AI 中文总结
该研究构造了特定参数的埃尔米特自对偶最大距离可分码，通过非二元稳定子得到多种绝对最大纠缠态，利用精确域算术等完成认证，为纠缠态的构造与验证提供了新方法。
AI 中文摘要
我们构造了参数为$[12,6,7]_{25}$、$[18,9,10]_{121}$和$[18,9,10]_{169}$的埃尔米特自对偶最大距离可分码。通过非二元稳定子构造，这些码定义了$\text{AME}(12,5)$、$\text{AME}(18,11)$和$\text{AMD}(18,13)$态，而单粒子投影可得到$\text{AME}(17,11)$和$\text{AME}(17,13)$态。参数为$[12,6,7]_{25}$的码不等价于广义里德-所罗门码，它是通过无预设坐标对称性的系统搜索得到的；其单项式半线性自同构群在9个坐标上作用出正则$\boldsymbol{\text{Z}_3^2}$轨道。将该作用施加于两个9坐标轨道，可将长度为18的搜索简化为9元群代数核，其埃尔米特特征分解将不变自对偶性约束拆分为独立块并给出精确族计数；在$q=13$时，范数1坐标缩放消除了因子$q+1=14$。埃尔米特自对偶性和MDS性质通过精确域算术和完全平方子式枚举得到认证，图态割秩计算（包括长度为18的图的所有单顶点删除）提供了额外的精确校验。
英文摘要
We give explicit Hermitian self-dual maximum distance separable codes with parameters $[12,6,7]_{25}$, $[18,9,10]_{121}$, and $[18,9,10]_{169}$. The stabilizer construction proves the existence of ${\rm AME}(12,5)$, ${\rm AME}(18,11)$, and ${\rm AME}(18,13)$ states; one-party projection also gives ${\rm AME}(17,11)$ and ${\rm AME}(17,13)$. The first code was found by a direct search. A regular $\mathbb{Z}_3^2$ coordinate orbit of its automorphism group suggested a group-circulant form that reduces each length-eighteen construction to a nine-entry kernel. The printed matrices are certified by exact Hermitian products and complete square-minor enumeration.
Comments4 pages. Results were found and written up with the help of Claude Fable 5 (Anthropic) and ChatGPT 5.6 Sol (OpenAI)
