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Cyclic and constacyclic codes over a non-chain ring

Year 2014, , 1 - 12, 01.03.2014

Ayşegül Bayram , Irfan Siap

https://doi.org/10.13069/jacodesmath.31486
Cited By: 11

Abstract

In this study, we consider linear and especially cyclic codes over the non-chain ring Zp[v]/ vp− vwhere p is a prime. This is a generalization of the case p = 3. Further, in this work the structure ofconstacyclic codes are studied as well. This study takes advantage mainly from a Gray map whichpreserves the distance between codes over this ring and p-ary codes and moreover this map enlightensthe structure of these codes. Furthermore, a MacWilliams type identity is presented together withsome illustrative examples

Keywords

Non-chain rings Linear codes Cyclic codes Constacycllic codes MacWilliams type identity

References

  • T. Abualrub, I. Siap, On the Construction of Cyclic Codes over the Ring Z2+ uZ2, WSEAS Trans. on Math., 5(6), 750-756, 2006.
  • T. Abulraub, I. Siap, Cyclic Codes over the Rings Z2+ uZ and Z+ uZ2+ u2Z2, Designs, Codes and Cryptography, 3(42), 273-287, 2007.
  • M. Al-Ashker, M. Hamoudeh, Cyclic codes over Z2+ uZ2+ u2Z2+ · · · + uk−1Z2, Turkish J. Math., 35(4), 737-749, 2011.
  • A. Bayram, I. Siap, Structure of Codes over the Ring Z3[v]/ v3− v , Applicable Algebra in Engi- neering, Communication and Computing, 24(5), 369-386, 2013.
  • K. Betsumiya, M. Harada, Optimal self-dual codes over F× F, with respect to the Hamming weight, IEEE Transactions on Information Theory, 50(2), 356-358, 2004.
  • A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over F2+uF2, IEEE Trans. Inf. Theory, 45(4), 1250-1255, 1999.
  • S.T. Dougherty, B. Yildiz and S. Karadeniz, Codes over R, Gray Maps and their Binary Images, Finite Fields and Their Applications, 17(3), 205-219, 2011.
  • J. Gao, Y. Wang, Some results on linear codes over Fp+ vFp+ v3Fp, Journal of Applied Mathematics and Computing, May 2014.
  • A.R. Hammons, Jr., P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, and P. Sole, The Z4-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inf. Theory, 40(2), 301-319, 1994.
  • F.J. MacWilliams and N.J.A. Sloane, The Theory of Error Correcting Codes, North-Holland, Ams- terdam, The Netherlands, 1977.
  • M. Ozen and I. Siap, Linear Codes Over Fq[u]/(us) with Respect to the Rosenbloom-Tsfasman Metric, Designs, Codes and Cryptography, 38(1), 17-29, 2006.
  • Y. H. Park, Modular independence and generator matrices for codes over Zm, Des. Codes Crypt., 50(2), 147-162, 2009.
  • J.-F. Qian, L.-N. Zhang, and S.-X. Zhu, Constacyclic and Cyclic Codes over F+ uF2+ uF, IEICE Trans. Fundamentals, 89(6), 1863-1865, 2006.
  • B. Yildiz, S. Karadeniz, Linear Codes over F2+ uF2+ vF+ uvF, Des. Codes Crypt., 54(1), 61-81, 20 B. Yildiz, S. Karadeniz, Cyclic Codes over F2+ uF+ vF+ uvF2, Des. Codes Crypt., 58(3), 221-234, 20 S.-X. Zhu, Y. Wang, M.-J. Shi, Cyclic codes over F+ vF2, ISIT’09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory, 3, 1719-1722, 2009.

