Construction of MDS Entanglement-Assisted Quantum Error-Correcting 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。该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，确定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。该文所构造的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。该研究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 an effective way to protect quantum information by using pre-shared entanglement between the sender and receiver. Existing constructions of EAQECCs mainly rely on classical cyclic or constacyclic codes and often require strong algebraic constraints, which limit the range of achievable parameters. This paper develops a general and systematic framework for constructing new families of EAQECCs from Twisted Reed-Solomon (TRS) codes over finite fields. The study has two aims. The first is to extend classical Reed-Solomon-based code design to the twisted setting so that richer algebraic structures can be used. The second is to determine the exact number of maximally entangled pairs required to attain the quantum Singleton bound. The final objective is to construct Maximum-Distance Separable (MDS) EAQECCs with greater flexibility and broader parameter ranges than existing methods. Methods The proposed method starts from the definition of TRS codes over finite fields. A twist parameter is introduced into the generator matrix, which changes the structure of the corresponding parity-check matrices. By systematically analyzing the associated coset-sum matrices in the twisted and untwisted cases, the rank of the relevant matrix product is determined. This rank equals the number of required entangled pairs and therefore provides the theoretical basis for the construction of EAQECCs. A detailed algebraic analysis shows that the matrix contains a submatrix with entries $ {M}_{l,j}=\displaystyle\sum\nolimits_{y\in W}{\left({\xi }^{j}y\right)}^{tl} $, which simplifies to $ t\zeta^{jl} $under suitable group-theoretic conditions. The resulting matrix is a Vandermonde matrix, and its full rank gives an explicit characterization of the entanglement structure. This property is then used to construct MDS EAQECCs. Based on these results, two families of EAQECCs are derived according to the number of entangled pairs. The corresponding parameters are tabulated and are shown to satisfy the quantum Singleton bound with equality, which confirms that the constructed codes are MDS. Results and Discussions Comprehensive parameter analysis and explicit examples verify the theoretical results. Comparative analysis further shows the flexibility of the proposed framework. Unlike previous constructions that require divisibility conditions such as $ a\mid (q+1) $and $ a\mid (q-1) $, the present approach remains applicable under broader algebraic settings and thus extends the feasible range of code parameters. This difference is summarized in the remark section and verified numerically. A systematic comparison with existing MDS EAQECCs ( Table 4 ) reveals several new parameter regimes that are not accessible with classical or cyclic-code-based constructions. In particular, the proposed method yields larger code lengths and more flexible entanglement consumption rates $ \dfrac{c}{n} $, which improves both the efficiency and the generality of EAQECCs. The algebraic consistency observed across all tested cases supports the correctness and general applicability of the TRS-based framework.Conclusions This study establishes an algebraic framework for constructing MDS EAQECCs from TRS codes. By rigorously analyzing the rank properties of coset-sum matrices, the required entanglement is determined precisely, and the conditions under which the constructed codes attain the quantum Singleton bound are identified. Two broad classes of MDS EAQECCs are obtained, corresponding to $ a\mid \left(q+1\right) $ and $ a\mid \left(q-1\right) $, respectively, and both are verified by explicit examples and tabulated results. Compared with existing studies, the proposed approach not only generalizes earlier constructions but also extends the achievable parameter space to cases not covered by Reed-Solomon-code- or cyclic-code-based frameworks. The derived codes show improved structural flexibility, clearer algebraic characterization, and potential value for high-performance quantum information systems. This work therefore provides a unified perspective for the development of algebraically optimized EAQECCs and offers a basis for future studies of TRS-based quantum code 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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