An additive code is an $\mathbb{F}_q$-linear subspace of $\mathbb{F}_{q^m}^n$ over $\mathbb{F}_{q^m}$, which is not a linear subspace over $\mathbb{F}_{q^m}$. Linear complementary pairs (LCP) of codes have important roles in cryptography, such as increasing the speed and capacity of digital communication and strengthening security by improving the encryption necessities to resist cryptanalytic attacks. This paper studies an algebraic structure of additive complementary pairs (ACP) of codes over $\mathbb{F}_{q^m}$. Further, we characterize an ACP of codes in analogous generator matrices and parity check matrices. Additionally, we identify a necessary condition for an ACP of codes. Besides, we present some constructions of an ACP of codes over $\mathbb{F}_{q^m}$ from LCP codes over $\mathbb{F}_{q^m}$ and also from an LCP of codes over $\mathbb{F}_{q}$. Finally, we study the constacyclic ACP of codes over $\mathbb{F}_{q^m}$ and the counting of the constacyclic ACP of codes.

翻译：加法码是 $\mathbb{F}_{q^m}^n$ 在 $\mathbb{F}_q$ 上的线性子空间，但不是 $\mathbb{F}_{q^m}$ 上的线性子空间。线性互补码对（LCP）在密码学中具有重要作用，例如提高数字通信的速度和容量，以及通过改进加密需求以抵抗密码分析攻击来增强安全性。本文研究了 $\mathbb{F}_{q^m}$ 上加法互补码对（ACP）的代数结构。进一步，我们利用类似的生成矩阵和校验矩阵刻画了加法互补码对。此外，我们给出了加法互补码对的一个必要条件。另外，我们提出了一些从 $\mathbb{F}_{q^m}$ 上的 LCP 码以及从 $\mathbb{F}_{q}$ 上的 LCP 码构造 $\mathbb{F}_{q^m}$ 上加法互补码对的方法。最后，我们研究了 $\mathbb{F}_{q^m}$ 上的常循环加法互补码对及其计数问题。