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

PAN Xin; GAO Jian

doi:10.11999/JEIT250258

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PAN Xin, GAO Jian. Constructing Two Classes of Maximum Distance Separable Entanglement-Assisted Quantum Error-Correcting Codes by Using Twisted Generalized Reed-Solomon Codes[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT250258

Citation:

PAN Xin, GAO Jian. Constructing Two Classes of Maximum Distance Separable Entanglement-Assisted Quantum Error-Correcting Codes by Using Twisted Generalized Reed-Solomon Codes[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT250258

Citation:

PAN Xin, GAO Jian. Constructing Two Classes of Maximum Distance Separable Entanglement-Assisted Quantum Error-Correcting Codes by Using Twisted Generalized Reed-Solomon Codes[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT250258

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

doi: 10.11999/JEIT250258 cstr: 32379.14.JEIT250258

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)

Received Date: 2025-04-11
Rev Recd Date: 2025-07-27

Available Online: 2025-08-06

Abstract

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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References(36)

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