A class of 2D skew-cyclic codes over \({\mathbb {F}}_{q}+u{\mathbb {F}}_{q}\)

  • Amit SharmaEmail author
  • Maheshanand Bhaintwal
Original Paper


In this paper we present a class of 2D skew-cyclic codes over \(R={\mathbb {F}}_{q}+u{\mathbb {F}}_{q}, u^2=1\), using the bivariate skew polynomial ring \(R[x,y,\theta ,\sigma ]\), where \({\mathbb {F}}_q\) is a finite field, and \(\theta \) and \(\sigma \) are two commuting automorphisms of R. After defining a partial order on \(R[x,y,\theta ,\sigma ],\) we obtain division algorithm for \(R[x,y,\theta ,\sigma ]\) under two different conditions. The structure of 2D skew-cyclic codes over R is obtained in terms of their generating sets. For this, we have classified these codes into different classes, based on certain conditions they satisfy, and accordingly obtained their generating sets in each case separately. A decomposition of a 2D skew-cyclic code C over R into 2D skew-cyclic codes over \({\mathbb {F}}_{q}\) is studied and some examples are given to illustrate the results.


Skew polynomial rings Skew-cyclic codes 2D skew-cyclic codes Automorphisms Generating sets 



This work was partially supported by DST, Govt. of India, under Grant No. SB/S4/MS: 893/14. Also, the first author would like to thank the Council of Scientific & Industrial Research (CSIR), India for providing financial support.


  1. 1.
    Abualrub, T., Siap, I., Aydin, N.: \({\mathbb{Z}}_2{\mathbb{Z}}_4\)-additive cyclic codes. IEEE Trans. Inf. Theory 60, 1508–1514 (2014)CrossRefGoogle Scholar
  2. 2.
    Abualrub, T., Siap, I., Aydogdu, I.: \({\mathbb{Z}}_2 ({\mathbb{Z}}_2 + u{\mathbb{Z}}_2)\)- linear cyclic codes. In: International MultiConference of Engineers and Computer Scientists (IMECS’2014), vol. II, Hong Kong (2014)Google Scholar
  3. 3.
    Aydogdu, I., Siap, I.: On \({\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}\)-additive codes. Linear Multilinear Algebra 63, 2089–2102 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Boucher, D., Ulmer, F.: Coding with skew polynomial rings. J. Symb. Comput. 44, 1644–1656 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Boucher, D., Geiselmann, W., Ulmer, F.: Skew cyclic codes. Appl. Algebra Eng. Commun. Comput. 18, 379–389 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Boucher, D., Ulmer, F.: Codes as modules over skew polynomial rings. In: Proceedings of 12th IMA International Conference, Cryptography and Coding, Cirencester, UK, LNCS 5921, pp. 38–55 (2009)CrossRefGoogle Scholar
  7. 7.
    Boucher, D., Solé, P., Ulmer, F.: Skew constacyclic codes over Galois rings. Adv. Math. Commun. 2, 273–292 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ikai, T., Kosako, H., Kojima, Y.: Two-dimensional cyclic codes. Electron. Commun. Jpn. 57A, 27–35 (1975)MathSciNetGoogle Scholar
  9. 9.
    Imai, H.: A theory of two-dimensional cyclic codes. Inf. Control 34, 1–21 (1977)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ribenboim, P.: Sur la localisation des anneaux non commutatifs, Seminaire Dubreil. Algebre et theorie des nombres, tome 24 (1970–1971)Google Scholar
  11. 11.
    Sharma, A., Bhaintwal, M.: \(F_3R\)-skew cyclic codes. Int. J. Inf. Coding Theory 3(3), 234–251 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Xiuli, L., Hongyan, L.: 2-D skew cyclic codes over \({\mathbb{F}}_{q}[x, y;\rho,\theta ]\). Finite Fields Appl. 25, 49–63 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Yildiz, B., Aydin, N.: On cyclic codes over \({\mathbb{Z}}_4 + u{\mathbb{Z}}_4\) and their \({\mathbb{Z}}_4\)-images. Int. J. Inf. Coding Theory 2, 226–237 (2014)MathSciNetCrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Indian Institute of Technology RoorkeeRoorkeeIndia

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