The quantity \(g_2^{(k)}(v)\) is the minimum number of blocks in a family of blocks from a \(v\)-set that covers all \(\binom{v}{2}\) pairs exactly twice, given the restriction that the longest block in the covering family has length \(k\) (there may be many blocks of length \(k\)). We give certain results for the case \(k = 4\).