Packing pairs by quintuples with index 2: \(v\) odd, \(v\not\equiv 13\pmod{20}\) (Q1318789)

A \((v,k,\lambda)\) packing design is a collection \(\beta\) of \(k\)-element subsets, called blocks, of a \(v\)-set \(V\) such that every 2-element subset of \(V\) occurs in at most \(\lambda\) blocks. The problem is to find the maximum number \(\sigma (v,k,\lambda)\) of blocks in a \((v,k,\lambda)\) packing design. The authors determine \(\sigma (v,5,2)\) for all odd \(v\) with \(v \not \equiv 13 \pmod{20}\), with the possible exception of \(v=19\), 27, 137, 139, 147.