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

Zhihui Li

doi:doi:10.11648/j.engmath.20190302.11

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

Zhihui Li

Published in Engineering Mathematics (Volume 3, Issue 2)

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 Z₂^α×(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), 2024. Published by Science Publishing Group

Keywords

Additive Codes, Cyclic Codes, Minimal Generating Set

References

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[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.
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[7]	Abualrub T., Siap I. and Aydin N., Z₂Z₄-Additive cyclic codes, IEEE Transactions on Information Theory. 60 (3), 1508–1514, 2014.
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[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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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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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
PY  - 2019
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

Zhihui Li

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

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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}
}

Copy | Download

TY  - JOUR
T1  - Z2 (Z2+uZ2)(Z2+uZ2+u2Z2)-Additive Cyclic Codes
AU  - Zhihui Li
Y1  - 2019/10/25
PY  - 2019
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  -

Copy | Download