On the Classification of Resolvable \(2\)-\((12, 6, 5c)\) Designs

In this paper, we describe a backtrack search over parallel classes with a partial isomorph rejection to classify resolvable \(2\)-(12, 6, \(5c\)) designs. We use the intersection pattern between the parallel classes and the fact that any resolvable \(2\)-(12, 6, \(5c\)) design is also a resolvable \(3\)-(12, 6, \(2c\)) design to effectively guide the search. The method was able to enumerate all nonsimple resolutions and a subfamily of simple resolutions of a \(2\)-(12, 6, 15) design. The method is also used to confirm the computer classification of the resolvable \(2\)-(12, 6, \(5c\)) designs for \(c \in \{1, 2\}\). A consistency checking based on the principle of double counting is used to verify the computation results.