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.

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