Steiner system ($S(t,k,v)$)

A Steiner system $S(t,k,v)$ is a $v$-point set with $k$-point blocks such that every $t$-point subset is contained in exactly one block. Necessarily the number of blocks is $\left(\genfrac{}{}{0ex}{}{v}{t}\right)/\left(\genfrac{}{}{0ex}{}{k}{t}\right)$. This count is a necessary consistency check, not a substitute for proving exact containment. The small Witt design is $S(5,6,12)$.

A designated subset in a Steiner system is a Steiner block. In a system $S(t,k,v)$, each Steiner block contains $k$ points and each $t$-point subset belongs to exactly one Steiner block. For $t=2$, this is the pair-coverage condition used in a cyclotomic construction of a Steiner 2-design.

In a finite additive group, a family of $k$-subsets is a difference family of index one if each nonzero group element occurs exactly once among all their ordered within-block differences. Translating its Steiner blocks by every group element gives a Steiner system with parameters $2$-$(v,k,1)$: the unique representation of a difference determines the unique translated Steiner block through two specified points.

Put $m=\left(\genfrac{}{}{0ex}{}{k}{2}\right)$, take odd $q\equiv 1(\mathrm{mod}2m)$, and let $H$ be the index-$m$ multiplicative subgroup. If a $k$-set $A$ has its unordered pair differences in distinct $H$-cosets, choose $S\subset H$ containing exactly one element from each pair $\{h,-h\}$. Then $\{sA:s\in S\}$ is a difference family for a Steiner 2-design. Indeed one unordered pair supplies each coset, and its two difference orientations multiplied by $S$ cover that coset once. Translations yield a $2$-$(q,k,1)$ Steiner system. The cyclotomic pattern extension by a second-moment bound supplies suitable $A$ for all sufficiently large $q$ in this progression.

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.