Construction of Entanglement-Assisted Quantum MDS Codes
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摘要: 量子通信作为未来信息安全传输的核心技术，在实际应用中高度依赖高效可靠的量子纠错编码方案。纠缠辅助量子纠错码(EAQECCs)通过引入预共享纠缠态，突破了传统量子纠错码对经典纠错码自正交性的条件限制，显著提升了编码设计的灵活性与性能。本文基于有限域上的扭曲Reed–Solomon(TRS)码，系统构造了几类新参数的极大距离可分(MDS)EAQECCs。通过分析TRS码关联的陪集和矩阵的秩，本文确定了EAQECCs所需预共享纠缠态数量的精确条件，借助群的代数结构与陪集和严格证明了构造的EAQECCs具备最优的纠错性能。本文所构造的MDS EAQECCs码长突破了部分文献中所构造MDS EAQECCs码长选择的局限性，拓展了MDS EAQECCs的码长范围。本文的结果不仅深化了人们对MDS EAQECCs代数结构的认知，而且为面向量子通信系统高性能量子纠错码的设计提供了新的理论工具。
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关键词:
- 极大距离可分纠缠辅助量子纠错码 /
- 扭曲Reed-Solomon码 /
- 生成矩阵
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. -
表 1 定理3构造的MDS EAQECCs
q a b $\left[\kern-0.15em\left[ { n,k,d;c} \right]\kern-0.15em\right]_{q} $ 8 9 5 $\left[\kern-0.15em\left[ { 42{,}36{,}7;5} \right]\kern-0.15em\right]_{8} $ 11 12 8 $\left[\kern-0.15em\left[ { 90{,}81{,}10;8} \right]\kern-0.15em\right]_{11} $ 16 17 9 $\left[\kern-0.15em\left[ { 150{,}136{,}13;9} \right]\kern-0.15em\right]_{16} $ 17 18 6 $\left[\kern-0.15em\left[ { 112{,}97{,}12;6} \right]\kern-0.15em\right]_{17} $ 9 10 3 $\left[\kern-0.15em\left[ { 32{,}24{,}7;5} \right]\kern-0.15em\right]_{9} $ 13 7 3 $\left[\kern-0.15em\left[ { 96{,}80{,}11;4} \right]\kern-0.15em\right]_{13} $ 17 18 9 $\left[\kern-0.15em\left[ { 160{,}144{,}14;10} \right]\kern-0.15em\right]_{19} $ 19 20 11 $\left[\kern-0.15em\left[ { 216{,}198{,}16;12} \right]\kern-0.15em\right]_{19} $ 
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表 2 定理4构造的MDS EAQECCs
q a b $ \left[\kern-0.15em\left[ { n,k,d;c} \right]\kern-0.15em\right]_{q} $ 16 17 12 $ \left[\kern-0.15em\left[ { 195{,}170{,}15;11} \right]\kern-0.15em\right]_{16} $ 17 18 10 $ \left[\kern-0.15em\left[ { 176{,}161{,}14;9} \right]\kern-0.15em\right]_{17} $ 19 20 13 $ \left[\kern-0.15em\left[ { 252{,}234{,}17;12} \right]\kern-0.15em\right]_{19} $ 19 20 14 $ \left[\kern-0.15em\left[ { 270{,}170{,}15;6} \right]\kern-0.15em\right]_{19} $ 19 20 16 $ \left[\kern-0.15em\left[ { 302{,}289{,}18;15} \right]\kern-0.15em\right]_{19} $ 23 24 15 $ \left[\kern-0.15em\left[ { 352{,}330{,}20;14} \right]\kern-0.15em\right]_{23} $
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表 3 定理5构造的MDS EAQECCs
q a b $ \left[\kern-0.15em\left[ { n,k,d;c} \right]\kern-0.15em\right]_{q} $ 9 8 7 $ \left[\kern-0.15em\left[ { 70{,}63{,}8;6} \right]\kern-0.15em\right]_{9} $ 11 10 8 $ \left[\kern-0.15em\left[ { 96{,}88{,}9;7} \right]\kern-0.15em\right]_{11} $ 11 10 9 $ \left[\kern-0.15em\left[ { 108{,}99{,}10;8} \right]\kern-0.15em\right]_{11} $ 13 12 9 $ \left[\kern-0.15em\left[ { 126{,}117{,}10;9} \right]\kern-0.15em\right]_{13} $ 13 12 10 $ \left[\kern-0.15em\left[ { 140{,}128{,}11;10} \right]\kern-0.15em\right]_{13} $ 16 15 11 $ \left[\kern-0.15em\left[ { 187{,}176{,}12;10} \right]\kern-0.15em\right]_{16} $ 16 15 10 $ \left[\kern-0.15em\left[ { 170{,}160{,}11;10} \right]\kern-0.15em\right]_{16} $
