Some structural properties of mixed orthogonal arrays and their irredundancy

TL;DR

This paper explores the structural properties of mixed orthogonal arrays, establishing bounds, duality concepts, and their connection to error-block codes and quantum states, advancing understanding in combinatorial design and quantum information theory.

Contribution

It introduces new structural results for mixed orthogonal arrays, including bounds, duality, and a link to error-block codes and quantum states, which were not previously known.

Findings

Proved a Singleton-type upper bound for MOAs.

Established a trace duality and correspondence with error-block codes.

Characterized irredundant MOAs and their relation to quantum states.

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.