Unions of Orthogonal Arrays and their aberrations via Hilbert bases

TL;DR

This paper presents a method to generate and analyze unions of orthogonal arrays using Hilbert bases, enabling efficient identification of optimal arrays based on aberration criteria.

Contribution

It introduces a formula for the Word Length Pattern of unions of OAs using polynomial counting functions, facilitating the search for optimal arrays.

Findings

Complete description of classes of OAs with 5 binary factors, strength 2, sizes 16 and 20.

Efficient computation of best OAs according to the Generalized Minimum Aberration criterion.

Method to generate all OAs of a given size and strength as unions of minimal sets.

Abstract

We generate all the Orthogonal Arrays (OAs) of a given size n and strength t as the union of a collection of OAs which belong to an inclusion-minimal set of OAs. We derive a formula for computing the (Generalized) Word Length Pattern of a union of OAs that makes use of their polynomial counting functions. In this way the best OAs according to the Generalized Minimum Aberration criterion can be found by simply exploring a relatively small set of counting functions. The classes of OAs with 5 binary factors, strength 2, and sizes 16 and 20 are fully described.