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On the linear codes over the ring Z₄+v₁Z₄+...+v_{t}Z₄

Year 2019, Volume: 68 Issue: 1, 809 - 823, 01.02.2019
https://doi.org/10.31801/cfsuasmas.478655

Abstract

Some results on linear codes over the ring Z₄+uZ₄+vZ₄,u²=u,v²=v,uv=vu=0 in [6,7] are generalized to the ring D_{t}=Z₄+v₁Z₄+...+v_{t}Z₄,v_{i}²=v_{i},v_{i}v_{j}=v_{j}v_{i}=0 for i≠j,1≤i,j≤t. A Gray map Φ_{t} from D_{t}ⁿ to Z₄^{(t+1)n} is defined. The Gray images of the cyclic, constacyclic and quasi-cyclic codes over D_{t} are determined. The cyclic DNA codes over D_{t} are introduced. The binary images of them are determined. The nontrivial automorphism on D_{i} for i=2,3,...,t is given. The skew cyclic, skew constacyclic and skew quasi-cyclic codes over D_{t} are introduced. The Gray images of them are determined. The skew cyclic DNA codes over D_{t} are introduced. Moreover, some properties of MDS codes over D_{t} are discussed.

References

  • Abualrub, T., Ghrayeb, A. and Zeng, X., Construction of cyclic codes over GF(4) for DNA computing, J. Franklin Institute, 343, (2006) 448-457. Abualrub, T. and Seneviratne, P., On θ-cyclic codes over F₂+vF₂, Australasian Journal of Com., 54, (2012) 115-126.
  • Bandi, R. K. and Bhaintwal, M., Codes over Z₄+vZ₄, IEEE, (2014) 978-14799-3080-7.
  • Bhaintwal, M., Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., DOI: 0.1007/s10623-011-9494-0.
  • Cengellenmis, Y., Dertli, A. and Dougherty, S.T., Codes over an infinite family of rings with a Gray map, Designs, Codes and Cryptography, 72, (2014) 559-580.
  • Dertli, A. and Cengellenmis, Y., On the codes over the ring Z₄+uZ₄+vZ₄ cyclic, constacyclic, quasi-cyclic codes, their skew codes, cyclic DNA and skew cyclic DNA codes, Journal of Science, to be submitted.
  • Dertli, A. and Cengellenmis, Y., Some results on linear codes over the finite ring Z₄+uZ₄+vZ₄; MacWilliams identities, MDS codes, Journal of Science and Arts, (2016) 36, 209-215.
  • Dertli, A. and Cengellenmis, Y. and Eren, S., On the Codes over a Semilocal Finite Ring, Inter. Journal of Advanced Computer Science & Applications, 6, (2015) 283-292.
  • Dertli, A. and Cengellenmis, Y. and Eren, S., On Skew Cyclic and Quasi-cyclic Codes Over F₂+uF₂+u²F₂, Palestine Journal of Mathematics, 4, (2015) 540--546.
  • Dougherty, S. T. and Shiromoto, K., MDS codes over Z_{k}, IEEE Trans. Inform. Theory, (2000) 46, 265-269.
  • Dougherty, S. T. and Shiromoto, K., Maximum distance codes over rings of order 4, IEEE Trans. Inform. Theory, 47, (2001) 400-404.
  • Gaborit, P. and King, O. D., Linear construction for DNA codes, Theor. Computer Science, (2005) 334, 99-113.
  • Gao, J. and Gao, Y., Some Results on Linear Codes over Z₄+vZ₄, arXiv:1402.6 771v1, (2014).
  • Guenda, K. and Gulliver, T. A., Construction of cyclic codes over F₂+uF₂ for DNA computing, AAECC, 24, (2013) 445-459.
  • Hammons, A. R., Kumar, V., Calderbank, A. R., Sloane, N. J. A. and Sole, P., The Z₄-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inf. Theory, 40, (1994) 301-319.
  • Jitman, S., Ling, S. and Udomkovanich, P., Skew constacyclic codes over finite chain rings, AIMS Journal.
  • Li, P., Guo, X. and Zhu, S., Some results of linear codes over the ring Z₄+uZ₄+vZ₄+uvZ₄, arXiv:1601.04453v1, (2016).
  • Pattanayak, S. and Singh, A. K., On cyclic DNA codes over the ring Z₄+uZ₄, arXiv:1508.02015v1, (2015).
  • Shiromoto, K., Singleton bounds for codes over finite rings, Journal of Algebraic Combinatorics, 12, (2000) 95-99.
  • Wu, M., Skew cyclic and quasi-cyclic codes of arbitrary length over Galois rings, International Journal of Algebra, 7, (2013) 803-807.
  • Yildiz, B. and Karadeniz, S., Linear codes over Z₄+uZ₄, MacWilliams identities, projections and formally self-dual codes, Finite fields and their applications, 27, (2014) 24-40.
  • Zhu, S. and Wang, L., A class of constacyclic codes over F_{p}+vF_{p} and their Gray images, Discrete Math. 311, (2011), 2677-2682.
Year 2019, Volume: 68 Issue: 1, 809 - 823, 01.02.2019
https://doi.org/10.31801/cfsuasmas.478655

