The Lattice of N-Run Orthogonal Arrays
E. M. Rains and N. J. A. Sloane
Information Sciences Research Center
AT&T Shannon Lab
Florham Park, New Jersey 07932-0971
and
John Stufken
Department of Statistics
Iowa State University
Ames, IA 50011
April 20, 2000
ABSTRACT
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.
For the full version, see
http://NeilSloane.com/doc/rao.pdf or
http://NeilSloane.com/doc/rao.ps
A slightly different version of this paper will appear in
J. Statistical Planning and Inference, 2001.