In this work, we investigate additive complementary dual (ACD) codes and their construction over finite fields \(\mathbb {F}_{q^2}\) with respect to the trace inner products, where q is a prime power. First, we associate an additive code with a matrix known as a generator matrix. After that, we describe ACD codes in terms of generator matrices for the trace Hermitian and the trace Euclidean inner products. We also construct ACD codes over \(\mathbb {F}_{q^2}\) from linear codes over \(\mathbb {F}_q.\) Additionally, we present techniques for constructing ACD codes with various parameters from a given ACD code over \(\mathbb {F}_{q^2}.\) By applying these methods, we construct numbers of trace Euclidean and trace Hermitian ACD codes that exhibit better parameters compared to the best known linear codes over \(\mathbb {F}_9\) and \(\mathbb {F}_4\) of the same size and length.
