PBIB-designs and association schemes arising from minimum bi-connected dominating sets of hypercubes

In this paper we disprove a result on the PBIB designs arising from γbc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{bc}$$\end{document}-sets in Q4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{4}$$\end{document}, with m=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m = 4$$\end{document} association schemes proved by Chaluvaraju et al. (Afr Math 29:47–63, 2018). In Ref. [11], the authors had posed an open problem to find the PBIB designs in hypercubes of dimension higher than 5. Here, we settle the problem for the first few values, and give a general iterative proof technique for higher dimensional hypercubes.