AI 中文总结
本文研究有限域上码的ℓ秩加法相交对（RAIP），建立其充要条件，证明q>2时任意加法码对单项式等价于ACP、q>3时任意加法码单项式等价于ACD，提出由自正交加法码构造ℓ-RAIP的通用方法并给出显式构造。
AI 中文摘要
本文针对给定正整数ℓ，引入并研究有限域上码的ℓ秩加法相交对（RAIP）这一概念。ℓ-RAIP提供了一个统一框架，可推广加法互补对偶（ACD）码、码的加法互补对（ACP）以及加法码的核。我们建立了刻画一对加法码构成ℓ-RAIP的充要条件。此外，当q>2时，我们证明任意一对加法码都单项式等价于码的ACP；由此可得，当q>3时，每个加法码都单项式等价于ACD码。本文的一项关键贡献是提出了一种由自正交加法码推导码的ℓ-RAIP的通用构造方法，在合适条件下可生成（ℓ+1）-RAIP码族。此外，还给出了ℓ-RAIP码的若干显式构造。
英文摘要
This paper introduces and studies \(\ell\)-rank additive intersection pairs (RAIP) of codes over finite fields for a given positive integer \(\ell\). The notion of \(\ell\)-RAIP provides a common framework that generalizes additive complementary dual (ACD) codes, additive complementary pairs (ACP) of codes, and the hull of an additive code. We establish necessary and sufficient conditions characterizing when a pair of additive codes forms an \(\ell\)-RAIP. Furthermore, for (q>2), we prove that any pair of additive codes is monomially equivalent to an ACP of codes. As a consequence, for (q>3), every additive code is monomially equivalent to an ACD code. { A key contribution of the paper is a general construction method for \(\ell\)-RAIP of codes derived from self-orthogonal additive codes, which yields families of \((\ell+1)\)-RAIP of codes under suitable conditions.} In addition, several explicit constructions of \(\ell\)-RAIP of codes are presented.
