混合正交数组的一些结构性质及其不可约性
Some structural properties of mixed orthogonal arrays and their irredundancy
AI总结：
本文研究混合正交数组的结构特性，推导了单利型上界，建立了MDS和近MDS混合正交数组的特征，并探讨了不可约混合正交数组在构造t-均匀和绝对最大纠缠量子态中的作用。
摘要：
混合（非对称）正交数组（MOAs）通过允许不同字母表的列扩展了经典正交数组。然而，其研究需要不同于对称正交数组的结构工具，因为对称设置中的几个关键特性在混合情况下不再可用，包括欧几里得对偶性、唯一的全局索引以及某些经典界限。本文为混合正交数组建立了三个结构结果。首先，我们证明了单利型上界，并获得了MDS和近MDS混合正交数组的特征。其次，我们引入了$\mathbb{F}_q$-线性MOA在$\prod_{i=1}^{s} \mathbb{F}_{q^{n_i}}$上的迹对偶性，并建立了与$\mathbb{F}_q$-线性误差块码的对应关系，通过关联的误差块码的对偶距离确定MOA的强度。最后，我们发展了不可约混合正交数组（IrMOAs）的结构理论，受其在构造$t$-均匀和绝对最大纠缠（AME）量子态中的作用启发。在极值情况$t=\lfloor s/2\rfloor$下，我们证明了具有最小索引1的$\mathbb{F}_q$-线性IrMOA（产生最小支持的AME态）等同于$\mathbb{F}_q$-线性误差块MDS码。
英文摘要：
Mixed (asymmetric) orthogonal arrays (MOAs) generalize classical orthogonal arrays by allowing columns over different alphabets. However, their study requires very different structural tools than those used for symmetric orthogonal arrays (OAs), since several key features of the symmetric setting are no longer available in the mixed case, including Euclidean duality, a unique global index, and certain classical bounds. In this paper, we establish three structural results for mixed orthogonal arrays. First, we prove a Singleton-type upper bound and obtain a characterization of MDS and almost-MDS mixed orthogonal arrays. Second, we introduce a trace duality for $\mathbb{F}_q$-linear MOAs over $\prod_{i=1}^{s} \mathbb{F}_{q^{n_i}}$ and establish a correspondence with $\mathbb{F}_q$-linear error-block codes that determines the strength of the MOA via the dual distance of the associated error-block code. Finally, we develop a structural theory of irredundant mixed orthogonal arrays (IrMOAs), motivated by their role in the construction of $t$-uniform and absolutely maximally entangled (AME) quantum states. In the extremal case $t=\lfloor s/2\rfloor$, we prove that $\mathbb{F}_q$-linear IrMOAs with minimum index $1$ (yielding AME states of minimal support) are equivalent to $\mathbb{F}_q$-linear error-block MDS codes.
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