利用扭曲的Reed-Solomon码构造两类极大距离可分纠缠辅助量子纠错码

潘鑫; 高健

doi:10.11999/JEIT250258

利用扭曲的Reed-Solomon码构造两类极大距离可分纠缠辅助量子纠错码

doi: 10.11999/JEIT250258 cstr: 32379.14.JEIT250258

潘鑫,
高健^,

山东理工大学数学与统计学院淄博 255000

基金项目: 山东省自然科学基金(ZR2024YQ057, ZR2022MA024)，国家自然科学基金(12071264)

详细信息

作者简介:
潘鑫：女，硕士生，研究方向为编码理论及其应用

高健：男，副教授，博士，博士生导师，研究方向为编码理论及其应用

通讯作者:
高健　dezhougaojian@163.com

中图分类号: TN911.22
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- 被引次数: 0
出版历程
- 收稿日期: 2025-04-11
- 修回日期: 2025-07-27
- 网络出版日期: 2025-08-06

Constructing Two Classes of Maximum Distance Separable Entanglement-Assisted Quantum Error-Correcting Codes by Using Twisted Generalized Reed-Solomon Codes

PAN Xin,
GAO Jian^,

School of Mathematics and Statistics, Shandong University of Technology, Zibo 255000, China

Funds: The Natural Science Foundation of Shandong Province (ZR2024YQ057, ZR2022MA024), The National Natural Science Foundation of China (12071264)

摘要

摘要: 随着量子通信和量子计算技术的飞速发展，高效量子纠错编码技术已成为保障量子系统可靠性的核心需求。传统量子纠错码在参数灵活性和最小距离约束方面存在显著局限性，难以适应复杂量子通信场景中的动态需求。该文基于扭曲的Reed-Solomon(TGRS)码，根据码长中$ i $的奇偶性的不同具体讨论矩阵$ \boldsymbol{G}{\boldsymbol{G}}^{\mathrm{H}} $的秩，进一步通过分析该矩阵的秩确定厄米特正交包的维数，从而得到两类极大距离可分纠缠辅助量子纠错码(MDS EAQECCs)。研究发现，通过特定的扭曲操作，所构造的两类MDS EAQECCs不仅能够灵活调整码长，还能显著提升最小距离，突破了传统理论界限。此外，该文利用扭曲操作将两类MDS EAQECCs提升为最大纠缠态极大距离可分纠缠辅助量子纠错码 (ME-MDS EAQECCs)。该文研究成果不仅为量子纠错码设计提供了更广泛的参数选择，还为动态量子通信场景中的高效纠错提供了理论支撑。
- 纠缠辅助量子码 /
- 扭曲的Reed-Solomon码 /
- 厄米特正交包
Abstract: Objective With the rapid development of quantum communication and computation, efficient error-correction technologies are critical for ensuring the reliability of quantum systems. Maximal Distance Separable (MDS) Entanglement-Assisted Quantum Error-Correcting Codes (EAQECCs) with flexible code lengths and larger minimum distances offer significant advantages in enhancing quantum system robustness. However, classical quantum codes face limitations in parameter flexibility and minimum distance constraints. This study addresses these challenges by leveraging Twisted Generalized Reed-Solomon (TGRS) codes to construct novel MDS EAQECCs, aiming to improve performance in complex quantum communication scenarios. In this paper, TGRS codes are innovatively applied to construct MDS EAQECCs. Different from polynomial construction methods, we determine the dimension of the Hermitian Hull through special matrix rank analysis, and successfully construct $ q $-ary MDS EAQECCs with a minimum distance exceeding$ q+1 $. Two construction schemes with flexible code lengths are proposed. Notably, the minimum distances of the constructed $ q $-ary codes all exceed $ q+1 $. Table 2 systematically summarizes the known MDS EAQECCs with a length less than $ {q}^{2} $ and a minimum distance exceeding $ q+1 $. The novelty of the construction schemes in this paper is highlighted through parameter comparison. Reasoning shows that our schemes achieve flexible adjustment of the code length while maintaining the advantage of the minimum distance. Methods The proposed approach integrates TGRS codes with algebraic techniques to design MDS EAQECCs. Two families of MDS EAQECCs are constructed by using TGRS codes over finite fields. The key steps are as follows: (1) TGRS Code Construction: Utilizing twisted polynomials and hook parameters to generate TGRS codes with adjustable dimensions and minimum distances. (2) Hermitian Hull Analysis: Applying matrix rank analysis to determine the dimensions of the Hermitian Hull of the constructed codes, which is crucial for satisfying the Singleton bound in quantum codes. (3) Twisted Operation Optimization: Employing twisted operations to transform the constructed MDS EAQECCs into ME-MDS EAQECCs. Results and Discussions This paper constructs two families of MDS EAQECCs using TGRS codes and gives certain twisted conditions under which the codes are ME-MDS EAQECCs. Compared with other known codes, these new codes have more flexible parameters and significant advantages in terms of code length, minimum distance, and maximum entanglement state. This paper constructs $ q $-ary EAQECCs with $ [[i(q-1), $$ i(q-1)-2j-q-t+2v+2,q+j+1;q-t+2v+2]{]}_{q} $ when $ i $ is odd and $ [[i(q-1),i(q-1)- 2j-q-t+ $$ 2v+3,q+j+1;q-t+2v+3]{]}_{q} $ when $ i $ is even. Based on Theorem 1 and Theorem 2, several types of MDS EAQECCs are obtained, and their parameters are listed in Table 1.Theorems 3-6 give the existence conditions of specific $ q $-ary EAQECCs under different parameter settings. Furthermore, this paper upgrades the MDS EAQECCs to ME-MDS EAQECCs with $ \left[\right[i(q-1),i(q-1)-q-j,q+j+1;q+j]{]}_{q} $ by a twisted operation. Meanwhile, this paper constructs $ q $-ary EAQECCs with $ \left[\right[i(q+1),(i-1)(q+1)-s,q+2;q-s+1]{]}_{q} $. Moreover, this paper upgrades the MDS EAQECCs to ME-MDS EAQECCs with $ [[i(q+1),(i-1) $$ (q+1),q+2;q+1]{]}_{q} $ by a twisted operation. Theorem 7 gives the dimension of the Hermitian Hull of this RS code after the twisted operation on its generating matrix. Similar to the first construction, a new twisted operation is applied, which upgrades the MDS EAQECCs to ME-MDS EAQECCs with specific parameters. Theorems 8, 9 give the existence conditions of specific $ q $-ary EAQECCs under different parameter settings based on this construction method. Conclusions Two families of MDS EAQECCs are constructed using TGRS codes, and the parameters of known MDS EAQECCs are systematically summarized. Comparative analysis reveals that the EAQECCs proposed in this paper offer the following advantages: Compared with the known $ q $-ary MDS EAQECCs (Table 2), the parameters of the MDS EAQECCs constructed here are new and have not been covered by previous studies; The codes not only enable flexible code-length adjustments but also achieve a minimum distance that significantly exceeds traditional theoretical bounds; Under specific twisted conditions, the constructed MDS EAQECCs are upgraded to ME-MDS EAQECCs. By introducing the twisted operation on RS codes, more flexible parameter combinations are obtained. This provides greater flexibility in the design of quantum error-correcting codes, enabling better adaptation to different quantum communication requirements. This improvement further optimizes the performance of quantum error-correcting codes, enhancing the entanglement-assisted error-correcting ability and improving the overall efficiency of quantum systems. These research results indicate that TGRS codes are important theoretical tools for constructing high-performance EAQECCs with excellent parameters. They play a pivotal part in advancing the development of quantum communication technology. Moreover, they offer a firm theoretical underpinning for the practical implementation of quantum error-correcting codes. Future research can focus on further exploring the potential of TGRS codes in constructing more advanced quantum error-correcting codes and expanding their applications in different quantum communication scenarios.
- Entanglement-assisted quantum codes /
- Twisted generalized reed-solomon codes /
- Hermitian hull

