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.

On a six-point set $X$ and its six totals of synthemes $Y$, each duad determines a partition of $Y$ into three pairs according to their shared synthemes. Include both whole halves, the 45 blocks $D\cup (Y\setminus P)$ for these duad-pair incidences, and their complements. For each partition $X=A\bigsqcup A^c$ into triples, cross synthemes give two triangular components on $Y$; include the four unions of a part of $X$ with a component. These 132 blocks form the small Witt design.