In this paper, constructions of frameproof codes, separable codes, and B2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_2$$\\end{document} codes are obtained. For each family of codes, the Lovász Local Lemmais used to establish lower bounds for the codes. The obtained bounds match all best known bounds in the literature. Our strategy allows us to present constructions of the aforementioned codes, by using the variable framework for the Lovász Local Lemma.