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Z2 (Z2+uZ2)(Z2+uZ2+u2Z2)-Additive Cyclic Codes

Received: 14 June 2019     Accepted: 11 October 2019     Published: 25 October 2019
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Abstract

In this paper, we introduce the algebraic structure of Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -additive codes and Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -additive cyclic codes. Compared to the Z2Z4Z8-additive codes, the Gray image of any Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -linear code will always be a linear binary code. Therefore, we consider the Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -additive cyclic codes as a (Z2+uZ2+u2Z2) [x] -submodule of Z2α×(Z2+uZ2)β×(Z2+uZ2+u2Z2)θ. We give the definition of Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -additive codes with generator matrices and parity-check matrices. Furthermore, we give the fundamental result on considering their additive cyclic codes with generator polynomials and spanning sets.

Published in Engineering Mathematics (Volume 3, Issue 2)
DOI 10.11648/j.engmath.20190302.11
Page(s) 30-39
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Additive Codes, Cyclic Codes, Minimal Generating Set

References
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[2] Dougherty S. T., Fernndez-Crdoba C., Codes over Z2k, Gray Map Self-Dual Codes Adv. Math. Commun. 5 (4), 571-588, 2011.
[3] Greferath M., Schmidt S. E., Gray isometries for finite chain rings and a nonlinear ternary (36, 3/sup 12/, 15) code, IEEE Transactions on Information Theory. 45 (7), 2522–2524, 1999.
[4] Honold T., Landjev I., Linear Codes over Finite Chain Rings, JOURNAL OF COMBINATORICS. 7 (1), R11–R11, 2001.
[5] Delsarte P., Levenshtein V. I., Association schemes and coding theory, IEEE Transactions on Information Theory. 44 (6), 2477–2504, 1998.
[6] Borges J., Fernndez-Crdoba C., Pujol J., Rif J., Villanueva M., Z2Z4-Linear codes: generator matrices and duality. Designs, Codes Cryptogr. 54 (2), 167–179, 2010.
[7] Abualrub T., Siap I. and Aydin N., Z2Z4-Additive cyclic codes, IEEE Transactions on Information Theory. 60 (3), 1508–1514, 2014.
[8] Aydogudu I., Abualrub T. and Siap I., On Z2Z2 [u] -additive codes, Int. J. Comput. Math. 92 (9), 1806C-1814, 2015.
[9] Borges J., Fernndez-Crdoba C., Ten-Valls R., Z2Z4-Additive cyclic codes, generator polynomials and dual codes, IEEE Transactions on Information Theory. 62 (11), 6348–6354, 2016.
[10] Aydogdu I., Gursoy F., Z2Z4Z8-Cyclic codes, Journal of Applied Mathematics andm Computing. 60 (1–2), 327–341, 2019.
[11] Joaquim B., Dougherty S. T., Cristina F. C., et. al., Binary Images of Z2Z4-Additive cyclic codes, IEEE Transactions on Information Theory. 64 (12), 7551–7556, 2018.
[12] Ismail A., Taher A., The structure of Z2Z4s-additive cyclic codes, Discrete Mathematics, Algorithms and Applications. 10 (04), 1850048, 2018.
[13] Wan Z. X., Quaternary codes, World Scientific. 8, 1997.
[14] Abualrub T. and Siap I., Cyclic codes over the rings Z2+uZ2 and Z2+uZ2+u2Z2 , Designs Codes and Cryptography. 42 (3), 273–287, 2007.
[15] Macwilliams F. J., Sloane N. J. A., The theory of error-correcting codes, Elsevier. 16, 1977.
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  • APA Style

    Zhihui Li. (2019). Z2 (Z2+uZ2)(Z2+uZ2+u2Z2)-Additive Cyclic Codes. Engineering Mathematics, 3(2), 30-39. https://doi.org/10.11648/j.engmath.20190302.11

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    ACS Style

    Zhihui Li. Z2 (Z2+uZ2)(Z2+uZ2+u2Z2)-Additive Cyclic Codes. Eng. Math. 2019, 3(2), 30-39. doi: 10.11648/j.engmath.20190302.11

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    AMA Style

    Zhihui Li. Z2 (Z2+uZ2)(Z2+uZ2+u2Z2)-Additive Cyclic Codes. Eng Math. 2019;3(2):30-39. doi: 10.11648/j.engmath.20190302.11

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  • @article{10.11648/j.engmath.20190302.11,
      author = {Zhihui Li},
      title = {Z2 (Z2+uZ2)(Z2+uZ2+u2Z2)-Additive Cyclic Codes},
      journal = {Engineering Mathematics},
      volume = {3},
      number = {2},
      pages = {30-39},
      doi = {10.11648/j.engmath.20190302.11},
      url = {https://doi.org/10.11648/j.engmath.20190302.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20190302.11},
      abstract = {In this paper, we introduce the algebraic structure of Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -additive codes and Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -additive cyclic codes. Compared to the Z2Z4Z8-additive codes, the Gray image of any Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -linear code will always be a linear binary code. Therefore, we consider the Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -additive cyclic codes as a (Z2+uZ2+u2Z2) [x] -submodule of Z2α×(Z2+uZ2)β×(Z2+uZ2+u2Z2)θ. We give the definition of Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -additive codes with generator matrices and parity-check matrices. Furthermore, we give the fundamental result on considering their additive cyclic codes with generator polynomials and spanning sets.},
     year = {2019}
    }
    

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  • TY  - JOUR
    T1  - Z2 (Z2+uZ2)(Z2+uZ2+u2Z2)-Additive Cyclic Codes
    AU  - Zhihui Li
    Y1  - 2019/10/25
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    N1  - https://doi.org/10.11648/j.engmath.20190302.11
    DO  - 10.11648/j.engmath.20190302.11
    T2  - Engineering Mathematics
    JF  - Engineering Mathematics
    JO  - Engineering Mathematics
    SP  - 30
    EP  - 39
    PB  - Science Publishing Group
    SN  - 2640-088X
    UR  - https://doi.org/10.11648/j.engmath.20190302.11
    AB  - In this paper, we introduce the algebraic structure of Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -additive codes and Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -additive cyclic codes. Compared to the Z2Z4Z8-additive codes, the Gray image of any Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -linear code will always be a linear binary code. Therefore, we consider the Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -additive cyclic codes as a (Z2+uZ2+u2Z2) [x] -submodule of Z2α×(Z2+uZ2)β×(Z2+uZ2+u2Z2)θ. We give the definition of Z2 (Z2+uZ2) (Z2+uZ2+u2Z2) -additive codes with generator matrices and parity-check matrices. Furthermore, we give the fundamental result on considering their additive cyclic codes with generator polynomials and spanning sets.
    VL  - 3
    IS  - 2
    ER  - 

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Author Information
  • Department of Mathematics, School of Mathematics and Statistics, Shandong University of Technology, Zibo, China

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