一种2<i><sup>m</sup></i>元域上量子纠错码的构造方法

王玉; 开晓山; 朱士信

doi:10.11999/JEIT221145

一种2^m元域上量子纠错码的构造方法

doi: 10.11999/JEIT221145

1.
合肥学院人工智能与大数据学院合肥 230601
2.
合肥工业大学数学学院合肥 230601

基金项目: 国家自然科学基金(12171134, U21A20428)，安徽省高校优秀青年人才支持计划项目(gxyqZD2021137)

详细信息

作者简介:
王玉：男，副教授，博士，研究方向为代数编码

开晓山：男，教授，博士生导师，研究方向为代数编码

朱士信：男，教授，博士生导师，研究方向为代数编码理论、信息安全与序列密码等

通讯作者:
王玉　wangyu351@hfuu.edu.cn

中图分类号: TN911.22
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- 文章访问数: 377
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出版历程
- 收稿日期: 2022-09-01
- 修回日期: 2022-11-27
- 网络出版日期: 2022-12-02
- 刊出日期: 2023-05-10

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

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)

摘要

摘要: 构造具有良好参数的量子码是量子纠错码研究的重要内容。该文利用有限非链环 $R = {F_{{4^m}}} + v{F_{{4^m}}}$ 上的厄米特对偶包含常循环码来构造 ${2^m}$ 元量子码。定义了一种新的Gray 映射 $\phi$ ，能够将环 $R$ 上线性码 $C$ 的厄米特对偶包含性保持到 $\phi (C)$ 上。研究了环 $R$ 上常循环码是厄米特对偶包含码的条件。给出了一种构造 ${2^m}$ 元量子码的方法，并构造了一些新的4元和8元量子码。
- 常循环码 /
- 量子码 /
- 有限非链环 /
- Gray映射
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

HTML全文

表 1 码长为34的新的4元量子码

${g_1}(x)$	${g_2}(x)$	$\phi (C)$	${[[n,k,d]]_4}$
$1{\omega ^3}1$	$(1{\omega ^2}{\omega ^3})(1{\omega ^3}{\omega ^3})$	[34,28,5]₁₆	[[34,22, $\ge$ 5]]₄
$(1{\omega ^3}1)(1{\omega ^6}1)$	$(1{\omega ^2}{\omega ^3})(1{\omega ^3}{\omega ^3})$	[34,26,6]₁₆	[[34,18, $\ge$ 6]]₄
$(1{\omega ^3}1)(1{\omega ^6}1)$	$(1{\omega ^3}{\omega ^3})(1{\omega ^{11}}{\omega ^3})(1{\omega ^{13}}{\omega ^3})$	[34,24,7]₁₆	[[34,14, $\ge$ 7]]₄
$(1{\omega ^3}1)(1{\omega ^6}1)(1\omega 1)$	$(1{\omega ^3}{\omega ^3})(1{\omega ^{11}}{\omega ^3})(1{\omega ^{13}}{\omega ^3})$	[34,22,8]₁₆	[[34,10, $\ge$ 8]]₄
$(1{\omega ^3}1)(1{\omega ^6}1)(1\omega 1)$	$(1{\omega ^3}{\omega ^3})(1{\omega ^{11}}{\omega ^3})(1{\omega ^{13}}{\omega ^3})(1{\omega ^6}{\omega ^3})$	[34,20,9]₁₆	[[34,6, $\ge$ 9]]₄

