American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Yuan Cao, Yonglin Cao, Hai Q. Dinh, Ramakrishna Bandi, Fang-Wei Fu. An explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020067
[2]

Joaquim Borges, Cristina Fernández-Córdoba, Roger Ten-Valls. On ${{\mathbb{Z}}}_{p^r}{{\mathbb{Z}}}_{p^s}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 169-179. doi: 10.3934/amc.2018011
[3]

Lingyu Diao, Jian Gao, Jiyong Lu. Some results on $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020029
[4]

Helena Rifà-Pous, Josep Rifà, Lorena Ronquillo. $\mathbb{Z}_2\mathbb{Z}_4$-additive perfect codes in Steganography. Advances in Mathematics of Communications, 2011, 5 (3) : 425-433. doi: 10.3934/amc.2011.5.425
[5]

Joaquim Borges, Steven T. Dougherty, Cristina Fernández-Córdoba. Characterization and constructions of self-dual codes over $\mathbb Z_2\times \mathbb Z_4$. Advances in Mathematics of Communications, 2012, 6 (3) : 287-303. doi: 10.3934/amc.2012.6.287
[6]

Makoto Araya, Masaaki Harada, Hiroki Ito, Ken Saito. On the classification of $\mathbb{Z}_4$-codes. Advances in Mathematics of Communications, 2017, 11 (4) : 747-756. doi: 10.3934/amc.2017054
[7]

Amit Sharma, Maheshanand Bhaintwal. A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation. Advances in Mathematics of Communications, 2018, 12 (4) : 723-739. doi: 10.3934/amc.2018043
[8]

Masaaki Harada. Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights. Advances in Mathematics of Communications, 2016, 10 (4) : 695-706. doi: 10.3934/amc.2016035
[9]

Thomas Feulner. Canonization of linear codes over $\mathbb Z$4. Advances in Mathematics of Communications, 2011, 5 (2) : 245-266. doi: 10.3934/amc.2011.5.245
[10]

Steven T. Dougherty, Cristina Fernández-Córdoba. Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes. Advances in Mathematics of Communications, 2011, 5 (4) : 571-588. doi: 10.3934/amc.2011.5.571
[11]

Michael Kiermaier, Johannes Zwanzger. A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval. Advances in Mathematics of Communications, 2011, 5 (2) : 275-286. doi: 10.3934/amc.2011.5.275
[12]

Jean Bourgain. On random Schrödinger operators on $\mathbb Z^2$. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 1-15. doi: 10.3934/dcds.2002.8.1
[13]

Joackim Bernier. Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on $ h\mathbb{Z} $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3179-3195. doi: 10.3934/dcds.2019131
[14]

Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from $ {{\mathbb{R}}^{2}} $ to $ {{\mathbb{S}}^{2}} $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3215-3233. doi: 10.3934/dcds.2019228
[15]

Minjia Shi, Yaqi Lu. Cyclic DNA codes over $ \mathbb{F}_2[u,v]/\langle u^3, v^2-v, vu-uv\rangle$. Advances in Mathematics of Communications, 2019, 13 (1) : 157-164. doi: 10.3934/amc.2019009
[16]

Lakehal Belarbi. Ricci solitons of the $ \mathbb{H}^{2} \times \mathbb{R} $ Lie group. Electronic Research Archive, 2020, 28 (1) : 157-163. doi: 10.3934/era.2020010
[17]

Jonathan Meddaugh, Brian E. Raines. The structure of limit sets for $\mathbb{Z}^d$ actions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4765-4780. doi: 10.3934/dcds.2014.34.4765
[18]

Kwangseok Choe, Hyungjin Huh. Chern-Simons gauged sigma model into $ \mathbb{H}^2 $ and its self-dual equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4613-4646. doi: 10.3934/dcds.2019189
[19]

Murat Şahİn, Hayrullah Özİmamoğlu. Additive Toeplitz codes over $ \mathbb{F}_{4} $. Advances in Mathematics of Communications, 2020, 14 (2) : 379-395. doi: 10.3934/amc.2020026
[20]

Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ are inclined. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1131-1143. doi: 10.3934/dcdss.2020067

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (166)
  • HTML views (370)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences