On the complete weight distributions of quantum error-correcting codes

doi:10.1088/1674-1056/acac09

中国物理B ›› 2023, Vol. 32 ›› Issue (5): 50307-050307.doi: 10.1088/1674-1056/acac09

On the complete weight distributions of quantum error-correcting codes

Chao Du(杜超)^1,2, Zhi Ma(马智)^1,2,†, and Maosheng Xiong(熊茂胜)^3,‡

1 State Key Laboratory of Mathematical Engineering and Advanced Computing, Zhenghzhou 450001, China;
2 Henan Key Laboratory of Network Cryptography Technology, Zhenghzhou 450001, China;
3 Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China

收稿日期:2022-09-07 修回日期:2022-12-15 接受日期:2022-12-16 出版日期:2023-04-21 发布日期:2023-05-05
通讯作者: Zhi Ma, Maosheng Xiong E-mail:ma_zhi@163.com;mamsxiong@ust.hk
基金资助:
Project supported by the National Natural Science Foundation of China (Grant Nos. 61972413, 61901525, and 62002385), the National Key R&D Program of China (Grant No. 2021YFB3100100), and RGC under Grant No. N HKUST619/17 from Hong Kong, China.

On the complete weight distributions of quantum error-correcting codes

Chao Du(杜超)^1,2, Zhi Ma(马智)^1,2,†, and Maosheng Xiong(熊茂胜)^3,‡

1 State Key Laboratory of Mathematical Engineering and Advanced Computing, Zhenghzhou 450001, China;
2 Henan Key Laboratory of Network Cryptography Technology, Zhenghzhou 450001, China;
3 Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China

Received:2022-09-07 Revised:2022-12-15 Accepted:2022-12-16 Online:2023-04-21 Published:2023-05-05
Contact: Zhi Ma, Maosheng Xiong E-mail:ma_zhi@163.com;mamsxiong@ust.hk
Supported by:
Project supported by the National Natural Science Foundation of China (Grant Nos. 61972413, 61901525, and 62002385), the National Key R&D Program of China (Grant No. 2021YFB3100100), and RGC under Grant No. N HKUST619/17 from Hong Kong, China.

摘要/Abstract

摘要： In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the MacWilliams identities, and as applications they showed how such weight distributions may be used to obtain the singleton-type and hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the MacWilliams identities of the complete weight distributions of quantum codes. We obtain new information about the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes. We get new identities involving the complete weight distributions of two different quantum codes. We estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.

关键词: quantum codes, complete weight distributions, MacWilliams identities, BCH codes

Abstract: In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the MacWilliams identities, and as applications they showed how such weight distributions may be used to obtain the singleton-type and hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the MacWilliams identities of the complete weight distributions of quantum codes. We obtain new information about the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes. We get new identities involving the complete weight distributions of two different quantum codes. We estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.

Key words: quantum codes, complete weight distributions, MacWilliams identities, BCH codes

中图分类号: (Quantum computation architectures and implementations)

03.67.Lx

03.67.Pp (Quantum error correction and other methods for protection against decoherence)

Chao Du(杜超), Zhi Ma(马智), and Maosheng Xiong(熊茂胜). On the complete weight distributions of quantum error-correcting codes[J]. 中国物理B, 2023, 32(5): 50307-050307.

Chao Du(杜超), Zhi Ma(马智), and Maosheng Xiong(熊茂胜). On the complete weight distributions of quantum error-correcting codes[J]. Chin. Phys. B, 2023, 32(5): 50307-050307.

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On the complete weight distributions of quantum error-correcting codes

On the complete weight distributions of quantum error-correcting codes

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