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表 2 新的4元量子码

$n$	$\lambda$	${g_1}(x)$	${g_2}(x)$	$\phi (C)$	${[[n,k,d]]_4}$	${[[n',k',d']]_4}$
3	$1 + v + v{\omega ^3}$	$1{\omega ^5}$	$1\omega$	${[6,4,3]_{16}}$	${[[6,2,3]]_4}$	MDS
7	$1 + v + v{\omega ^3}$	$1011$	$1{\omega ^9}0{\omega ^{12}}$	${[14,8,6]_{16}}$	${[[14,2, \ge 6]]_4}$	${[[14,0,4]]_4}{\text{ }}$ ^[15]
11	$1 + v + v{\omega ^3}$	$1{\omega ^5}11{\omega ^{10}}1$	$1{\omega ^8}{\omega ^6}{\omega ^9}{\omega ^7}1$	${[22,12,7]_{16}}$	${[[22,2, \ge 7]]_4}$	${[[24,0,6]]_4}$ ^[17]
15	$1$	$(1\omega )(1{\omega ^2})$	$1{\omega ^4}$	${[30,27,3]_{16}}$	${[[30,24, \ge 3]]_4}$	${[[31,21,3]]_4}$ ^[17]
17	$1$	$1{\omega ^3}1$	$1{\omega ^6}1$	${[34,30,4]_{16}}$	${[[34,26, \ge 4]]_4}$	${[[34,24,4]]_4}$ ^[15]
19	$1 + v + v{\omega ^3}$	$1{\omega ^{10}}0{\omega ^{10}}{\omega ^{10}}{\omega ^5}{\omega ^5}{\omega ^5}1$	$1{\omega ^7}0\omega {\omega ^{13}}{\omega ^5}{\omega ^2}{\omega ^{11}}{\omega ^8}$	${[38,20,11]_{16}}$	${[[38,2, \ge 11]]_4}$	${[[40,2,8]]_4}$ ^[17]
45	$1$	$(1\omega )(100{\omega ^4})$	$(1{\omega ^2})(100{\omega ^5})$	${[90,82,4]_{16}}$	${[[90,74, \ge 4]]_4}$	${[[90,66,4]]_4}$ ^[17]
63	$1 + v + v{\omega ^3}$	$111{\omega ^5}$	$1\omega$	${[126,122,3]_{16}}$	${[[126,118, \ge 3]]_4}$	${[[127,113,3]]_4}$ ^[17]
77	$1$	$1{\omega ^5}11{\omega ^{10}}1$	$1011$	${[154,146,4]_{16}}$	${[[154,138, \ge 4]]_4}$	${[[154,128,4]]_4}$ ^[17]
85	$1$	$(1{\omega ^2}{\omega ^3})(1{\omega ^4}{\omega ^6})$	$(1{\omega ^9}{\omega ^9})(1{\omega ^8}{\omega ^{12}})$	${[170,162,4]_{16}}$	${[[170,154, \ge 4]]_4}$	${[[171,151,4]]_4}$ ^[17]
91	$1 + v + v{\omega ^3}$	$1{\omega ^4}{\omega ^{13}}1$	$(1{\omega ^3}{\omega ^8}{\omega ^9})(1{\omega ^7}{\omega ^4}{\omega ^9})$	${[182,173,5]_{16}}$	${[[182,164, \ge 5]]_4}$	$[[185,149,5]]_4$ ^[17]

