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.

翻译：混合（非对称）正交数组通过允许列使用不同字母表，推广了经典正交数组。然而，其研究需要使用与对称正交数组截然不同的结构工具，因为对称情形下的若干关键特征在混合情形中不再适用，包括欧几里得对偶、唯一的全局指标以及某些经典界。本文建立了混合正交数组的三个结构结果。首先，我们证明了一个Singleton型上界，并得到了MDS和几乎MDS混合正交数组的表征。其次，我们在$\prod_{i=1}^{s} \mathbb{F}_{q^{n_i}}$上引入$\mathbb{F}_q$-线性MOA的迹对偶，并建立与$\mathbb{F}_q$-线性纠错块码的对应关系，该关系通过关联纠错块码的对偶距离确定MOA的强度。最后，我们发展了非冗余混合正交数组的结构理论，其动机源于它们在构造$t$-均匀和绝对最大纠缠量子态中的作用。在极值情形$t=\lfloor s/2\rfloor$下，我们证明了具有最小指标$1$（产生最小支撑的AME态）的$\mathbb{F}_q$-线性IrMOA等价于$\mathbb{F}_q$-线性纠错块MDS码。