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表 1 本文构造的MDS EAQECCs

构造类	构造参数	参数来源
1	$ \left[\right[120, 58, 37; 10]{]}_{25} $	定理3
	$ \left[\right[156, 106, 40; 28]{]}_{27} $	定理3
	$ \left[\right[288, 194, 73; 50]{]}_{49} $	定理3
	$ \left[\right[160, 2, 121; 82]{]}_{81} $	定理4
	$ \left[\right[78, 11, 40; 11]{]}_{27} $	定理5
	$ \left[\right[72, 22, 40; 28]{]}_{25} $	定理5
	$ \left[\right[496, 250, 187; 126]{]}_{125} $	定理6
	$ \left[\right[32, 18, 13; 10]{]}_{9} $	定理6
	$ \left[\right[32, 20, 13; 12]{]}_{9} $	定理3的ME提升
	$ \left[\right[78, 39, 40; 39]{]}_{25} $	定理3的ME提升
2	$ \left[\right[130, 103, 27; 25]{]}_{25} $	定理8
	$ \left[\right[30, 23, 7; 5]{]}_{5} $	定理8
	$ \left[\right[630, 503, 127; 125]{]}_{125} $	定理8
	$ \left[\right[84, 55, 29; 27]{]}_{27} $	定理9
	$ \left[\right[12, 7, 5; 3]{]}_{3} $	定理9
	$ \left[\right[30, 19, 11; 9]{]}_{9} $	定理9
	$ \left[\right[12, 8, 5; 4]{]}_{3} $	定理8的ME提升
	$ \left[\right[30, 24, 7; 6]{]}_{5} $	定理8的ME提升

