A Construction Method of Quantum Error-correcting Codes over <inline-formula><tex-math id="M2">\begin{document}${\boldsymbol F_{{2^m}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="221145_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="221145_M2.png"/></alternatives></inline-formula>

WANG Yu; KAI Xiaoshan; ZHU Shixin

doi:10.11999/JEIT221145

Volume 45 Issue 5

May 2023

Turn off MathJax

Article Contents

Article Navigation > Journal of Electronics & Information Technology > 2023 > 45(5): 1731-1736

WANG Yu, KAI Xiaoshan, ZHU Shixin. A Construction Method of Quantum Error-correcting Codes over ${\boldsymbol F_{{2^m}}}$[J]. Journal of Electronics & Information Technology, 2023, 45(5): 1731-1736. doi: 10.11999/JEIT221145

Citation:

WANG Yu, KAI Xiaoshan, ZHU Shixin. A Construction Method of Quantum Error-correcting Codes over ${\boldsymbol F_{{2^m}}}$[J]. Journal of Electronics & Information Technology, 2023, 45(5): 1731-1736. doi: 10.11999/JEIT221145

Citation:

PDF( 712 KB)

A Construction Method of Quantum Error-correcting Codes over ${\boldsymbol F_{{2^m}}}$

doi: 10.11999/JEIT221145

1.
School of Artificial Intelligence and Big Data, Hefei University, Hefei 230601, China
2.
School of Mathematics, Hefei University of Technology, Hefei 230601, China

Funds: The National Natural Science Foundation of China (12171134, U21A20428), The Key Project of Support Program for Outstanding Young Talents in University of Anhui Province (gxyqZD2021137)

Received Date: 2022-09-01
Rev Recd Date: 2022-11-27

Available Online: 2022-12-02

Publish Date: 2023-05-10

Abstract

Abstract

Constructing quantum codes with good parameters is an important part of quantum error-correcting codes research. In this paper, $ {2^m} $-ary quantum codes are derived through Hermitian dual-containing constacyclic codes over finite non-chain ring $ R = {F_{{4^m}}} + v{F_{{4^m}}} $. A new Gray map $ \phi $ is defined, which is Hermitian dual-containing preserving from a linear code C over R to $ \phi (C) $. The condition for constacyclic codes over R to be Hermitian dual-containing is studied. A method of constructing $ {2^m} $-ary quantum codes is presented, and some new 4-ary and 8-ary quantum codes are obtained.
- Constacyclic codes,
- Quantum codes,
- Finite non-chain ring,
- Gray map

FullText(HTML)

References(17)

References

[1]	CALDERBANK A R, RAINS E M, SHOR P M, et al. Quantum error correction via codes over GF(4)[J]. IEEE Transactions on Information Theory, 1998, 44(4): 1369–1387. doi: 10.1109/18.681315
[2]	ASHIKHMIN A and KNILL E. Nonbinary quantum stabilizer codes[J]. IEEE Transactions on Information Theory, 2001, 47(7): 3065–3072. doi: 10.1109/18.959288
[3]	KAI Xiaoshan, ZHU Shixin, and LI Ping. Constacyclic codes and some new quantum MDS codes[J]. IEEE Transactions on Information Theory, 2014, 60(4): 2080–2086. doi: 10.1109/TIT.2014.2308180
[4]	CHEN Bocong, LING San, and ZHANG Guanghui. Application of constacyclic codes to quantum MDS codes[J]. IEEE Transactions on Information Theory, 2015, 61(3): 1474–1484. doi: 10.1109/TIT.2015.2388576
[5]	朱士信, 黄山, 李锦. 有限域上常循环厄密特对偶包含码及其应用[J]. 电子与信息学报, 2018, 40(5): 1072–1078. doi: 10.11999/JEIT170735 ZHU Shixin, HUANG Shan, and LI Jin. Constacyclic Hermitian dual-containing codes over finite fields and their application[J]. Journal of Electronics &Information Technology, 2018, 40(5): 1072–1078. doi: 10.11999/JEIT170735
[6]	QIAN Jianfa, MA Wenping, and GUO Wangmei. Quantum codes from cyclic codes over finite ring[J]. International Journal of Quantum Information, 2009, 7(6): 1277–1283. doi: 10.1142/S0219749909005560
[7]	QIAN Jianfa. Quantum codes from cyclic codes over $ {F_2} + v{F_2} $ [J]. Journal of Information and Computational Science, 2013, 10(6): 1715–1722. doi: 10.12733/jics20101705
[8]	ASHRAF M and MOHAMMAD G. Construction of quantum codes from cyclic codes over $ {F_p} + v{F_p} $ [J]. International Journal of Information and Coding Theory, 2015, 3(2): 137–144. doi: 10.1504/IJICOT.2015.072627
[9]	GAO Jian and WANG Yongkang. $ u $ -Constacyclic codes over $ F_{P}+u F_{p} $ and their applications of constructing new non-binary quantum codes[J]. Quantum Information Processing, 2018, 17(1): 4. doi: 10.1007/s11128-017-1775-8
[10]	ALAHMADI A, ISLAM H, PRAKASH O, et al. New quantum codes from constacyclic codes over a non-chain ring[J]. Quantum Information Processing, 2021, 20(2): 60. doi: 10.1007/s11128-020-02977-y
[11]	ISLAM H, PRAKASH O, and VERMA R K. New quantum codes from constacyclic codes over the ring $ {R_{k, m}} $ [J]. Advances in Mathematics of Communications, 2022, 16(1): 17–35. doi: 10.3934/amc.2020097
[12]	WANG Yu, KAI Xiaoshan, SUN Zhonghua, et al. Quantum codes from Hermitian dual-containing constacyclic codes over $ \mathbb{F}_{q^{2}}+v \mathbb{F}_{q^{2}} $ [J]. Quantum Information Processing, 2021, 20(3): 122. doi: 10.1007/s11128-021-03052-w
[13]	SHI Xiaoping, HUANG Xinmei, and YUE Qin. Construction of new quantum codes derived from constacyclic codes over $ F_{q^{2}}+u F_{q^{2}}+\cdots+u^{r-1} F_{q^{2}} $ [J]. Applicable Algebra in Engineering, Communication and Computing, 2021, 32(5): 603–620. doi: 10.1007/s00200-020-00415-1
[14]	ISLAM H, PATEL S, PRAKASH O, et al. A family of constacyclic codes over a class of non-chain rings $ {A_{q, r}} $ and new quantum codes[J]. Journal of Applied Mathematics and Computing, 2022, 68(4): 2493–2514. doi: 10.1007/s12190-021-01623-9
[15]	TANG Yongsheng, YAO Ting, SUN Zhonghua, et al. Nonbinary quantum codes from constacyclic codes over polynomial residue rings[J]. Quantum Information Processing, 2020, 19(3): 84. doi: 10.1007/s11128-020-2584-z
[16]	LIU Yan, SHI Minjia, SEPASDAR Z, et al. Construction of Hermitian self-dual constacyclic codes over $ {F_{{q^2}}} + v{F_{{q^2}}} $ [J]. Applied and Computational Mathematics, 2016, 15(3): 359–369.
[17]	EDEL Y. Some good quantum twisted codes[EB/OL]. https://www.mathi.uni-heidelberg.de/～yves/Matritzen/QTBCH/QTBCHIndex.html, 2022.