Intersection Problem for Simple 2-fold (3n, n, 3) Group Divisible Designs

In this paper, we will give intersection numbers for two simple 2-fold (3n, n, 3) group divisible designs. More precisely, we will develop constructions which show that there exists two simple 2-fold (3n, n, 3) group divisible designs which intersect in precisely k∈{0,1,2,…,2n2}\{2n2-1,2n2-2,2n2-3,2n2-5}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${k \in \{0, 1, 2, \ldots, 2n^2\}{\setminus} \{2n^2-1, 2n^2-2, 2n^2-3, 2n^2-5\}}$$\end{document} triples for n≥5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \geq 5}$$\end{document}. There are some exceptions for n = 2, 3, 4.