Encyclopaedia of DesignTheory: An association scheme

This example is the (ordinary) cube. The labels are the coordinates of the vertices and each colour represents the positions where one of the four matrices forming the association scheme has the entry 1.

If white, yellow, black and red label matrices A₀, A₁, A₂, A₃ respectively, then we have the equations

A0² = A0	A0A1 = A1	A0A2 = A2	A0A3 = A3
A1A0 = A1	A1² = 3A0 + 2A2	A1A2 = 2A1 + 3A3	A1A3 = A2
A2A0 = A2	A2A1 = 2A1 + 3A3	A2² = 3A0 + 2A2	A2A3 = A1
A3A0 = A3	A3A1 = A2	A3A2 = A1	A3² = A0

The algebra spanned by these four 8×8 matrices is the Bose-Mesner algebra of the association scheme.

This association scheme is the Hamming scheme H(3,2).

The six faces of the cube form a partially balanced design based on this association scheme. In detail: the blocks are

000, 001, 010, 011

100, 101, 110, 111

000, 001, 100, 101

010, 011, 110, 111

000, 010, 100, 110

001, 011, 101, 111

We see that every point (every vertex of the cube) lies in three blocks (faces of the cube). Moreover:

• If two points are first associates (i.e. two vertices on an edge of the cube), they lie in two blocks;
• If two points are second associates (i.e. opposite vertices of a face), they lie in one block;
• If two points are third associates (i.e. opposite vertices of the cube), they lie in no blocks.

For more pictures like this one, see the Web page for the book Association Schemes: Designed experiments, algebra and combinatorics by R. A. Bailey, Cambridge University Press, Cambridge, 2004.

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R. A. Bailey, Peter J. Cameron
20 November 2002