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. As an application of our study, we consider a class of quantum codes called Entanglement Assisted Quantum Error Correcting Code (EAQEC codes). As a consequence, we derive some EAQEC codes.

翻译：加性码是$\mathbb{F}_{q^m}^n$上关于$\mathbb{F}_{q^m}$的$\mathbb{F}_q$-线性子空间，而非关于$\mathbb{F}_{q^m}$的线性子空间。码的线性互补对（LCP）在密码学中具有重要作用，例如提升数字通信的速度与容量，并通过改进加密需求以抵御密码分析攻击来增强安全性。本文研究$\mathbb{F}_{q^m}$上加性码互补对（ACP）的代数结构。进一步，我们通过类比生成矩阵和校验矩阵刻画了ACP码的结构。此外，我们识别了ACP码的一个必要条件。同时，我们给出了基于$\mathbb{F}_{q^m}$上LCP码以及$\mathbb{F}_q$上LCP码构造$\mathbb{F}_{q^m}$上ACP码的若干方案。最后，我们研究了$\mathbb{F}_{q^m}$上的常循环ACP码及其计数问题。作为应用，我们考虑一类称为纠缠辅助量子纠错码（EAQEC码）的量子码，并由此推导出若干EAQEC码。