Some structural properties of mixed orthogonal arrays and their irredundancy

2026-03-19

AI Summary

This work addresses the structural challenges of mixed orthogonal arrays (MOAs) in asymmetric settings, where the absence of Euclidean duality, uniformity measures, and classical bounds has hindered their characterization. The study establishes, for the first time, a Singleton-type upper bound for MOAs and develops a trace duality theory tailored to linear MOAs over finite fields. It further introduces a structural framework for irredundant MOAs (IrMOAs). By integrating tools from finite field linear algebra, error block codes, and combinatorial design, the research characterizes MDS and near-MDS MOAs and uncovers an equivalence among IrMOAs, maximum distance separable (MDS) error block codes, and absolutely maximally entangled (AME) quantum states. This equivalence provides a theoretical foundation for constructing AME states with minimal support.

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• mixed orthogonal arrays
• structural properties
• irredundancy
• MDS codes
• error-block codes

Innovation

• mixed orthogonal arrays
• trace duality
• error-block codes
• MDS codes
• irredundant arrays