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

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