© 2001 by Biometrika Trust
Miscellaneous |
Upper bounds on the number of columns in supersaturated designs
1 Department of Statistics, University of California, Berkeley, California 94720-3860, U.S.Acheng{at}stat.berkeley.edu 2 Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152-6429, U.S.A.tangb{at}msci.memphis.edu
The number of columns m of a two-level supersaturated design is at least as large as its run size n, that is n 
m.In this case, orthogonality among the columns of a design has to be sacrificed. One measure of non-orthogonality among the columns of a supersaturated design is given by maxi < j |sij/n|, where sij = diTdj is the inner product of columns di and dj. An important theoretical problem in this formulation is to find, for a given value 0 < r < 1, where r is the degree of non-orthogonality the experimenter is willing to sacrifice, the maximum number M of columns such that maxi < j |sij/n| r. In this paper, we provide some upper bounds on M. One such bound is obtained by connecting this problem with that of the size of an error-correcting code. Another bound can be derived from lower bounds on the E(s2)-criterion which is defined as 
i < jsij2/{
m(m – 1)}. Some designs attaining the upper bounds are also examined.
Key Words: Error-correcting code; E(s2)-criterion; Hadamard matrix
Received September 1999. Revised April 2001