A note on the minimum size of an orthogonal array

TL;DR

This paper explores the minimum size constraints of orthogonal arrays, examining when smaller arrays than the complete factorial design are possible and unique, especially in mixed-level cases.

Contribution

It provides new insights into the lower bounds of orthogonal array sizes and characterizes conditions for their uniqueness and existence in mixed-level scenarios.

Findings

L_t is a lower bound on array size for strength t.

Complete factorial design is unique at size L_k.

Constructs of proper fractions of strength k-1 are presented.

Abstract

It is an elementary fact that the size of an orthogonal array of strength t on k factors must be a multiple of a certain number, say L_t, that depends on the orders of the factors. Thus L_t is a lower bound on the size of arrays of strength t on those factors, and is no larger than L_k, the size of the complete factorial design. We investigate the relationship between the numbers L_t, and two questions in particular: For what t is L_t < L_k? And when L_t = L_k, is the complete factorial design the only array of that size and strength t? Arrays are assumed to be mixed-level. We refer to an array of size less than L_k as a proper fraction. Guided by our main result, we construct a variety of mixed-level proper fractions of strength k-1 that also satisfy a certain group-theoretic condition.