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表 2 与已知$ q $-元MDS EAQECCs的参数对比

构造类	$ n $	$ d $	$ c $	限制条件	文献
1	$ 2\mathrm{\lambda }\left(q-1\right) $	$ d $	$ 2i $	$ i\in \left\{\mathrm{1,2}\right\} $, $ \mathrm{\lambda } $为奇数, $ \mathrm{\lambda }\|\left(q-1\right) $, $ 8\|\left(q+1\right) $ $ \dfrac{q-1}{2}\left(i-1\right)+4\mathrm{\lambda }+1\le d\le \dfrac{q-1}{2}+2\left(i+1\right)\mathrm{\lambda } $	[30]
2	$ \dfrac{{q}^{2}-1}{\mathrm{\lambda }} $	$ y+1 $	$ y $	$ 1\le y\le \left\lfloor\dfrac{{q}^{2}-1}{2\mathrm{\lambda }}\right\rfloor $	[31]
3	$ {q}^{2}+i $	$ d $	1	$ 2\le d\le 2q+i-1),i\in \{-\mathrm{1,1}\} $,	[32]
	$ \dfrac{{q}^{2}-1}{i} $	$ d $	2	$ \dfrac{q+1}{2}+2\le d\le \dfrac{3}{2}q-\dfrac{1}{2} $
4	$ {q}^{2}+1 $	$ 2q+2 $	5	$ q\equiv 3\left(\mathrm{m}\mathrm{o}\mathrm{d}4\right) $	[33]
	$ {q}^{2}+1 $	$ 2\mathrm{\lambda }+2 $	9	$ q\equiv 3\left(\mathrm{m}\mathrm{o}\mathrm{d}4\right) $, $ q+1\le \mathrm{\lambda }\le 2q-2 $
	$ \dfrac{{q}^{2}-1}{4} $	$ d $	4	$ q\equiv 3\left(\mathrm{m}\mathrm{o}\mathrm{d}4\right) $, $ \dfrac{3q+7}{4}\le d\le \dfrac{5q+1}{4} $
5	$ \dfrac{\left(q-1\right)\left(q+1\right)}{8} $	$ d $	8	$ q+1\le d\le \dfrac{9q-1}{8} $	[34]
6	$ \dfrac{{q}^{2}-1}{i} $	$ d $	$ i $	$ i\in \{\mathrm{3,5},7\},q+1\le d\le \dfrac{\left(i+1\right)\left(q+1\right)}{j}-1 $	[35]
	$ \dfrac{{q}^{2}-1}{j} $	$ d $	7	$ d=\dfrac{8\left(q+1\right)}{j}-1 $
7	$ \dfrac{{q}^{2}+1}{2} $	$ 3q-1 $	13	$ q\ge 5 $	[36]
		$ 4q- 2 $	25	$ q\ge 7 $
		$ 6q- 4 $	61	$ q\ge 11 $
8	$ i\left(q-1\right) $	$ q+j+1 $	$ q-t+2v+2 $	$ {i}^{2}\equiv 1 \left(\text{mod} p\right) $或$ i\equiv 0 \left(\text{mod} p\right) $, $ i $为奇数, $ j=\dfrac{q-3}{2} $, $ v=\left\lfloor\dfrac{\left(j+1\right)t}{q+1}\right\rfloor $, $ t=\mathrm{gcd}\left(i,q+1\right) $	定理3
	$ i\left(q-1\right) $	$ q+j+1 $	$ q-t+2v+3 $	$ {i}^{2}\equiv 1 \left(\text{mod} p\right) $或$ i\equiv 0 \left(\text{mod} p\right) $, $ i $为偶数, $ j=\dfrac{q-3}{2} $, $ v=\left\lfloor\dfrac{\left(j+1\right)t}{q+1}\right\rfloor $, $ t=\mathrm{gcd}\left(i,q+1\right) $
	$ i\left(q-1\right) $	$ q+j+1 $	$ q+j $	$ c=d-1 $	ME提升
	$ 2\left(q-1\right) $	$ q+j+1 $	$ q-t+2v+3 $	$ c=d-1 $，$ i=2 $
	$ 3\left(q-1\right) $	$ q+j+1 $	$ q-t+2v+2 $	$ c=d-1 $，$ i=3 $
9	$ i\left(q+1\right) $	$ q+2 $	$ q-s+1 $	$ i $为奇数, $ {i}^{2}\equiv 1 \left(\text{mod} p\right) $或$ i\equiv 0 \left(\text{mod} p\right) $，$ 2\le i\le q-2 $, $ s=\mathrm{gcd}\left(i,q-1\right) $	定理8
	$ i\left(q+1\right) $	$ q+2 $	$ c=d-1 $		ME提升
	$ 3\left(q+1\right) $	$ q+2 $	$ q $	$ c=d-1 $，$ i=3 $

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