On the linear $\ell$-intersection pair of codes over a finite principal ideal ring

Generalizing the linear complementary dual, the linear complementary pair and the hull of codes, we introduce linear $\ell$-intersection pair of codes over a finite principal ideal ring $R,$ for some positive integer $\ell$. Two linear codes are said to be a linear $\ell$-intersection pair of codes over $R$ if the cardinality of the intersection of two linear codes are equal to $q^{\ell}$, where $q$ is the cardinality of the radical of $R.$ In this paper, we study linear $\ell$-intersection pair of codes over $R$ in a very general setting by a uniform method. We provide a necessary and sufficient condition for a non-free (or free) linear $\ell$-intersection pair of codes over $R.$