Orthogonal designs II

Orthogonal designs are a natural generalization of the Baumert-Hall arrays which have been used to construct Hadamard matrices. We continue our investigation of these designs and show that orthogonal designs of type (1, k) and order n exist for every k < n when n = 2t+2. 3 and n = 2t+2.5 (where t is a positive integer). We also find orthogonal designs that exist in every order 2n and others that exist in every order 4n. Coupled with some results of earlier work, this means that the weighing matrix conjecture 'For every order n = 0 (mod4) there is, for each kt = kIn ' is resolved in the affirmative for all orders n = 2t+1.3, n = 2t+1. 5 (t a positive integer). The fact that the matrices we find are skew-symmetric for all k < n when n = 0(mod 8) and because of other considerations we pose three other conjectures about weighing matrices having additional structure and resolve these conjectures affirmatively in a few cases. In an appendix we give a table of the known results for orders <64