siap, On $ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive codes, Linear and Multilinear Algebra, 63 (2015), 2089-2102. doi: 10.1080/03081087.2014.952728.
[4]	J. Borges, C. Fernández-Córdoba, J. Pujol and J. Rifà, $ \mathbb{Z}_2\mathbb{Z}_4$-linear codes: Geneartor matrices and duality, Des. Codes Cryptogr., 54 (2009), 167-179. doi: 10.1007/s10623-009-9316-9.
[5]	J. Borges, C. Fernández-Córdoba and R. Ten-Valls, $ \mathbb{Z}_2$-double cyclic codes, Des.Codes Cryptogr., 86 (2018), 463-479. doi: 10.1007/s10623-017-0334-8.
[6]	P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Reports Suppl., 10 (1973), vi+97 pp.
[7]	C. Fernández-Córdoba, J. Pujol and M. Villanueva, $ {\mathbb{Z}}_{2}{\mathbb{Z}}_{4}$-linear coes: Rank and kernel, Des. Codes Cryptogr., 56 (2010), 43-59. doi: 10.1007/s10623-009-9340-9.
[8]	J. Gao, M. Shi, T. Wu and F.-W. Fu, On double cyclic codes over $ \mathbb{Z}_4$, Finite Fields Appl., 39 (2016), 233-250. doi: 10.1016/j.ffa.2016.02.003.
[9]	W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003. doi: 10.1017/CBO9780511807077.
[10]	H. Mostafanasab, Triple cyclic codes over $ \mathbb{Z}_2$, Palest. J. Math., 6 (2017), Special Issue Ⅱ, 123–134, arXiv: 1509.05351.
[11]	M. Shi, P. Solé and B. Wu, Cyclic codes and the weight enumerators of linear codes over $ \mathbb{F}_2 + v\mathbb{F}_2 + v^2\mathbb{F}_2$, Applied and Computational Mathematics, 12 (2013), 247-255.
[12]	M. Shi and Y. Zhang, Quasi-twisted codes with constacyclic constituent codes, Finite Fields Appl., 39 (2016), 159-178. doi: 10.1016/j.ffa.2016.01.010.
[13]	M. Shi, L. Qian, S. Lin, 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. doi: 10.1016/j.ffa.2016.11.016.
[14]	Z.-X. Wan, Quaternary Codes, World Scientific, Singapore, 1997. doi: 10.1142/3603.

show all references

References:

[1]	T. Abualrub, I. Siap and N. Aydin, $ \mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes, IEEE Trans. Inform. Theory, 60 (2014), 1508-1514. doi: 10.1109/TIT.2014.2299791.
[2]	I. Aydogdu, T. Abualrub and I. Siap, $ \mathbb{Z}_2\mathbb{Z}_2[u]$-additive codes, Int. J. Comput. Math., 92 (2015), 1806-1814. doi: 10.1080/00207160.2013.859854.
[3]	I. Aydogdu and I. Siap, On $ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive codes, Linear and Multilinear Algebra, 63 (2015), 2089-2102. doi: 10.1080/03081087.2014.952728.
[4]	J. Borges, C. Fernández-Córdoba, J. Pujol and J. Rifà, $ \mathbb{Z}_2\mathbb{Z}_4$-linear codes: Geneartor matrices and duality, Des. Codes Cryptogr., 54 (2009), 167-179. doi: 10.1007/s10623-009-9316-9.
[5]	J. Borges, C. Fernández-Córdoba and R. Ten-Valls, $ \mathbb{Z}_2$-double cyclic codes, Des.Codes Cryptogr., 86 (2018), 463-479. doi: 10.1007/s10623-017-0334-8.
[6]	P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Reports Suppl., 10 (1973), vi+97 pp.
[7]	C. Fernández-Córdoba, J. Pujol and M. Villanueva, $ {\mathbb{Z}}_{2}{\mathbb{Z}}_{4}$-linear coes: Rank and kernel, Des. Codes Cryptogr., 56 (2010), 43-59. doi: 10.1007/s10623-009-9340-9.
[8]	J. Gao, M. Shi, T. Wu and F.-W. Fu, On double cyclic codes over $ \mathbb{Z}_4$, Finite Fields Appl., 39 (2016), 233-250. doi: 10.1016/j.ffa.2016.02.003.
[9]	W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003. doi: 10.1017/CBO9780511807077.
[10]	H. Mostafanasab, Triple cyclic codes over $ \mathbb{Z}_2$, Palest. J. Math., 6 (2017), Special Issue Ⅱ, 123–134, arXiv: 1509.05351.
[11]	M. Shi, P. Solé and B. Wu, Cyclic codes and the weight enumerators of linear codes over $ \mathbb{F}_2 + v\mathbb{F}_2 + v^2\mathbb{F}_2$, Applied and Computational Mathematics, 12 (2013), 247-255.
[12]	M. Shi and Y. Zhang, Quasi-twisted codes with constacyclic constituent codes, Finite Fields Appl., 39 (2016), 159-178. doi: 10.1016/j.ffa.2016.01.010.
[13]	M. Shi, L. Qian, S. Lin, 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. doi: 10.1016/j.ffa.2016.11.016.
[14]	Z.-X. Wan, Quaternary Codes, World Scientific, Singapore, 1997. doi: 10.1142/3603.

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$ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes

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