Linear ℓ-intersection pairs of MDS codes and their applications to AEAQECCs

Two linear codes are said to be a linear ℓ-intersection pair if their intersection has dimension ℓ. Guenda et al. (Des Codes Cryptogr. 88, 133–152, 2020) constructed most of the linear ℓ-intersection pairs of MDS codes and we complement their results by constructing some linear ℓ-intersection pairs of MDS codes over Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{q}$$\end{document} of lengths n = q,q + 1. Furthermore, we construct all the possible linear ℓ-intersection pairs of MDS codes over F2m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{2^{m}}$$\end{document} of length n = 2m + 2 ≥ 6. As a consequence, linear ℓ-intersection pairs of MDS codes for all possible parameters are given. Moveover, we can apply our results to asymmetric entanglement-assisted quantum error-correcting codes (AEAQECCs) and obtain all the possible pure MDS CSS-type AEAQECCs.