Some New Constructions for Orthogonal Designs

We give three new constructions for orthogonal designs using amicable orthogonal designs. These are then used to show (i) all possible n-tuples, n ~ 5 , are the types of orthogonal designs in order 16 and (ii) all possible n-tuples, n ~ 3 are the types of orthogonal designs in order 32 , (iii) all 4-tuples, (e, f, g, 32-e-f-g) , o ~ e T f T g ~ 32 are the types of orthogonal designs in order 32. These resultg are used in a paper by Peter J. Robinson, "Orthogonal designs of order sixteen", in this same volume, to fully update the status of the existence of orthogonal designs in order 16