Some structural properties of mixed orthogonal arrays and their irredundancy

Abstract

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.