An application of modified group divisible designs (Q1336444)

Let \(\kappa\), \(\lambda\), \(m\), \(\upsilon\) be positive integers. A group divisible design of order \(\kappa\), \(\lambda\), \(m\), \(\upsilon\) consists of a finite set \(V\) of points, a partition \(\gamma= \{G_ 1,\dots, G_ r\}\) of \(V\) into \(r\) subsets of \(V\) called groups and a collection \(\beta= \{B_ 1, B_ 2,\dots\}\) consisting of \(\kappa\)-subsets of \(V\) called blocks satisfying the following properties: (1) \(| G_ i|= m\), \(i= 1,\dots, r\). (2) \(| B\cap G|\leq 1\) for \(B\in \beta\) and \(G\in \gamma\). (3) Every 2-subset \(\{x,y\}\subset V\) whose elements \(x\), \(y\) belong to different groups \(G_ i\), \(G_ j\) is contained in exactly \(\lambda\) blocks. Group divisible designs with block sizes 3 and 4 are known to exist. In this paper the author constructs group divisible designs with block size 5 and several other covering designs and packing designs.