On Covering Designs with Block Size \(5\) and Index \(4\)

A \((v, k, \lambda)\) covering design of order \(v\), block size \(k\), and index \(\lambda\) is a collection of \(k\)-element subsets, called blocks of a set \(V\) such that every \(2\)-subset of \(V\) occurs in at least \(\lambda\) blocks. The covering problem is to determine the minimum number of blocks in a covering design. In this paper we solve the covering problem with \(k = 5\) and \(\lambda = 4\) and all positive integers \(v\) with the possible exception of \(v = 17, 18, 19, 22, 24, 27, 28, 78, 98\).