On Covering Pairs by Quintuples: The Cases \(v \equiv 3\) or \(11\) modulo \(20\)

Let \(C(v)\) denote the least number of quintuples of a \(v\)-set \(V\) with the property that every pair of distinct elements of \(V\) occurs in at least one quintuple. It is shown, for \(v \equiv 3 \text{ or } 11\; \text{modulo} \;20\) and \(v \geq 11\), that \(C(v) = \lceil(v-1)/{4}\rceil\) with the possible exception of \(v \in \{83, 131\}\).