The Lattice of N-Run Orthogonal Arrays
<|reference_start|>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 [c(N-1)], 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.<|reference_end|>
@article{rains2002the,
    title={The Lattice of N-Run Orthogonal Arrays},
    author={E. M. Rains, N. J. A. Sloane, John Stufken},
    journal={J. Statistical Planning and Inference, Vol. 102 (2002), pp.
  477-500},
    year={2002},
    archivePrefix={arXiv},
    eprint={math/0205299},
    primaryClass={math.CO cs.IT math.IT}
}