Construction of Entanglement-Assisted Quantum MDS Codes

QU Yuanyue; GAO Jian

doi:10.11999/JEIT251251

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QU Yuanyue, GAO Jian. Construction of Entanglement-Assisted Quantum MDS Codes[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT251251

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QU Yuanyue, GAO Jian. Construction of Entanglement-Assisted Quantum MDS Codes[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT251251

QU Yuanyue, GAO Jian. Construction of Entanglement-Assisted Quantum MDS Codes[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT251251

Citation:

QU Yuanyue, GAO Jian. Construction of Entanglement-Assisted Quantum MDS Codes[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT251251

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Construction of Entanglement-Assisted Quantum MDS Codes

doi: 10.11999/JEIT251251 cstr: 32379.14.JEIT251251

QU Yuanyue,
GAO Jian^,

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

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

Received Date: 2025-11-26
Accepted Date: 2026-03-09
Rev Recd Date: 2026-03-09

Available Online: 2026-03-26

Abstract

Abstract

Objective Entanglement-assisted quantum error-correcting codes (EAQECCs) provide a powerful mechanism for protecting quantum information through the use of pre-shared entanglement between sender and receiver. Traditional constructions of EAQECCs mainly rely on classical cyclic or constacyclic codes and often require strong algebraic constraints that limit the range of achievable parameters. This paper aims to develop a general and systematic framework for constructing new families of EAQECCs derived from twisted Reed-Solomon (TRS) codes over finite fields. The motivation is twofold: first, to extend the classical Reed–Solomon-based code design to its twisted form so as to capture richer algebraic structures; and second, to determine the exact number of maximally entangled pairs required for achieving the quantum Singleton bound. The ultimate goal is to produce maximum-distance separable (MDS) EAQECCs that outperform existing constructions in flexibility and parameter diversity. Methods The proposed method begins with the definition of TRS codes over finite fields, which introduce a “twist” parameter into the generator matrix, thereby altering the structure of their parity-check matrices. By systematically analyzing the associated coset-sum matrices and corresponding to twisted and untwisted cases, the rank of their product is determined. This rank directly equals the number of required entangled states, which forms the theoretical basis of our EAQECCs design.A detailed algebraic analysis shows that contains a submatrix with entries $ {M}_{l,j}=\displaystyle\sum\nolimits_{y\in W}{\left({\xi }^{j}y\right)}^{tl} $, which simplifies to under certain group-theoretic conditions. The resulting matrix, which is a Vandermonde matrix, ensures full rank and thus provides an explicit characterization of the entanglement structure. This establishes the rank-preserving property crucial to constructing MDS EAQECCs. Based on these results, we derive two families of EAQECCs characterized by the number of entangled pairs. The corresponding parameters are tabulated and expressed as which satisfy the quantum Singleton bound with equality, confirming the MDS nature of the constructed codes. Results and Discussions Comprehensive parameter analyses and explicit examples verify the theoretical findings. Comparative studies further demonstrate the flexibility of the proposed framework. Unlike previous constructions that require divisibility conditions such as $ a\mid (q+1) $and $ a\mid (q-1) $, our approach remains valid under broader algebraic configurations, thereby significantly extending the feasible range of codes parameters. This difference is conceptually summarized in the remark section and verified numerically. A systematic comparison of our results with existing MDS EAQECCs(Tables 4)reveals several new parameter regimes previously inaccessible to classical or cyclic-code-based constructions. Particularly, our method yields larger code lengths and more adaptable entanglement consumption rates $ \dfrac{c}{n} $, improving both the efficiency and generality of EAQECCs. The algebraic consistency across all tested cases confirms the correctness and universality of the TRS-based framework. Conclusions This study establishes a comprehensive algebraic framework for constructing MDS EAQECCs derived from twisted Reed–Solomon codes. By rigorously analyzing the rank properties of coset-sum matrices, we precisely determine the entanglement requirement and identify conditions under which the constructed codes achieve the quantum Singleton bound. Two broad classes of MDS EAQECCs are obtained, corresponding to $ a\mid \left(q+1\right) $ and $ a\mid \left(q-1\right) $, respectively, both verified through explicit examples and tabulated results. Compared with existing papers, the proposed approach not only generalizes prior constructions but also extends the achievable parameter space to cases not covered by Reed–Solomon codes or cyclic codes frameworks. The derived codes exhibit improved structural flexibility, theoretical clarity, and potential applicability to high-performance quantum information systems. This work thus provides a novel and unified perspective for developing algebraically optimized EAQECCs, laying the foundation for future research on TRS-based quantum codes families and their efficient encoding implementations.
- Entanglement-assisted quantum error-correcting MDS codes,
- Twisted Reed–Solomon codes,
- Generator matrices.

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References(33)

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