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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.

000 001 010 011 100 101 110 111
000                
001                
010                
011                
100                
101                
110                
111                

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).

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R. A. Bailey, Peter J. Cameron
7 August 2002