On the Amicability of Orthogonal Designs

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 entries and with the total number of variables equal or nearly equal to 2n+2. In doing so, we make a surprising connection between the two concepts of amicable sets of matrices and an amicable pair of matrices. With the recent discovery of a link between the theory of amicable orthogonal designs and space-time codes, this paper may have applications in space-time codes. AMS Subject Classification: Primary 05B20.