Existence of optimal strong partially balanced designs with block size five. (Q1428521)

A partially balanced \(t\)-design is a set \(X\) of \(v\) elements, and a collection \(\mathcal B\) of \(b\) \(k\)-subsets of \(X\) ({blocks}) so that every \(t\)-subset of \(X\) appears either in zero or in exactly \(\lambda\) of the blocks. It is {strong} if it is also a partially balanced \(s\)-design for \(0 < s < t\). Further it is {optimal} if \(b\) is the largest number of blocks achieved in any strong partially balanced \(t\)-design with the same \(v\), \(k\), and \(\lambda\). In this paper, optimal strong partially balanced \(t\)-designs are examined for \(t=2\), \(k=5\), and \(\lambda=1\).