Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON QUANTUM CODES FROM CYCLIC CODES OVER A CLASS OF NONCHAIN RINGS

  • Sari, Mustafa (Department of Mathematics Faculty of Art and Sciences Yildiz Technical University) ;
  • Siap, Irfan (Department of Mathematics Faculty of Art and Sciences Yildiz Technical University)
  • Received : 2015.07.09
  • Published : 2016.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we extend the results given in [3] to a nonchain ring $R_p={\mathbb{F}}_p+v{\mathbb{F}}_p+{\cdots}+v^{p-1}{\mathbb{F}}_p$, where $v^p=v$ and p is a prime. We determine the structure of the cyclic codes of arbitrary length over the ring $R_p$ and study the structure of their duals. We classify cyclic codes containing their duals over $R_p$ by giving necessary and sufficient conditions. Further, by taking advantage of the Gray map ${\pi}$ defined in [4], we give the parameters of the quantum codes of length pn over ${\mathbb{F}}_p$ which are obtained from cyclic codes over $R_p$. Finally, we illustrate the results by giving some examples.

Keywords

Acknowledgement

Supported by : Yildiz Technical University

References

  1. S. A. Aly, A. Klappenecker, and P. K. Sarvepalli, Primitive quantum BCH codes over finite fields, Proc. IEEE Int. Symp. Inf. Theory, Seattle, WA, pp. 1105-1108, 2006.
  2. A. Ashikhmin and E. Knill, Nonbinary quantum stabilizer codes, IEEE Trans. Inform. Theory 47 (2000), 3065-3072.
  3. M. Ashraf and G. Mohammad, Quantum codes from cyclic codes over ${\mathbb{F}}_3+v{\mathbb{F}}_3$, Int. J. Quantum Inf. 12 (2014), no. 6, 1450042, 8 pp.
  4. A. Bayram and I. Siap, Cyclic and constacyclic codes over a non-chain ring, J. Algebra Comb. Discrete Struct. Appl. 1 (2014), no. 1, 1-12.
  5. A. R. Calderbank, E. M. Rains, P. M. Shor, and N. J. A. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory 44 (1998), no. 4, 1369-1387. https://doi.org/10.1109/18.681315
  6. A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54 (1996), no. 2, 1098-1105. https://doi.org/10.1103/PhysRevA.54.1098
  7. A. Dertli, Y. Cengellenmis, and S. Eren, On quantum codes obtained from cyclic codes over ${\mathbb{F}}_2+v{\mathbb{F}}_2+v^2{\mathbb{F}}_2$, http://arxiv.org/abs/1407.1232v1.
  8. S. J. Devitt, W. J. Munro, and K. Nemoto, Quantum Error Correction for Beginners, http://arxiv.org/pdf/0905.2794v4.pdf. https://doi.org/10.1088/0034-4885/76/7/076001
  9. D. Gottesman, Stabilizer codes and quantum error correction, Caltech Ph. D. Thesis, eprint:quant-ph/9705052.
  10. A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli, Nonbinary stabilizer codes over finite fields, IEEE Trans. Inform. Theory 52 (2006), no. 11, 4892-4914. https://doi.org/10.1109/TIT.2006.883612
  11. C. Y. Lai and C. C. Lu, A construction of quantum stabilizer codes based on syndrome assignment by classical parity-check matrices, IEEE Trans. Inform. Theory 57 (2011), no. 10, 7163-7179. https://doi.org/10.1109/TIT.2011.2165812
  12. J. Qian, Quantum codes from cyclic codes over ${\mathbb{F}}_2+v{\mathbb{F}}_2$, J. Inform. Comput. Sci. 10 (2013), no. 6, 1715-1722. https://doi.org/10.12733/jics20101705
  13. P. W. Shor, Scheme for reducing decoherence in quantum memory, Phys. Rev. A 52 (1995), 2493-2496. https://doi.org/10.1103/PhysRevA.52.R2493
  14. A. M. Steane, Enlargement of Calderbank-Shor-Steane quantum codes, IEEE Trans. Inform. Theory 45 (1999), no. 7, 2492-2495. https://doi.org/10.1109/18.796388
  15. V. D. Tonchev, The existence of optimal quaternary [28, 20, 6] and quantum [[28, 12, 6]] codes, J. Algebra Comb. Discrete Appl. 1 (2014), no. 1, 13-17.
  16. Y. Xunru and M. Wenping, Gray map and quantum codes over the ring ${\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2$, in Proc. IEEE Int. Conf. on Trust, Security and Privacy in Computing and Communications, Changsha, China(IEEE Computer Society Press), pp. 897-899, 2011.

Cited by

  1. Quantum codes over Fp from cyclic codes over Fp[u, v]/〈u2 − 1, v3 − v, uv − vu〉 pp.1936-2455, 2018, https://doi.org/10.1007/s12095-018-0299-0