The applications of additive codes mainly lie in quantum error correction and quantum computing. Due to their applications in quantum codes, additive codes have grown in importance. In addition to this, additive codes allow the implementation of a variety of dualities. The article begins by developing the properties of Additive Complementary Dual (ACD) codes with respect to arbitrary dualities over finite abelian groups. Further, we introduce a subclass of non-symmetric dualities referred to as skew-symmetric dualities. Then, we precisely count symmetric and skew-symmetric dualities over finite fields. Two conditions have been obtained: one is a necessary and sufficient condition, and the other is a necessary condition. The necessary and sufficient condition is for an additive code to be an ACD code over arbitrary dualities. The necessary condition is on the generator matrix of an ACD code over skew-symmetric dualities. We provide bounds for the highest possible minimum distance of ACD codes over skew-symmetric dualities. Finally, we find some new quaternary ACD codes over non-symmetric dualities with better parameters than the symmetric ones.

翻译：加性码的应用主要在于量子纠错和量子计算。由于其应用于量子码，加性码的重要性日益增长。此外，加性码允许实现多种对偶性。本文首先发展了有限阿贝尔群上关于任意对偶性的加性互补对偶（ACD）码的性质。进一步，我们引入了一类非对称对偶的子类，称为斜对称对偶。随后，我们精确计算了有限域上的对称对偶和斜对称对偶的数量。得到了两个条件：一个是充要条件，另一个是必要条件。充要条件是关于任意对偶性的加性码成为ACD码的条件。必要条件是关于斜对称对偶的ACD码的生成矩阵。我们给出了斜对称对偶上ACD码的最高可能最小距离的界。最后，我们找到了一些在非对称对偶上具有比对称对偶更优参数的新四元ACD码。