Small Witt design ($S(5,6,12)$)

The small Witt design is a Steiner system $S(5,6,12)$ with 132 blocks. A six-set duad-syntheme duality constructs it on two disjoint six-sets: two whole-half blocks, forty-five blocks of each of the types $(2,4)$ and $(4,2)$, and forty corresponding-partition blocks of type $(3,3)$. The internal-duad/cross-syntheme incidence rule proves that each five-set lies in exactly one of these blocks.