Computational Statistics & Data Analysis 41 (2002) 171 – 184 www.elsevier.com/locate/csda On good matrices, skew Hadamard matrices and optimal designs S. Georgiou, C. Koukouvinos ∗ , S. Stylianou Department of Mathematics, National Technical University of Athens, Zografou 15773, Athens, Greece Abstract In two-level factorial experiments and in other linear models the coecients 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 n 6 35 and n = 127: In this paper, we give for the rst time all non-equivalent circulant good matrices of odd order 33 6 n 6 39: 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 dierence sets. In the sequel we use good matrices to construct some skew Hadamard matrices and orthogonal designs. c  2002 Elsevier Science B.V. All rights reserved. Keywords: Good matrices; Supplementary dierence sets; Skew Hadamard matrices; Linear models; Optimal designs; Orthogonal designs 1. Introduction In certain linear models the coecients of the unknown parameters can take one out of two values, which can be taken as +1 and -1: The problem we face is how to design the experiment so that the estimators of the unknown parameters are “ecient” in some sense. * Corresponding author. E-mail address: ckoukouv@math.ntua.gr (C. Koukouvinos). 0167-9473/02/$ - see front matter c  2002 Elsevier Science B.V. All rights reserved. PII:S0167-9473(02)00067-1