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表 4 一些已知的MDS EAQECCs
参数 限制条件 参考文献 $ \left[\kern-0.15em\left[ { n=\dfrac{{q}^{2}-1}{2}+\dfrac{{q}^{2}-1}{2},n-2d+c+2,d;c} \right]\kern-0.15em\right]_{q} $ $ q> 3,b\mid \left(q+1\right),b\equiv 2\left(mod4\right),2\leq d\leq \dfrac{3\left(q+1\right)}{4}+\dfrac{q+1}{2b},c=\dfrac{b}{2}+1 $ [13] $ \left[\kern-0.15em\left[ { n=\dfrac{{q}^{2}+1}{a},n-2d+4k-2,d;4k-1} \right]\kern-0.15em\right]_{q} $ $ a=4{h}^{2}+{(4h+1)}^{2},q=(2t-1)a-10h-2,h\geq 2,t\in {\mathbb{Z}}^{+};2f(1)-4t-4\leq $
$d\leq 2f(2)\text{偶数} ,k=2\mathrm{或}2f(k-1)+2\leq d\leq 2f(k)\text{偶数},k=3{,}4 $[14] $ \left[\kern-0.15em\left[ { n=\dfrac{{q}^{2}+1}{a},n-2d+4k-1,d;1} \right]\kern-0.15em\right]_{q} $ $ a={h}^{2}+{(3h+1)}^{2},,q=2ta-1-h-3,h\geq 2,t\in {\mathbb{Z}}^{+};2\leq d\leq 2f(1)\text{偶数} $ [14] $ \left[\kern-0.15em\left[ { n=\dfrac{{q}^{2}+1}{a},n-2d+4k-1,d;1+4\left(k-1\right)} \right]\kern-0.15em\right]_{q} $ $ a={h}^{2}+{(3h+1)}^{2},q=2ta-1-h-3,h\geq 2,t\in {\mathbb{Z}}^{+};$
$2f(k-1)+2\leq d\leq 2f(k)\text{偶数},k=2{,}3 $[14] $ \left[\kern-0.15em\left[ { n=\dfrac{t(p-l)}{{p}^{e-l}-1},k-h,n-k+1;n-k-h} \right]\kern-0.15em\right]_{q} $ $ {q=p}^{e},2(e-l)\mid e,1\leq t\leq {p}^{e-l}-1{,}1\leq k\leq \left\lfloor \dfrac{{p}^{e}+n}{{p}^{e}+1}\right\rfloor ,0\leq h\leq k-1 $ [15] $ \left[\kern-0.15em\left[ { \begin{matrix}n=\dfrac{({t}_{1}+1)({q}^{2}-1)}{{h}_{1}}+\dfrac{(2{t}_{2}+2)({q}^{2}-1)}{{h}_{2}},\\ n-2k+c,k+1;c\\ \end{matrix}} \right]\kern-0.15em\right]_{q} $ $ {h}_{1},{h}_{2}\text{偶数},{K}_{1}=\dfrac{({t}_{1}+1)({q}^{2}-1)}{{h}_{1}}+\dfrac{q-1}{2},{K}_{2}=(\dfrac{{h}_{2}}{2}+{t}_{2}+1)\cdot \dfrac{q+1}{{h}_{2}}-2, $
$ 0\leq {t}_{1}\leq \dfrac{{h}_{1}-1}{2},0\leq {t}_{2}\leq \dfrac{{h}_{2}-3}{4},min\{{K}_{1},{K}_{2}\}\leq k\leq max\{{K}_{1},{K}_{2}\} $[16] $ \left[\kern-0.15em\left[ { n=\dfrac{{q}^{2}-1}{t},n-2d+t+2,d;t} \right]\kern-0.15em\right]_{q} $ $ q\text{奇数},t\mid (q+1),t\geq 3\text{奇数},\dfrac{(t-1)(q+1)}{t}+2\leq d\leq \dfrac{(t-1)(q+1)}{t}-2 $ [17] $ \left[\kern-0.15em\left[ { n=\dfrac{b\left({q}^{2}-1\right)}{a}+\dfrac{{q}^{2}-1}{a},n-2d+c+3,d;c} \right]\kern-0.15em\right]_{q} $ $ a\geq 3,a\mid (q+1),b\leq min\{a-3,q-3\},2\leq d\leq \dfrac{a+b+1}{2}\cdot \dfrac{q+1}{a},$
$c=b\mathrm{或}b+1 $定理3 $ \left[\kern-0.15em\left[ { n=\dfrac{b\left({q}^{2}-1\right)}{a}+\dfrac{{q}^{2}-1}{a},n-2d+c+3,d;c} \right]\kern-0.15em\right]_{q} $ $ a\geq 3,a\mid (q+1),b\leq min\{a-3,q-3\},2\leq d\leq \dfrac{a+b+1}{2}\cdot \dfrac{q+1}{a}, $
$ c=b-1\mathrm{或}b\mathrm{或}b+1 $定理4 $ \left[\kern-0.15em\left[ { n=\dfrac{b\left({q}^{2}-1\right)}{a},n-2d+c+3,d;c} \right]\kern-0.15em\right]_{q} $ $ b\leq a,a\mid (q-1),2\leq d\leq \dfrac{b(q+1)}{a}+1,c=b-1\mathrm{或}b $ 定理5
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