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The q-analog of the Kramer-Mesner matrix

The next step in the construction is the evaluation of the entries of the Kramer-Mesner matrices. We achieve this goal by using the information obtained during the ladder game.

We can restrict attention to Kramer-Mesner matrices , because of

(The analogous equation -- with binomial coefficients instead of the Gaussian numbers -- holds for designs on sets, the proof of it is easily generalized to the -analog.

The matrix can be obtained as follows: We assume the representatives of the double cosets of and , representatives of the double cosets of and and the corresponding orbit graph. Now the entry of indexed by the orbits

is

where the sum is over all double cosets which are connected with the double coset and which are connected with the double coset at the same time.

If we take our last example then we get the Kramer-Mesner matrix:

 (3,0,4) (3,1,2,) (3,2,6) (3,3,6) (3,4,8) (3,5,4) (5,0,6) 6/2 6/2 0 6/6 0 0 (5,1,4) 4/4 4/2 4/2 0 4/4 4/4 (5,2,6) 0 6/2 0 6/6 0 6/2 (5,3,24) 0 0 0 24/6 24/8 0

,

and thus the -analog of the Kramer-Mesner matrix turns out to be

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