BibTex RIS Cite

Cyclic and constacyclic codes over a non-chain ring

Year 2014, Volume: 1 Issue: 1, 1 - 12, 01.03.2014

Ayşegül Bayram , Irfan Siap

https://doi.org/10.13069/jacodesmath.31486
https://izlik.org/JA97DJ57MT

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, Volume: 1 Issue: 1, 1 - 12, 01.03.2014

Ayşegül Bayram , Irfan Siap

https://doi.org/10.13069/jacodesmath.31486
https://izlik.org/JA97DJ57MT

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
Authors

Ayşegül Bayram

Irfan Siap

Publication Date March 1, 2014
DOI https://doi.org/10.13069/jacodesmath.31486
IZ https://izlik.org/JA97DJ57MT
Published in Issue Year 2014 Volume: 1 Issue: 1

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 1.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. doi:10.13069/jacodesmath.31486
Chicago Bayram, Ayşegül, and Irfan Siap. 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.
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 [1]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, Mar. 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 1, 2014): 1-12. https://doi.org/10.13069/jacodesmath.31486.
JAMA 1.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, Mar. 2014, pp. 1-12, doi:10.13069/jacodesmath.31486.
Vancouver 1.Ayşegül Bayram, Irfan Siap. Cyclic and constacyclic codes over a non-chain ring. Journal of Algebra Combinatorics Discrete Structures and Applications. 2014 Mar. 1;1(1):1-12. doi:10.13069/jacodesmath.31486

Cited By

https://doi.org/
On Constacyclic Codes Over <inline-formula> <tex-math notation="LaTeX">$\mathbb{Z}_4[v]/\langle v^2-v \rangle$ </tex-math> </inline-formula> and Their Gray Images
IEEE Communications Letters
https://doi.org/10.1109/LCOMM.2018.2848942
Polyadic Cyclic Codes over a Non-Chain Ring
Journal of Computer and Communications
https://doi.org/10.4236/jcc.2021.95004
Quadratic residue codes over the ring 𝔽 p [ u ] / 〈 u m − u 〉 $\mathbb {F}_{p}[u]/\langle u^{m}-u\rangle $ and their Gray images
Cryptography and Communications
https://doi.org/10.1007/s12095-017-0223-z
On the linear codes over the ring Rp
Discrete Mathematics, Algorithms and Applications
https://doi.org/10.1142/S1793830916500361
(1−2u 3)-constacyclic codes and quadratic residue codes over F p [ u ] / 〈 u 4 − u 〉 $\mathbb {F}_{p}[u]/\langle u^{4}-u\rangle $
Cryptography and Communications
https://doi.org/10.1007/s12095-016-0184-7
Constacyclic codes over finite commutative semi-simple rings
Finite Fields and Their Applications
https://doi.org/10.1016/j.ffa.2016.11.008
Polyadic constacyclic codes over a non-chain ring $$\mathbb {F}_{q}[u,v]/\langle f(u),g(v), uv-vu\rangle $$
Journal of Applied Mathematics and Computing
https://doi.org/10.1007/s12190-019-01290-x
DNA computing over the ring ℤ4[v]/〈v2 − v〉
International Journal of Biomathematics
https://doi.org/10.1142/S1793524518500900
Some results on $${\mathbb {F}}_4[v]$$-double cyclic codes
Computational and Applied Mathematics
https://doi.org/10.1007/s40314-021-01428-3
Cyclic codes over a non-chain ring R, and their application to LCD codes
Discrete Mathematics
https://doi.org/10.1016/j.disc.2021.112545
Some results on cyclic codes over ℤq + uℤq
Discrete Mathematics, Algorithms and Applications
https://doi.org/10.1142/S1793830915500585
Constacyclic codes over general mixed alphabets and their applications
Designs, Codes and Cryptography
https://doi.org/10.1007/s10623-025-01652-6
Cyclic DNA codes over $$\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2$$ F 2 + u F 2 + v F 2 + u v F 2 and their applications
Journal of Applied Mathematics and Computing
https://doi.org/10.1007/s12190-016-1046-3
ON QUANTUM CODES FROM CYCLIC CODES OVER A CLASS OF NONCHAIN RINGS
Bulletin of the Korean Mathematical Society
https://doi.org/10.4134/BKMS.b150544