A note on the minimum size of an orthogonal array

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.

KEYWORDS:

• Conjugacy
• Factorial design
• Mixed-level array
• Multiset
• Orthogonal array
• Strength.

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 62K15
• Secondary 05B15

Acknowledgments

We thank the associate editor and the referee for their helpful comments.

Notes

1 The condition assumed in Beder and Beder (2014) is that the counting function of the multiset D be constant on conjugacy classes. Since we are constructing proper fractions, D is a set and its counting function is its ordinary indicator function, which is constant on conjugacy classes iff D is a union of such classes.