Tbilisi Mathematical Journal

On skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$

Mohammad Ashraf and Ghulam Mohammad

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In the present paper, we study skew cyclic codes over the ring $F_{q}+vF_{q}+v^2F_{q}$, where $v^3=v,~q=p^m$ and $p$ is an odd prime. The structural properties of skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$ have been studied by using decomposition method. By defining a Gray map from $F_{q}+vF_{q}+v^2F_{q}$ to $F_{q}^3$, it has been proved that the Gray image of a skew cyclic code of length $n$ over $F_{q}+vF_{q}+v^2F_{q}$ is a skew $3$-quasi cyclic code of length $3n$ over $F_{q}$. Further, it is shown that the skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$ are principally generated. Finally, the idempotent generators of skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$ have also been studied.


The authors are thankful to the anonymous referees for their careful reading of the paper and valuable comments.

Article information

Tbilisi Math. J., Volume 11, Issue 2 (2018), 35-45.

Received: 16 June 2017
Accepted: 25 January 2018
First available in Project Euclid: 20 June 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 94B05: Linear codes, general
Secondary: 94B15: Cyclic codes

dual codes quasi cyclic codes skew polynomial rings skew cyclic codes idempotent generators


Ashraf, Mohammad; Mohammad, Ghulam. On skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$. Tbilisi Math. J. 11 (2018), no. 2, 35--45. doi:10.32513/tbilisi/1529460020. https://projecteuclid.org/euclid.tbilisi/1529460020

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  • T. Abualrub, A. Ghrayeb, N. Aydin and I. Siap, On the construction of skew quasi cyclic codes , IEEE. Trans. Inform. Theory, 56(2010), 2081-2090.
  • T. Abualrub, N. Aydin and P. Seneviratne, On $\theta$-cyclic codes over $F_2+vF_2$, Australas. J. Combin., 54(2012), 115-126.
  • M. Ashraf and G. Mohammad, On skew cyclic codes over $F_3+vF_3$, Int. J. Inf. Coding Theory, 2(4)(2014), 218-225.
  • M. Ashraf and G. Mohammad, On skew cyclic codes over a semi-local ring, Discrete Math. Algorithm. Appl. 7(4), 1550042 (2015).
  • M. Bhaintwal, Skew quasi cyclic codes over Galois rings, Des. Codes Cryptogr., 62(1)(2012), 85-101.
  • D. Boucher, W. Geiselmann and F. Ulmer, Skew cyclic codes, Appl. Algebra Eng. Commun. Comput, 18(4)(2007), 379-389.
  • D. Boucher, P. Sole and F. Ulmer, Skew constacyclic codes over Galois ring, Adv. Math. Commun., 2(3)(2008), 273-292.
  • D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symb. Comput., 44(2009), 1644-1656.
  • J. Gao, Skew cyclic codes over $F_p+vF_p$, J. Appl. Math. and Informatics 31(2013), 337-342.
  • J. Gao, Some results on linear codes over $F_p+uF_p+u^2F_p$, J. Appl. Math. Comput., 47 (2015), 473-485.
  • F. Gursoy, I. Siap and B. Yildiz, Construction of skew cyclic codes over $F_q+vF_q$, Adv. Math. Commun., 8 (2014), 313-322.
  • A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Sole, The $\mathbb{Z}_4$-linearty of Kerdock, Preparata, Goethals and Related codes, IEEE. Trans. Inform. Theory, 40(1994), 301-319.
  • S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6(2012), 29-63.
  • Z. Odemis Ozger, U. U. Kara and B. Yildiz, Linear, cyclic and constacyclic codes over $S_4=F_2 + u F_2 + u ^2 F_2 + u^3 F_2$, Filomat, 28(2014), 897-906.
  • I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory, 2(2011), 10-20.