Generalizing the linear complementary duals, the linear complementary pairs and the hull of codes, we introduce the concept of $\ell$-dimension linear intersection pairs ($\ell$-DLIPs) of codes over a finite commutative ring $(R)$, for some positive integer $\ell$. In this paper, we study $\ell$-DLIP of codes over $R$ in a very general setting by a uniform method. Besides, we provide a necessary and sufficient condition for the existence of a non-free (or free) $\ell$-DLIP of codes over a finite commutative Frobenius ring. In addition, we obtain a generator set of the intersection of two constacyclic codes over a finite chain ring, which helps us to get an important characterization of $\ell$-DLIP of constacyclic codes. Finally, the $\ell$-DLIP of constacyclic codes over a finite chain ring are used to construct new entanglement-assisted quantum error correcting (EAQEC) codes.

翻译：推广线性互补对偶、线性互补对以及码的核的概念，本文针对有限交换环$(R)$上的码，引入$\ell$-维线性交对（$\ell$-DLIPs）的概念，其中$\ell$为正整数。本文采用统一方法在极其一般的框架下研究$R$上码的$\ell$-DLIP。此外，我们给出了有限交换Frobenius环上存在非自由（或自由）$\ell$-DLIP的充要条件。进一步，我们得到了有限链环上两个常循环码交集的生成元集，这有助于获得常循环码$\ell$-DLIP的重要刻画。最后，利用有限链环上常循环码的$\ell$-DLIP构造了新型纠缠辅助量子纠错码。