On Covering Designs with Block Size \(5\) and Index \(7\)

Let \(V\) be a finite set of order \(v\). A \((v, k, \lambda)\) covering design of index \(\lambda\) and block size \(k\) is a collection of \(k\)-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, \(\alpha(v, k, \lambda)\), in a covering design. It is well known that \(\alpha(v,k,\lambda) \geq \lceil\frac{v}{k}\lceil\frac{v-1}{k-1}\lambda\rceil\rceil = \phi(v, k, \lambda)\), where \(\lceil x \rceil\) is the smallest integer satisfying \(x \leq \lceil x \rceil\). It is shown here that \(\alpha(v,5,7) = \phi(v, 5, 7)\) for all positive integers \(v \geq 5\) with the possible exception of \(v = 22, 28, 142, 162\).