On the decomposition of orthogonal arrays∗

When an orthogonal array is projected on a small number of factors, as is done in screening experiments, the question of interest is the structure of the projected design, by which we mean its decomposition in terms of smaller arrays of the same strength. In this paper we investigate the decomposition of arrays of strength t having t + 1 factors. The decomposition problem is well-understood for symmetric arrays on s = 2 symbols. In this paper we derive some general results on decomposition, with particular attention to arrays on s = 3 symbols. We give a new proof of the regularity of arrays of index 1 when s = 2 or 3, and show by counterexample that the result doesn’t extend to larger s. For s = 3 we also construct an indecomposable array of index 2. Finally, we determine the structure of completely decomposable arrays on 3 symbols having strength 2 and index 2, 3 or 4.