The concept of balancedly splittable orthogonal designs is introduced along with a recursive construction. As an application, equiangular tight frames over the real, complex, and quaternions meeting Delsarte–Goethals–Seidel upper bound are obtained.

Although it is known that the maximum number of variables in two amicable orthogonal designs of order 2n p, where p is an odd integer, never exceeds 2n+2, not much is known about the existence of amicable orthogonal designs lacking zero entries that have 2n+2 variables in total. In this paper we develop two methods to construct amicable orthogonal designs of order 2n p where p odd, with no zero...