Year 2014, , 1 - 12, 01.03.2014

Ayşegül Bayram , Irfan Siap

https://doi.org/10.13069/jacodesmath.31486
Cited By: 11

Abstract

References

  • T. Abualrub, I. Siap, On the Construction of Cyclic Codes over the Ring Z2+ uZ2, WSEAS Trans. on Math., 5(6), 750-756, 2006.
  • T. Abulraub, I. Siap, Cyclic Codes over the Rings Z2+ uZ and Z+ uZ2+ u2Z2, Designs, Codes and Cryptography, 3(42), 273-287, 2007.
  • M. Al-Ashker, M. Hamoudeh, Cyclic codes over Z2+ uZ2+ u2Z2+ · · · + uk−1Z2, Turkish J. Math., 35(4), 737-749, 2011.
  • A. Bayram, I. Siap, Structure of Codes over the Ring Z3[v]/ v3− v , Applicable Algebra in Engi- neering, Communication and Computing, 24(5), 369-386, 2013.
  • K. Betsumiya, M. Harada, Optimal self-dual codes over F× F, with respect to the Hamming weight, IEEE Transactions on Information Theory, 50(2), 356-358, 2004.
  • A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over F2+uF2, IEEE Trans. Inf. Theory, 45(4), 1250-1255, 1999.
  • S.T. Dougherty, B. Yildiz and S. Karadeniz, Codes over R, Gray Maps and their Binary Images, Finite Fields and Their Applications, 17(3), 205-219, 2011.
  • J. Gao, Y. Wang, Some results on linear codes over Fp+ vFp+ v3Fp, Journal of Applied Mathematics and Computing, May 2014.
  • A.R. Hammons, Jr., P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, and P. Sole, The Z4-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inf. Theory, 40(2), 301-319, 1994.
  • F.J. MacWilliams and N.J.A. Sloane, The Theory of Error Correcting Codes, North-Holland, Ams- terdam, The Netherlands, 1977.
  • M. Ozen and I. Siap, Linear Codes Over Fq[u]/(us) with Respect to the Rosenbloom-Tsfasman Metric, Designs, Codes and Cryptography, 38(1), 17-29, 2006.
  • Y. H. Park, Modular independence and generator matrices for codes over Zm, Des. Codes Crypt., 50(2), 147-162, 2009.
  • J.-F. Qian, L.-N. Zhang, and S.-X. Zhu, Constacyclic and Cyclic Codes over F+ uF2+ uF, IEICE Trans. Fundamentals, 89(6), 1863-1865, 2006.
  • B. Yildiz, S. Karadeniz, Linear Codes over F2+ uF2+ vF+ uvF, Des. Codes Crypt., 54(1), 61-81, 20 B. Yildiz, S. Karadeniz, Cyclic Codes over F2+ uF+ vF+ uvF2, Des. Codes Crypt., 58(3), 221-234, 20 S.-X. Zhu, Y. Wang, M.-J. Shi, Cyclic codes over F+ vF2, ISIT’09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory, 3, 1719-1722, 2009.
There are 14 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Ayşegül Bayram

Irfan Siap

Publication Date March 1, 2014
Published in Issue Year 2014

Cite

APA Bayram, A., & Siap, I. (2014). Cyclic and constacyclic codes over a non-chain ring. Journal of Algebra Combinatorics Discrete Structures and Applications, 1(1), 1-12. https://doi.org/10.13069/jacodesmath.31486
AMA Bayram A, Siap I. Cyclic and constacyclic codes over a non-chain ring. Journal of Algebra Combinatorics Discrete Structures and Applications. March 2014;1(1):1-12. doi:10.13069/jacodesmath.31486
Chicago Bayram, Ayşegül, and Irfan Siap. “Cyclic and Constacyclic Codes over a Non-Chain Ring”. Journal of Algebra Combinatorics Discrete Structures and Applications 1, no. 1 (March 2014): 1-12. https://doi.org/10.13069/jacodesmath.31486.
EndNote Bayram A, Siap I (March 1, 2014) Cyclic and constacyclic codes over a non-chain ring. Journal of Algebra Combinatorics Discrete Structures and Applications 1 1 1–12.
IEEE A. Bayram and I. Siap, “Cyclic and constacyclic codes over a non-chain ring”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 1, no. 1, pp. 1–12, 2014, doi: 10.13069/jacodesmath.31486.
ISNAD Bayram, Ayşegül - Siap, Irfan. “Cyclic and Constacyclic Codes over a Non-Chain Ring”. Journal of Algebra Combinatorics Discrete Structures and Applications 1/1 (March 2014), 1-12. https://doi.org/10.13069/jacodesmath.31486.
JAMA Bayram A, Siap I. Cyclic and constacyclic codes over a non-chain ring. Journal of Algebra Combinatorics Discrete Structures and Applications. 2014;1:1–12.
MLA Bayram, Ayşegül and Irfan Siap. “Cyclic and Constacyclic Codes over a Non-Chain Ring”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 1, no. 1, 2014, pp. 1-12, doi:10.13069/jacodesmath.31486.
Vancouver Bayram A, Siap I. Cyclic and constacyclic codes over a non-chain ring. Journal of Algebra Combinatorics Discrete Structures and Applications. 2014;1(1):1-12.

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