Michael G. Neubauer

NOTES ON D-OPTIMAL DESIGNS

Abstract The purpose of this paper is to exhibit new infinite families of D-optimal (0, 1)-matrices. We show that Hadamard designs lead to D-optimal matrices of size ( j , mj ) and ( j − 1, mj ), for certain integers j ≡ 3 (mod 4) and all positive integers m . For j a power of a prime and j ≡ 1 (mod 4), supplementary difference sets lead to D-optimal matrices of size ( j , 2mj ) and ( j − 1, 2mj ), for all positive integers m . We also show that for a given j and d sufficiently large, about half of the entries in each column of a D-optimal matrix are ones. This leads to a new relationship between D-optimality for (0, 1)-matrices and for (±1)-matrices. Known results about D-optimal (±1)-matrices are then used to obtain new D-optimal (0, 1)-matrices.

D-optimal weighing designs for four and five objects

For j = 4 and j = 5 and all d j, the maximum value of detXX , where X runs through all j d (0,1)-matrices, is determined along with a matrix X0 for which the maximum determinant is attained. In the theory of statistical designs, X0 is called a D-optimal design matrix. Design matrices that were previously thought to be D-optimal, are shown here to be D-optimal.

D-optimal weighing designs for six objects

Abstract. For all integers m≥6, we determine the maximum value of det XTX, where X is an m×6 (0, 1)-matrix, and exhibit (D-optimal) matrices X for which the maximum occurs. For D-optimal matrices X, the uniqueness of the Gram matrix XTX is discussed.

Maximalj-Simplices in the Reald-Dimensional Unit Cube

Department of Mathematics, California State University Northridge,Northridge, California 91330Communicated by the Managing EditorsReceived July 27, 1996For each positive even integer j there is an infinite arithmetic sequence of dimen-sions d for which we construct a j-simplex of maximum volume in the d-dimen-sional unit cube. For fixed d, all of these maximal j-simplices have the same Grammatrix, which is a multiple of I+J. For j even, a new upper bound for the volumeof a j-simplex in the d-dimensional unit cube is given.

D-optimal Designs for Seven Objects and a Large Number of Weighings

Let m ⩽ 7 be an integer and let M m ,7 (0,1) be the set of all m × 7 (0,1)-matrices. Define

AN INEQUALITY FOR POSITIVE DEFINITE MATRICES WITH APPLICATIONS TO COMBINATORIAL MATRICES

G(m,7) = \max \{ \det X^T X|X \in M_{m,7} (0,1)\} . \eqno (1)

n \equiv -1\ (\mod4)

A matrix X for which the maximum is achieved is said to be D- optimal . We determine G ( m , 7) for sufficiently large m and exhibit D-optimal matrices. For each m , the Gram matrices, X T X for D-optimal matrices X ε M m ,7 (0,1), are shown to be essentially unique.

Objects

Abstract Let M m , n (0,1) denote the set of all m × n (0,1)-matrices and let G(m,n)= max det X T X:X∈M m,n (0,1) . Inthis paper we exhibit some new formulas for G ( m , n ) where n≡−1( mod 4 ) . Specifically, for m = nt + r where 0⩽ r n , we show that for all sufficiently large t , G ( nt + r , n ) is a polynomial in t of degree n that depends on the characteristic polynomial of the adjacency matrix of a certain regular graph. Thus the problem of finding G ( nt + r , n ) for large t is equivalent to finding a regular graph, whose degree of regularity and number of vertices depend only on n and r , with a certain “trace-minimal” property. In particular we determine the appropriate trace-minimal graph and hence the formulas for G ( nt + r , n ) for n =11, 15, all r , and all sufficiently large t .