On the existence of \((v,4,3,1)\)-BHDs (Q1900191)

Consider a \(v\)-set \(V\) and \(\beta= (B_1, B_2,\dots, B_b)\) a set of blocks which are ordered \(k\)-subsets of \(V\), and consider \(B_i= (a_{i, 1}, a_{i, 2},\dots, a_{i, k})\) and \(t\leq k\). From \(B_i\) a set \(B^*_i\) of \((k- t+ 1)\) \(t\)-subsets of \(V\) can be created: \(B^*_i= \{\{a_{i, j}, a_{i, j+ 1},\dots, a_{i, j+ t- 1}\}\mid j= 1,\dots, k- t+ 1\}.\) If every element of \(V\) occurs in exactly \(r\) blocks and if every \(t\)-subset of \(V\) occurs in exactly \(\lambda\) sets such as \(B^*_i\), then \(\beta\) is a generalized handcuffed design, namely a \((v, k, t, \lambda)\)-HD. \(t\) is called the strength of the \((v, k, t, \lambda)\)-HD. Consider a \((v, k, t, \lambda)\)-HD \(\beta\) built on \(V\). If \(\beta\) is also a \((v, k, \theta, \lambda_\theta)\)-HD for all \(\theta\leq t\), then \(\beta\) is a balanced generalized handcuffed design, namely a \((v, k, t, \lambda)\)-BHD. \(t\) is called the strength of the \((v, k, t, \lambda)\)- BHD. It can be easily shown that \[ \begin{aligned} b & = \lambda {v!\over t!(v- t)! (k- t+ 1)},\tag{1}\\ \lambda_\theta & = \lambda {(v- \theta)! \theta! (k- \theta+ 1)\over (v- t)! t! (k- t+ 1)}\quad\text{for all}\quad \theta\leq t- 1\tag{2}\end{aligned} \] (\(\lambda_t= \lambda\) and \(\lambda_1= r\)). Therefore, for a \((v, k, t, \lambda)\)-BHD to exist it is necessary that \(b\) and \(\lambda_\theta\) in (1) and (2) are integers. For a \((v, 4, 3, 1)\)-BHD (balanced generalized handcuffed design) to exist the authors establish the necessary and sufficient condition that either \(v= 6n+ 2\) (\(n\) an integer \(\geq 1\)) or \(v= 6n+ 4\) (\(n\) an integer \(\geq 0\)).