Amicable complex orthogonal designs

In this letter, we define amicable complex orthogonal designs (ACOD) and propose two systematic methods to construct higher-order ACOD’s from lower-order ACOD’s. Although the upper bound on the number of variables of ACOD’s is found to be the same as that of amicable orthogonal designs (AOD), we show that certain types of AOD’s that were previously shown to be non-existent or undecided, such as AOD’s of order 8 with type (1, 1, 1, 1; 2, 2, 2, 2) and (1, 2, 2, 2; 1, 2, 2, 2), can be found from ACOD’s constructed using the proposed construction methods. The proposed methods can also be used to systematically construct new AOD’s that are of the same type as, but not equivalent to, those previously found by Zhao, Wang and Seberry using computer search. An interesting finding arising from this study is that an AOD or ACOD can be constructed from a lower-order amicable family (AF) or amicable complex family (ACF). This implies that the component matrices need not be disjoint in order to construct a higher-order AOD/ACOD. Index Terms — Orthogonal Designs, Amicable Orthogonal Designs, Complex Orthogonal Designs, Amicable Complex Orthogonal Designs.