A Census of Minimal Pair-Coverings with Restricted Largest Block Length

The quantity \(g^k(v)\) was introduced in \([6]\) as the minimum number of blocks necessary in a pairwise balanced design on \(v\) elements, subject to the condition that the longest block has length \(k\). Recently, we have needed to use all possibilities for such minimal covering designs, and we record all non-isomorphic solutions to the problem for \(v \leq 13\).