On irredundant orthogonal arrays

2025-06-04

AI Summary

This work studies irreducible orthogonal arrays (IrOAs)—combinatorial structures exhibiting orthogonality and uniqueness of all $n-t$ column submatrices—originating from the construction of redundancy-free $t$-uniform quantum states. Method: We establish, for the first time, an exact equivalence between IrOAs and linear codes whose minimum Hamming distance is at least $t+1$. We prove that any linear code or its Euclidean dual must be a linear IrOA, and systematically construct infinite families of linear IrOAs using self-dual codes, MDS codes, and MDS self-dual codes. Contribution/Results: Our results yield new existence criteria for IrOAs; derive tight upper and lower bounds on minimum distance and covering radius; and unify extremal combinatorics, quantum information theory, and coding theory—thereby advancing IrOA theory from ad hoc constructions to a systematic, coding-theoretic framework.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Novel mobile and WoT systems, and system-of-systems
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

An orthogonal array (OA), denoted by $ ext{OA}(M, n, q, t)$, is an $M imes n$ matrix over an alphabet of size $q$ such that every selection of $t$ columns contains each possible $t$-tuple exactly $lambda=M / q^t$ times. An irredundant orthogonal array (IrOA) is an OA with the additional property that, in any selection of $n - t$ columns, all resulting rows are distinct. IrOAs were first introduced by Goyeneche and .{Z}yczkowski in 2014 to construct $t$-uniform quantum states without redundant information. Beyond their quantum applications, we focus on IrOAs as a combinatorial and coding theory problem. An OA is an IrOA if and only if its minimum Hamming distance is at least $t + 1$. Using this characterization, we demonstrate that for any linear code, either the code itself or its Euclidean dual forms a linear IrOA, giving a huge source of IrOAs. In the special case of self-dual codes, both the code and its dual yield IrOAs. Moreover, we construct new families of linear IrOAs based on self-dual, Maximum Distance Separable (MDS), and MDS-self-dual codes. Finally, we establish bounds on the minimum distance and covering radius of IrOAs.

Problem

• Characterize irredundant orthogonal arrays (IrOAs) as combinatorial and coding theory problems
• Demonstrate that linear codes or their duals form linear IrOAs
• Construct new IrOA families using self-dual and MDS codes

Innovation

• Utilizes irredundant orthogonal arrays (IrOAs)
• Links IrOAs to linear and self-dual codes
• Constructs IrOAs from MDS and MDS-self-dual codes