Given a pick-7 lottery with 23 numbers that pays a prize to anyone matching at least 4 of the 7 numbers, there is a set of 253 tickets that guarantees a win. This set corresponds to the Witt design.
More formally, the Witt design on 23 points is a 4-(23,7,1) block design (Witt 1938). It is one of the most remarkable structures in all of combinatorics (Godsil and Royle 2001). It can be constructed by factoring over GF(2), into , where
The 2048 powers , , , ..., are then computed, mod . This set of vectors happens to be the [23,12,7] Golay code with 253 weight-7 vectors, 1288 weight-11 vectors, and 506 weight-15 vectors. For example, is a weight-7 vector.
The Witt design is the set of 253 weight-7 vectors acting on 23 points.
Consider as vertices the 253 vectors () and 23 points (). Set edges such that are adjacent if , and are adjacent if they share a single term. Select an arbitrary vertex. For all 176 neighbors of that vertex, change edges to non-edges, and non-edges to edges. Eliminate the now isolated vertex, and the remaining 275 vertex graph is the McLaughlin graph, a distance-regular graph.
From the Witt design, 77 vectors contain 0 (0 chosen arbitrarily from 0-22). Eliminating the 0's () gives the unique size 77 Steiner system on points 1 to 22. Consider as vertices the 77 vectors (), with adjacent if share no terms. This gives a 77-vertex graph known as the M22 graph.
Consider as vertices the 77 vectors (), 22 points (), and new symbol . Set edges such that is adjacent to all , are adjacent if , and are adjacent if share no terms. The resulting 100-vertex graph is the Higman-Sims graph, a distance-regular graph.
From on 77 vectors above, let the 56 vectors that do not contain the arbitrarily chosen point 22 be vertices, and connect disjoint vertices. This constructs the Gewirtz graph, a distance-regular graph on 56 nodes with intersection array .
The large Witt design is the 759 weight-8 vectors of the Golay code, frequently called the octads. The large Witt graph treats the 759 vectors of the large Witt design as vertices, with an edge connecting disjoint vectors. This is a distance-regular graph with intersection array .
The truncated Witt graph on 506 points is constructed by removing all vectors of the large Witt design containing an arbitrarily chosen symbol. This produces a distance-regular graph with intersection array .
The doubly truncated Witt graph on 330 points is constructed by removing all vectors of the large Witt design containing any two arbitrarily chosen symbols. This is a distance-regular graph with intersection array .
A similar construction factors over , producing the vectors of the Leech lattice (Conway and Sloane 1999).