Abstract

In two-level factorial experiments and in other linear models the coefficients of the unknown parameters can take one out of two values. When the number of observations is a multiple of four, the D-optimal design is a Hadamard matrix. Skew Hadamard matrices are of special interest due to their use, among others, in constructing D-optimal weighing designs for n≡3 (mod 4). A method is given for constructing skew Hadamard matrices which is based on the construction of good matrices. The construction is achieved through an algorithm which is also presented and relies on the discrete Fourier transform.

It is known that good matrices of order n, exist for all odd n35 and n=127. In this paper, we give for the first time all non-equivalent circulant good matrices of odd order 33n39. We note that no good matrices were previously known for orders 37 and 39. These are presented in a table in the form of the corresponding non-equivalent supplementary difference sets. In the sequel we use good matrices to construct some skew Hadamard matrices and orthogonal designs.

Author Keywords: Good matrices; Supplementary difference sets; Skew Hadamard matrices; Linear models; Optimal designs; Orthogonal designs

Article Outline

1. Introduction

2. Orthogonal designs and good matrices

3. Method of construction—the algorithm

3.1. Some preliminary results

3.2. The algorithm

3.3. Efficiency of the algorithm

4. Constructions using good matrices

5. The non-equivalent supplementary difference sets

References