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://www.research.att.com/~njas/doc/rao.pdf  or
http://www.research.att.com/~njas/doc/rao.ps



A slightly different version of this paper will appear in 
J. Statistical Planning and Inference, 2001.