Johnson scheme - Alchetron, The Free Social Encyclopedia

In mathematics, the Johnson scheme, named after Selmer M. Johnson, is also known as the triangular association scheme. It consists of the set of all binary vectors X of length ℓ and weight n, such that v = | X | = ( ℓ n ) . Two vectors x, y ∈ X are called ith associates if dist(x, y) = 2i for i = 0, 1, ..., n. The eigenvalues are given by

p i ( k ) = E i ( k ) , q k ( i ) = μ k v i E i ( k ) ,

where

μ i = ℓ − 2 i + 1 ℓ − i + 1 ( ℓ i ) ,

and E_k(x) is an Eberlein polynomial defined by

E k ( x ) = ∑ j = 0 k ( − 1 ) j ( x j ) ( n − x k − j ) ( ℓ − n − x k − j ) , k = 0 , … , n .

References

Johnson scheme Wikipedia

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