Johnson scheme

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 ${\displaystyle v=\left|X\right|={\binom {\ell }{n}}}$.[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

${\displaystyle p_{i}\left(k\right)=E_{i}\left(k\right),}$

${\displaystyle q_{k}\left(i\right)={\frac {\mu _{k}}{v_{i}}}E_{i}\left(k\right),}$

where

${\displaystyle \mu _{i}={\frac {\ell -2i+1}{\ell -i+1}}{\binom {\ell }{i}},}$

and E_k(x) is an Eberlein polynomial defined by

${\displaystyle E_{k}\left(x\right)=\sum _{j=0}^{k}(-1)^{j}{\binom {x}{j}}{\binom {n-x}{k-j}}{\binom {\ell -n-x}{k-j}},\qquad k=0,\ldots ,n.}$

References

1. P. Delsarte and V. I. Levenshtein, “Association schemes and coding theory,“ IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2477–2504, 1998.
2. 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.
3. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier, New York, 1978.