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

Abstract

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.