Volume 3, Issue 2, December 2019, Page: 30-39
Z2 (Z2+uZ2)(Z2+uZ2+u2Z2)-Additive Cyclic Codes
Zhihui Li, Department of Mathematics, School of Mathematics and Statistics, Shandong University of Technology, Zibo, China
Received: Jun. 14, 2019;       Accepted: Oct. 11, 2019;       Published: Oct. 25, 2019
DOI: 10.11648/j.engmath.20190302.11      View  45      Downloads  14
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.
Keywords
Additive Codes, Cyclic Codes, Minimal Generating Set
To cite this article
Zhihui Li, Z2 (Z2+uZ2)(Z2+uZ2+u2Z2)-Additive Cyclic Codes, Engineering Mathematics. Vol. 3, No. 2, 2019, pp. 30-39. doi: 10.11648/j.engmath.20190302.11
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
Hammons A. R., Kumar P. V., Calderbank A. R., Sloane N. J. A., Sol P., The Z4-linearity of Kerdock, Preparata, Goethals and related codes. IEEE Transactions on Information Theory. 40 (2), 301-319, 1994.
[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.
Browse journals by subject