American Institute of Mathematical Sciences

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Reports Suppl., 10 (1973), vi+97 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003. doi: 10.1017/CBO9780511807077.  Google Scholar

[10]

H. Mostafanasab, Triple cyclic codes over $ \mathbb{Z}_2$, Palest. J. Math., 6 (2017), Special Issue Ⅱ, 123–134, arXiv: 1509.05351.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

Z.-X. Wan, Quaternary Codes, World Scientific, Singapore, 1997. doi: 10.1142/3603.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Reports Suppl., 10 (1973), vi+97 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003. doi: 10.1017/CBO9780511807077.  Google Scholar

[10]

H. Mostafanasab, Triple cyclic codes over $ \mathbb{Z}_2$, Palest. J. Math., 6 (2017), Special Issue Ⅱ, 123–134, arXiv: 1509.05351.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

Z.-X. Wan, Quaternary Codes, World Scientific, Singapore, 1997. doi: 10.1142/3603.  Google Scholar

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