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.
| Published in | International Journal of Discrete Mathematics (Volume 4, Issue 1) |
| DOI | 10.11648/j.dmath.20190401.18 |
| Page(s) | 52-56 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Gray Map, Cyclic Codes, New Non-binary Quantum Codes
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APA Style
Leilei Gao. (2019). New Non-binary Quantum Codes Over Fq+uFq+vFq+uvFq. International Journal of Discrete Mathematics, 4(1), 52-56. https://doi.org/10.11648/j.dmath.20190401.18
ACS Style
Leilei Gao. New Non-binary Quantum Codes Over Fq+uFq+vFq+uvFq. Int. J. Discrete Math. 2019, 4(1), 52-56. doi: 10.11648/j.dmath.20190401.18
AMA Style
Leilei Gao. New Non-binary Quantum Codes Over Fq+uFq+vFq+uvFq. Int J Discrete Math. 2019;4(1):52-56. doi: 10.11648/j.dmath.20190401.18
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Download@article{10.11648/j.dmath.20190401.18,
author = {Leilei Gao},
title = {New Non-binary Quantum Codes Over Fq+uFq+vFq+uvFq},
journal = {International Journal of Discrete Mathematics},
volume = {4},
number = {1},
pages = {52-56},
doi = {10.11648/j.dmath.20190401.18},
url = {https://doi.org/10.11648/j.dmath.20190401.18},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20190401.18},
abstract = {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.},
year = {2019}
}
TY - JOUR T1 - New Non-binary Quantum Codes Over Fq+uFq+vFq+uvFq AU - Leilei Gao Y1 - 2019/05/06 PY - 2019 N1 - https://doi.org/10.11648/j.dmath.20190401.18 DO - 10.11648/j.dmath.20190401.18 T2 - International Journal of Discrete Mathematics JF - International Journal of Discrete Mathematics JO - International Journal of Discrete Mathematics SP - 52 EP - 56 PB - Science Publishing Group SN - 2578-9252 UR - https://doi.org/10.11648/j.dmath.20190401.18 AB - 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. VL - 4 IS - 1 ER -
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