On the Packing of Pairs by Quintuples With Index \(2\)

A packing design (briefly packing) of order \(v\), block size \(k\), and index \(\lambda\) is a pair \((X,\mathcal{D})\) where \(X\) is a \(v\)-set (of points) and \(\mathcal{D}\) is a collection of \(k\)-subsets of \(X\) (called blocks) with cardinality \(b\) such that every \(2\)-subset of \(X\) is contained in at most \(\lambda\) blocks of \(\mathcal{D}\). We denote it by \(\mathrm{SD}(k,\lambda; v,b)\). If no other such packing has more blocks, the packing is said to be maximum, and the number of blocks in \(\mathcal{D}\) is the packing number \(\mathrm{D}(k,\lambda;v)\). For fixed \(k\),\(\lambda\) and \(v\), the packing problem is to determine the packing number. In this paper, the values of \(\mathrm{D}(5,2; v)\) are determined for all \(v \geq 5\) except \(48\) values of \(v\).