A Steiner system is a -point set with -point blocks such that every -point subset is contained in exactly one block. Necessarily the number of blocks is . This count is a necessary consistency check, not a substitute for proving exact containment. The small Witt design is .
The small Witt design is a Steiner system 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 and , and forty corresponding-partition blocks of type . The internal-duad/cross-syntheme incidence rule proves that each five-set lies in exactly one of these blocks.
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A Steiner system is a specific type of combinatorial design that relates to the arrangement of points and subsets of those points. More formally, a Steiner system \( S(t, k, n) \) is defined by three parameters \( t \), \( k \), and \( n \), where: - \( n \) is the total number of points. - \( k \) is the size of each subset (often called a block).