Amicable complex orthogonal designs

In this paper, we define amicable complex orthogonal designs (ACOD) and propose two systematic methods to construct higher-order ACODs from lower-order ACODs. We found that the upper bound on the number of variables of an ACOD is the same as that of amicable orthogonal designs (AOD). We also show that certain types of AOD that were previously shown to be non-existent or undecided, such as AODs 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 ACODs constructed using our proposed construction methods. Our proposed methods can also be used to systematically construct new AODs 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 for constructing a higher-order AOD/ACOD need not be disjoint.