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表 3 新的8元量子码

$n$	$\lambda$	${g_1}(x)$	${g_2}(x)$	$\phi (C)$	${[[n,k,d]]_8}$	${[[n',k',d']]_8}$
$5$	$1 + v + v{\omega ^7}$	$1{\omega ^{42}}1$	$1{\omega ^{56}}{\omega ^{28}}$	${[10,6,5]_{64}}$	${[[10,2,5]]_8}$	MDS
$7$	$1 + v + v{\omega ^{21}}$	$1{\omega ^9}$	$1{\omega ^3}$	${[14,12,3]_{64}}$	${[[14,10,3]]_8}$	MDS
$7$	$1 + v + v{\omega ^{21}}$	$1{\omega ^9}$	$(1{\omega ^3})(1{\omega ^{12}})$	${[14,11,4]_{64}}$	${[[14,8,4]]_8}$	MDS
$21$	$1 + v + v{\omega ^{21}}$	$1{\omega ^3}$	$1\omega$	${[42,40,3]_{64}}$	${[[42,38,3]]_8}$	MDS
$35$	$1 + v + v{\omega ^{21}}$	$(1{\omega ^9})(1{\omega ^{57}}{\omega ^9})$	$1{\omega ^6}$	${[70,66,4]_{64}}$	${[[70,62, \ge 4]]_8}$	${[[70,46,3]]_8}$ ^[17]
$39$	$1 + v + v{\omega ^{21}}$	$(1{\omega ^{47}}{\omega ^{42}})(1{\omega ^{31}}{\omega ^{21}})$	$(1{\omega ^{27}}{\omega ^{35}})(1{\omega ^{45}}{\omega ^{14}})$	${[78,70,5]_{64}}$	${[[78,62, \ge 5]]_8}$	${[[78,46,5]]_8}$ ^[17]
$49$	$1 + v + v{\omega ^7}$	$1{\omega ^9}$	$(1{\omega ^{22}})(1{\omega ^{31}})$	${[98,95,3]_{64}}$	${[[98,92, \ge 3]]_8}$	${[[99,89,3]]_8}$ ^[17]
$63$	$1$	$(1\omega )(1{\omega ^2})$	$(1{\omega ^3})(1{\omega ^4})$	${[126,122,4]_{64}}$	${[[126,118, \ge 4]]_8}$	${[[127,113,3]]_8}$ ^[17]
$65$	$1 + v + v{\omega ^{21}}$	$(1{\omega ^4}1)(1{\omega ^8}1)$	$(1{\omega ^{52}}{\omega ^{21}})(1{\omega ^{19}}{\omega ^{21}})$	${[130,122,5]_{64}}$	${[[130,114, \ge 5]]_8}$	${[[133,113,5]]_8}$ ^[17]
$73$	$1 + v + v{\omega ^7}$	$1{\omega ^{36}}01$	$10{\omega ^{50}}{\omega ^{21}}$	${[146,140,4]_{64}}$	${[[146,134, \ge 4]]_8}$	${[[147,122,4]]_8}$ ^[17]
$91$	$1 + v + v{\omega ^7}$	$(1{\omega ^9})(1{\omega ^{31}}{\omega ^{36}})$	$(1{\omega ^{52}})(1{\omega ^{44}}{\omega ^{50}})(1{\omega ^{61}})$	${[182,175,5]_{64}}$	${[[182,168, \ge 5]]_8}$	${[[183,143,4]]_8}$ ^[17]

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参考文献(17)

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[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
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一种2^m元域上量子纠错码的构造方法

doi: 10.11999/JEIT221145

作者简介:
王玉：男，副教授，博士，研究方向为代数编码

开晓山：男，教授，博士生导师，研究方向为代数编码

朱士信：男，教授，博士生导师，研究方向为代数编码理论、信息安全与序列密码等

通讯作者:
王玉　wangyu351@hfuu.edu.cn

计量

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

计量

目录

留言板

一种2m元域上量子纠错码的构造方法

doi: 10.11999/JEIT221145

作者简介: 王玉：男，副教授，博士，研究方向为代数编码 开晓山：男，教授，博士生导师，研究方向为代数编码 朱士信：男，教授，博士生导师，研究方向为代数编码理论、信息安全与序列密码等

通讯作者: 王玉 wangyu351@hfuu.edu.cn

计量

出版历程

A Construction Method of Quantum Error-correcting Codes over F2m{\boldsymbol F_{{2^m}}}

计量

出版历程

目录

一种2^m元域上量子纠错码的构造方法

作者简介:
王玉：男，副教授，博士，研究方向为代数编码

开晓山：男，教授，博士生导师，研究方向为代数编码

朱士信：男，教授，博士生导师，研究方向为代数编码理论、信息安全与序列密码等

通讯作者:
王玉　wangyu351@hfuu.edu.cn

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