On $\mathrm{\ell }$-rank additive intersection pairs (RAIP) of codes

This paper introduces and studies $\mathrm{\ell }$-rank additive intersection pairs (RAIP) of codes over finite fields for a given positive integer $\mathrm{\ell }$. The notion of $\mathrm{\ell }$-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 $\mathrm{\ell }$-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 $\mathrm{\ell }$-RAIP of codes derived from self-orthogonal additive codes, which yields families of $(\mathrm{\ell }+1)$-RAIP of codes under suitable conditions.} In addition, several explicit constructions of $\mathrm{\ell }$-RAIP of codes are presented.