Volume 4, Issue 1, June 2019, Page: 52-56
New Non-binary Quantum Codes Over Fq+uFq+vFq+uvFq
Leilei Gao, Department of Mathematics, School of Mathematics and Statistics, Shandong University of Technology, Zibo, China
Received: Feb. 24, 2019;       Accepted: Apr. 10, 2019;       Published: May 6, 2019
DOI: 10.11648/j.dmath.20190401.18      View  816      Downloads  104
Let R= Fq+uFq+vFq+uvFq be a commutative ring with u2=u, v2=v, uv=vu, where q is a power of an odd prime. Ashraf and Mohammad constructed some new quantum codes from cyclic codes. Under this background, another Gray map from R to Fq4 is given. This map can be naturally extended to Rn. The problem on the ring turns to the field by this isomorphic map now. Therefore, This mapping is obviously a weight-preserving and distance-preserving map. The results show that the codes after mapping are self-orthogonal codes over Fq if they are self-orthogonal codes over R. Some computational examples show that some better non-binary quantum codes can be obtained under this Gray map. We discuss the structure of linear codes. On this basis, the structure of the generating matrix of linear codes is obtained. The structure of their dual codes is also obtained. The CSS construction guarantees the existence of quantum codes. Finally, with the help of the CSS construction, we get some good quantum codes. By comparison, our quantum codes have better parameters.
Gray Map, Cyclic Codes, New Non-binary Quantum Codes
To cite this article
Leilei Gao, New Non-binary Quantum Codes Over Fq+uFq+vFq+uvFq, International Journal of Discrete Mathematics. Vol. 4, No. 1, 2019, pp. 52-56. doi: 10.11648/j.dmath.20190401.18
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Shor, P. W.: Scheme for reducing decoherence in quantum memory, Phys. Rev. A, 52, 2493-2496 (1995).
Tang, Y., Zhu, S., Kai, X., Ding, J.: New quantum codes from dual-containing cyclic codes over finite rings, Quantum Inf. Process., 15, 4489-4500 (2016).
Qian, J., Ma, W., Gou, W.: Quantum codes from cyclic codes over finite ring, Int. J. Quantum Inf., 7, 1277-1283 (2009).
Kai, X., Zhu, S.: Quaternary construction of quantum codes from cyclic codes over F4+uF4, Int. J. Quantum Inf., 9, 689-700 (2011).
Ashraf, M., Mohammad, G.: Quantum codes from cyclic codes over F3+vF3, Int. J. Quantum Inf., 12, 1450042 (2014)
Ashraf, M., Mohammad, G.: Quantum codes from cyclic codes over Fp+vFp, Int. J. Inf. Coding Theory, 3 (2), 137-144 (2015)
Ashraf, M., Mohammad, G.: Quantum codes from cyclic codes over Fq+uFq+vFq+uvFq, Quantum Inf Process, 15, 4089-4098 (2016)
Gao, J.: Quantum codes from cyclic codes over Fq+vFq+v2Fq+v3Fq, Int. J. Quantum Inf., 13 (8), 1550063 (1-8) (2015).
Ozen, M., Ozzaim, N. T., Ince, H.: Quantum codes from cyclic codes over F3+uF3+vF3+uvF3, International Conference on Quantum Science and Applications, Journal of Physics: Conference Series, 766, 0120202, (2016).
Gao, J., Wang, Y.: Quantum codes derived from negacyclic codes, Int. J. Theor. Phys. 57, 682-686 (2018)
Gao, Y. Gao, J. Fu, F-W.: Quantum codes from cyclic codes over the ring Fq+v1Fq+…+ vrFq, Appl. Algebra Eng. Comm., doi: 10.1007/s00200-018-0366-y (2018).
La Guardia, G. G.: Quantum codes derived from cyclic codes. Int. J. Theor. Phys., 56, 2479-2484 (2017)
Ma, F., Gao, J., Fu, F-W.: Constacyclic codes over the ring Fq+vFq+V2Fq and their applications of constructing new non-binary quantum codes, Quantum Inf. Process., 17: 122 (2018)
Calderbank, A. R., Rains, E. M., Shor, P. M., Sloane, N. J. A.: Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44, 1369-1387 (1998)
Browse journals by subject