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Witt Design


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 x^(23)-1 over GF(2), into (x-1)g_2(x)(x^(11)g_2(x^(-1))), where

 g_2(x)=x^(11)+x^9+x^7+x^6+x^5+x+1.

The 2048 powers g_2^0(x), g_2^1(x), g_2^2(x), ..., g_2^(2047)(x) are then computed, mod x^(23)-1. 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, g_2(x)^1={0,1,5,6,7,9,11} 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 (v in V) and 23 points (p in P). Set edges such that p&v are adjacent if p not in v, and v_j&v_k 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 (0,1,5,6,7,9,11->1,5,6,7,9,11) gives the unique size 77 Steiner system S(3,6,22) on points 1 to 22. Consider as vertices the 77 vectors (v in V), with v_j&v_k adjacent if share no terms. This gives a 77-vertex graph known as the M22 graph.

Consider as vertices the 77 vectors (v in V), 22 points (p in P), and new symbol Omega. Set edges such that Omega is adjacent to all p, p&v are adjacent if p in v, and v_j&v_k are adjacent if share no terms. The resulting 100-vertex graph is the Higman-Sims graph, a distance-regular graph.

From V 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 {10,9;1,2}.

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 {30,28,24;1,3,15}.

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 {15,14,12;1,1,9}.

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 {7,6,4,4;1,1,1,6}.

A similar construction factors x^(23)-1 over GF(2^2), producing the 196560 vectors of the Leech lattice (Conway and Sloane 1999).


See also

Doubly Truncated Witt Graph, Gewirtz Graph, Golay Code, Higman-Sims Graph, Ivanov-Ivanov-Faradjev Graph, Large Witt Graph, Leech Lattice, M22 Graph, Mathieu Groups, McLaughlin Graph, Truncated Witt Graph

This entry contributed by Ed Pegg, Jr.

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References

Brouwer, A. E. and Haemers, W. H. "The Gewirtz Graph: An Exercise in the Theory of Graph Spectra." Eur. J. Comb. 14, 397-407, 1993.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. "Graphs Related to the Witt Designs." §11.4 in Distance-Regular Graphs. New York: Springer-Verlag, pp. 366-373, 1989.Cameron, P. J. and van Lint, J. H. Designs, Graph, Codes, and Their Links. Cambridge: Cambridge University Press, 1996.Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, 1993.Godsil, C. In Interesting Graphs and Their Colourings. Unpublished manuscript. 2004.Godsil, C. and Royle, G. "Witt Design on 23 Points." §10.11 in Algebraic Graph Theory. New York: Springer-Verlag, pp. 241-242, 2001.Havelicek, H. and Lenz, H. "Another Simple Proof for the Existence of the Small Witt Design." http://dmg.tuwien.ac.at/havlicek/anothersimple.pdf.Heumann, S. "Golay Codes." http://www.mdstud.chalmers.se/~md7sharo/coding/main/node34.html.van Lint, J. H. An Introduction to Coding Theory, 2nd ed. New York: Springer-Verlag, 1992.Witt, E. "Die 5-Fach transitiven Gruppen von Mathieu." Abh. Math. Sem. Univ. Hamburg 12, 256-264, 1938.

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Witt Design

Cite this as:

Pegg, Ed Jr. "Witt Design." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/WittDesign.html

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