# 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 and weight n, such that ${\displaystyle v=\left|X\right|={\binom {\ell }{n}}}$.[1][2][3] Two vectors xy ∈ X are called ith associates if dist(xy) = 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 Ek(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. Inform. 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.