On Covering Designs with Block Size \(5\) and Index \(6\)

Let \(V\) be a finite set of order \(v\). A \((v, \kappa, \lambda)\) covering design of index \(\lambda\) and block size \(\kappa\) is a collection of \(\kappa\)-element subsets, called blocks, such that every \(2\)-subset of \(V\) occurs in at least \(\lambda\) blocks. The covering problem is to determine the minimum number of blocks, \(\alpha(v, \kappa, \lambda)\), in a covering design. It is well known that
\(\alpha(v, \kappa, \lambda) \geq \lceil \frac{v}{\kappa}\lceil\frac{v-1}{\kappa -1}\lambda\rceil\rceil = \phi(v, \kappa, \lambda)\)
where \(\lceil x \rceil\) is the smallest integer satisfying \(x \leq \lceil x \rceil\). It is shown here that
\(\alpha(v, 5, 6) = \phi (v, 5, 6)\) for all positive integers \(v \geq 5\), with the possible exception of \(v = 18\).