The Lattice of N-Run Orthogonal Arrays

If the number of runs in a (mixed-level) orthogonal array of strength 2 is specified, what numbers of levels and factors are possible? The collection of possible sets of parameters for orthogonal arrays with N runs has a natural lattice structure, induced by the “expansive replacement” construction method. In particular the dual atoms in this lattice are the most important parameter sets, since any other parameter set for an N -run orthogonal array can be constructed from them. To get a sense for the number of dual atoms, and to begin to understand the lattice as a function of N , we investigate the height and the size of the lattice. It is shown that the height is at most bc(N − 1)c, where c = 1.4039 . . ., and that there is an infinite sequence of values of N for which this bound is attained. On the other hand, the number of nodes in the lattice is bounded above by a superpolynomial function of N (and superpolynomial growth does occur for certain sequences of values of N). Using a new construction based on “mixed spreads”, all parameter sets with 64 runs are determined. Four of these 64-run orthogonal arrays appear to be new.