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

TL;DR

This paper introduces linear 5-intersection pairs of codes, generalizing linear complementary pairs, with characterizations, constructions, and applications to quantum error correction, notably producing many new MDS entanglement-assisted quantum codes.

Contribution

It defines and characterizes linear 5-intersection pairs of codes, constructs such pairs over finite fields, and applies them to develop numerous new MDS entanglement-assisted quantum error-correcting codes.

Findings

Characterization of linear 5-intersection pairs via generator and parity-check matrices

Construction of MDS 5-intersection pairs over 5 finite fields for various parameters

Application to generate many new MDS entanglement-assisted quantum error-correcting codes

Abstract

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.