Johnson scheme explained

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=\left|X\right|=\binom{\ell}{n}

.^[1] ^[2] ^[3] 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\left(k\right)=E_i\left(k\right),

q_k\left(i\right)=

	\mu_k
	v_i

E_i\left(k\right),

where

\mu_i=

	\ell-2i+1
	\ell-i+1

\binom{\ell}{i},

and E_k(x) is an Eberlein polynomial defined by

E_k

	k
\left(x\right)=\sum
	j=0

(-1)^j\binom{x}{j}\binom{n-x}{k-j}\binom{\ell-n-x}{k-j}, k=0,\ldots,n.

Notes and References

P. Delsarte and V. I. Levenshtein, “Association schemes and coding theory,“ IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2477 - 2504, 1998.
P. Camion, "Codes and Association Schemes: Basic Properties of Association Schemes Relevant to Coding," in Handbook of Coding Theory, V. S. Pless and W. C. Huffman, Eds., Elsevier, The Netherlands, 1998.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier, New York, 1978.