发表机构
Air Force Engineering University（空军工程大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
介绍从\(\mathbb{F}_{q^2}\)上经典线性码构造\(q\)元量子稳定子码的约当标准型框架，通过衡量阻碍、降低埃尔米特内积矩阵秩来构造自正交码，应用此构造得到创纪录量子码，改进或补充已知参数。
AI 中文摘要
我们引入了一个约当标准型框架，用于从\(\mathbb{F}_{q^2}\)上的任意经典线性码构造\(q\)元量子稳定子码。该框架不要求经典线性码\(\mathcal{C}\)满足对偶包含条件（即自正交性）。给定具有校验矩阵\(H\)的经典码\(\mathcal{C}=[n,k,d]_{q^2}\)，我们通过秩\(r=(n - k)-\dim_{\mathbb{F}_{q^2}}(\mathcal{C}^{\perp_h}\cap\mathcal{C})\)来衡量对埃尔米特自正交性的阻碍。成分码\(\mathcal{C}\)是\(r -\)近对偶包含的，或者等价地，\(\mathcal{C}^{\perp_h}\)是\(r -\)近自正交的。通过沿着分解\(A = PJ_AP^{-1}\)的约当基\(W = P^{-1}\)进行秩一扰动来系统地降低埃尔米特内积矩阵\(A = HH^{\dagger}\)的秩，我们构造了一个明确的埃尔米特自正交码\(\mathcal{C}_{\mathrm{so}}=[n + r,n - k]_{q^2}\)。一个充分的距离保持准则保证了所得\(q\)元量子码的参数为\([[n + r,2k - n + r,\geq d]]_q\)。将此构造应用于经典码产生了几个创纪录的量子码，改进或补充了格拉斯尔表中最知名的参数。
英文摘要
We introduce a Jordan-canonical-form framework for constructing $q$-ary quantum stabilizer codes from arbitrary classical linear codes over $\F_{q^2}$. The framework does not require the classical linear code $\mathcal{C}$ to satisfy the dual-containing condition (i.e., self-orthogonality). Given a classical code $\mathcal{C}=[n,k,d]_{q^2}$ with parity-check matrix $H$, we measure the obstruction to Hermitian self-orthogonality by the rank $r=(n-k)-\dim_{\F_{q^2}}(\mathcal{C}^{\perp_h}\cap \mathcal{C})$. The ingredient code $\mathcal{C}$ is $r$-nearly dual containing, or, equivalently, $\mathcal{C}^{\perp_h}$ is $r$-nearly self-orthogonal, by which we mean that $r=\Rank(HH^{\dagger})=\dim_{\F_{q^2}}(\mathcal{C}^{\perp_h})-\dim_{\F_{q^2}}(\mathcal{C}^{\perp_h}\cap \mathcal{C})$. By systematically reducing the rank of the Hermitian inner-product matrix $A=HH^{\dagger}$ through rank-one perturbations along the Jordan basis $W=P^{-1}$ of the decomposition $A=PJ_AP^{-1}$, we construct an explicit Hermitian self-orthogonal code $\mathcal{C}_{\mathrm{so}}=[n+r,n-k]_{q^2}$. A sufficient distance-preservation criterion guarantees that the resulting $q$-ary quantum code has parameters $[[n+r,2k-n+r,\geq d]]_q$. Applying this construction to classical codes produces several record quantum codes that improve or supplement the best-known parameters in Grassl's tables.
Commentsv2: corrected the distance-preservation criterion (Theorem III.9) and the deficiency formula