Abstract

References

  • Abualrub, T., Ghrayeb, A. and Zeng, X., Construction of cyclic codes over GF(4) for DNA computing, J. Franklin Institute, 343, (2006) 448-457. Abualrub, T. and Seneviratne, P., On θ-cyclic codes over F₂+vF₂, Australasian Journal of Com., 54, (2012) 115-126.
  • Bandi, R. K. and Bhaintwal, M., Codes over Z₄+vZ₄, IEEE, (2014) 978-14799-3080-7.
  • Bhaintwal, M., Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., DOI: 0.1007/s10623-011-9494-0.
  • Cengellenmis, Y., Dertli, A. and Dougherty, S.T., Codes over an infinite family of rings with a Gray map, Designs, Codes and Cryptography, 72, (2014) 559-580.
  • Dertli, A. and Cengellenmis, Y., On the codes over the ring Z₄+uZ₄+vZ₄ cyclic, constacyclic, quasi-cyclic codes, their skew codes, cyclic DNA and skew cyclic DNA codes, Journal of Science, to be submitted.
  • Dertli, A. and Cengellenmis, Y., Some results on linear codes over the finite ring Z₄+uZ₄+vZ₄; MacWilliams identities, MDS codes, Journal of Science and Arts, (2016) 36, 209-215.
  • Dertli, A. and Cengellenmis, Y. and Eren, S., On the Codes over a Semilocal Finite Ring, Inter. Journal of Advanced Computer Science & Applications, 6, (2015) 283-292.
  • Dertli, A. and Cengellenmis, Y. and Eren, S., On Skew Cyclic and Quasi-cyclic Codes Over F₂+uF₂+u²F₂, Palestine Journal of Mathematics, 4, (2015) 540--546.
  • Dougherty, S. T. and Shiromoto, K., MDS codes over Z_{k}, IEEE Trans. Inform. Theory, (2000) 46, 265-269.
  • Dougherty, S. T. and Shiromoto, K., Maximum distance codes over rings of order 4, IEEE Trans. Inform. Theory, 47, (2001) 400-404.
  • Gaborit, P. and King, O. D., Linear construction for DNA codes, Theor. Computer Science, (2005) 334, 99-113.
  • Gao, J. and Gao, Y., Some Results on Linear Codes over Z₄+vZ₄, arXiv:1402.6 771v1, (2014).
  • Guenda, K. and Gulliver, T. A., Construction of cyclic codes over F₂+uF₂ for DNA computing, AAECC, 24, (2013) 445-459.
  • Hammons, A. R., Kumar, V., Calderbank, A. R., Sloane, N. J. A. and Sole, P., The Z₄-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inf. Theory, 40, (1994) 301-319.
  • Jitman, S., Ling, S. and Udomkovanich, P., Skew constacyclic codes over finite chain rings, AIMS Journal.
  • Li, P., Guo, X. and Zhu, S., Some results of linear codes over the ring Z₄+uZ₄+vZ₄+uvZ₄, arXiv:1601.04453v1, (2016).
  • Pattanayak, S. and Singh, A. K., On cyclic DNA codes over the ring Z₄+uZ₄, arXiv:1508.02015v1, (2015).
  • Shiromoto, K., Singleton bounds for codes over finite rings, Journal of Algebraic Combinatorics, 12, (2000) 95-99.
  • Wu, M., Skew cyclic and quasi-cyclic codes of arbitrary length over Galois rings, International Journal of Algebra, 7, (2013) 803-807.
  • Yildiz, B. and Karadeniz, S., Linear codes over Z₄+uZ₄, MacWilliams identities, projections and formally self-dual codes, Finite fields and their applications, 27, (2014) 24-40.
  • Zhu, S. and Wang, L., A class of constacyclic codes over F_{p}+vF_{p} and their Gray images, Discrete Math. 311, (2011), 2677-2682.
There are 21 citations in total.

Details

Primary Language English
Journal Section Review Articles
Authors

Abdullah Dertlı 0000-0001-8687-032X

Yasemin Cengellenmıs This is me 0000-0002-8133-9836

Publication Date February 1, 2019
Submission Date August 27, 2016
Acceptance Date December 30, 2017
Published in Issue Year 2019 Volume: 68 Issue: 1

Cite

APA Dertlı, A., & Cengellenmıs, Y. (2019). On the linear codes over the ring Z₄+v₁Z₄+...+v_{t}Z₄. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 809-823. https://doi.org/10.31801/cfsuasmas.478655
AMA Dertlı A, Cengellenmıs Y. On the linear codes over the ring Z₄+v₁Z₄+.+v_{t}Z₄. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. February 2019;68(1):809-823. doi:10.31801/cfsuasmas.478655
Chicago Dertlı, Abdullah, and Yasemin Cengellenmıs. “On the Linear Codes over the Ring Z₄+v₁Z₄+. +v_{t}Z₄”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 1 (February 2019): 809-23. https://doi.org/10.31801/cfsuasmas.478655.
EndNote Dertlı A, Cengellenmıs Y (February 1, 2019) On the linear codes over the ring Z₄+v₁Z₄+. +v_{t}Z₄. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 1 809–823.
IEEE A. Dertlı and Y. Cengellenmıs, “On the linear codes over the ring Z₄+v₁Z₄+...+v_{t}Z₄”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 1, pp. 809–823, 2019, doi: 10.31801/cfsuasmas.478655.
ISNAD Dertlı, Abdullah - Cengellenmıs, Yasemin. “On the Linear Codes over the Ring Z₄+v₁Z₄+. +v_{t}Z₄”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/1 (February 2019), 809-823. https://doi.org/10.31801/cfsuasmas.478655.
JAMA Dertlı A, Cengellenmıs Y. On the linear codes over the ring Z₄+v₁Z₄+. +v_{t}Z₄. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:809–823.
MLA Dertlı, Abdullah and Yasemin Cengellenmıs. “On the Linear Codes over the Ring Z₄+v₁Z₄+. +v_{t}Z₄”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 1, 2019, pp. 809-23, doi:10.31801/cfsuasmas.478655.
Vancouver Dertlı A, Cengellenmıs Y. On the linear codes over the ring Z₄+v₁Z₄+. +v_{t}Z₄. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(1):809-23.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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