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Skew cyclic codes over $F_{q}+uF_{q}+vF_{q}+uvF_{q}$

Year 2015, Volume: 2 Issue: 3, 163 - 168, 14.09.2015
https://doi.org/10.13069/jacodesmath.90080

Abstract

In this paper, we study skew cyclic codes over the ring $R=\mathbb{F}_{q}+u\mathbb{F}_{q}+v\mathbb{F}_{q}+uv\mathbb{F}_{q}$, where $u^{2}=u,v^{2}=v,uv=vu$, $q=p^{m}$ and $p$ is an odd prime. We investigate the structural properties of skew cyclic codes over $R$ through a decomposition theorem. Furthermore, we give a formula for the number of skew cyclic codes of length $n$ over $R.$

References

  • T. Abualrub and P. Seneviratne, Skew codes over rings, in Proc. IMECS, Hong Kong, II, 2010.
  • F. W. Anderson and K. R. Fuller, Rings and categories of modules, Springer, 1992.
  • D. Boucher, W. Geiselmann and F. Ulmer, Skew cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18(4), 379-389, 2007.
  • D. Boucher and F. Ulmer, Coding with skew polynomial ring, J. Symb. Comput., 44(12), 1644-1656, 200
  • D. Boucher, P. Sol´e and F. Ulmer, Skew constacyclic codes over Galois ring, Adv. Math. Commun., 2(3), 273-292, 2008.
  • J. Gao, Skew cyclic codes over Fp+ vFp, J. Appl. Math. Inform., 31(3,4), 337-342, 2013.
  • J. Gao, L. Z. Shen and F. W. Fu, Skew generalized quasi-cyclic codes over finite fields, arXiv preprint arXiv:1309.1621, 2013.
  • F. Gursoy, I. Siap and B. Yildiz, Construction of skew cyclic codes over Fq+ vFq, Adv. Math. Commun., 8(3), 313-322, 2014.
  • F. Hernando and D. Ruano, Sixteen new linear codes with Plotkin sum, arXiv preprint arXiv:0804.3507, 2008.
  • S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic over finite chain rings, Adv. Math. Commun., 6(1), 29-63, 2012.
  • A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The Z-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40(2), 301-319, 1994.
  • I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. Nat. Sci., 2(1), 10-20, 2011.
  • Y. T. Zhang, Research on constacyclic codes over some classes of finite non-chain rings, Master’s thesis, Hefei University of Technology, 2013.
Year 2015, Volume: 2 Issue: 3, 163 - 168, 14.09.2015
https://doi.org/10.13069/jacodesmath.90080

Abstract

References

  • T. Abualrub and P. Seneviratne, Skew codes over rings, in Proc. IMECS, Hong Kong, II, 2010.
  • F. W. Anderson and K. R. Fuller, Rings and categories of modules, Springer, 1992.
  • D. Boucher, W. Geiselmann and F. Ulmer, Skew cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18(4), 379-389, 2007.
  • D. Boucher and F. Ulmer, Coding with skew polynomial ring, J. Symb. Comput., 44(12), 1644-1656, 200
  • D. Boucher, P. Sol´e and F. Ulmer, Skew constacyclic codes over Galois ring, Adv. Math. Commun., 2(3), 273-292, 2008.
  • J. Gao, Skew cyclic codes over Fp+ vFp, J. Appl. Math. Inform., 31(3,4), 337-342, 2013.
  • J. Gao, L. Z. Shen and F. W. Fu, Skew generalized quasi-cyclic codes over finite fields, arXiv preprint arXiv:1309.1621, 2013.
  • F. Gursoy, I. Siap and B. Yildiz, Construction of skew cyclic codes over Fq+ vFq, Adv. Math. Commun., 8(3), 313-322, 2014.
  • F. Hernando and D. Ruano, Sixteen new linear codes with Plotkin sum, arXiv preprint arXiv:0804.3507, 2008.
  • S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic over finite chain rings, Adv. Math. Commun., 6(1), 29-63, 2012.
  • A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The Z-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40(2), 301-319, 1994.
  • I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. Nat. Sci., 2(1), 10-20, 2011.
  • Y. T. Zhang, Research on constacyclic codes over some classes of finite non-chain rings, Master’s thesis, Hefei University of Technology, 2013.
There are 13 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Ting Yao This is me

Minjia Shi This is me

Patrick Solé This is me

Publication Date September 14, 2015
Published in Issue Year 2015 Volume: 2 Issue: 3

Cite

APA Yao, T., Shi, M., & Solé, P. (2015). Skew cyclic codes over $F_{q}+uF_{q}+vF_{q}+uvF_{q}$. Journal of Algebra Combinatorics Discrete Structures and Applications, 2(3), 163-168. https://doi.org/10.13069/jacodesmath.90080
AMA Yao T, Shi M, Solé P. Skew cyclic codes over $F_{q}+uF_{q}+vF_{q}+uvF_{q}$. Journal of Algebra Combinatorics Discrete Structures and Applications. September 2015;2(3):163-168. doi:10.13069/jacodesmath.90080
Chicago Yao, Ting, Minjia Shi, and Patrick Solé. “Skew Cyclic Codes over $F_{q}+uF_{q}+vF_{q}+uvF_{q}$”. Journal of Algebra Combinatorics Discrete Structures and Applications 2, no. 3 (September 2015): 163-68. https://doi.org/10.13069/jacodesmath.90080.
EndNote Yao T, Shi M, Solé P (September 1, 2015) Skew cyclic codes over $F_{q}+uF_{q}+vF_{q}+uvF_{q}$. Journal of Algebra Combinatorics Discrete Structures and Applications 2 3 163–168.
IEEE T. Yao, M. Shi, and P. Solé, “Skew cyclic codes over $F_{q}+uF_{q}+vF_{q}+uvF_{q}$”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 2, no. 3, pp. 163–168, 2015, doi: 10.13069/jacodesmath.90080.
ISNAD Yao, Ting et al. “Skew Cyclic Codes over $F_{q}+uF_{q}+vF_{q}+uvF_{q}$”. Journal of Algebra Combinatorics Discrete Structures and Applications 2/3 (September 2015), 163-168. https://doi.org/10.13069/jacodesmath.90080.
JAMA Yao T, Shi M, Solé P. Skew cyclic codes over $F_{q}+uF_{q}+vF_{q}+uvF_{q}$. Journal of Algebra Combinatorics Discrete Structures and Applications. 2015;2:163–168.
MLA Yao, Ting et al. “Skew Cyclic Codes over $F_{q}+uF_{q}+vF_{q}+uvF_{q}$”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 2, no. 3, 2015, pp. 163-8, doi:10.13069/jacodesmath.90080.
Vancouver Yao T, Shi M, Solé P. Skew cyclic codes over $F_{q}+uF_{q}+vF_{q}+uvF_{q}$. Journal of Algebra Combinatorics Discrete Structures and Applications. 2015;2(3):163-8.