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...