Linear $ell$-Intersection Pairs of Codes and Their Applications

In this paper, a linear $ell$-intersection pair of codes is introduced as a generalization of linear complementary pairs of codes. Two linear codes are said to be a linear $ell$-intersection pair if their intersection has dimension $ell$. Characterizations and constructions of such pairs of codes are given in terms of the corresponding generator and parity-check matrices. Linear $ell$-intersection pairs of MDS codes over $mathbb{F}_q$ of length up to $q+1$ are given for all possible parameters. As an application, linear $ell$-intersection pairs of codes are used to construct entanglement-assisted quantum error correcting codes. This provides a large number of new MDS entanglement-assisted quantum error correcting codes.