In mathematics, the Johnson scheme, named after

Selmer M. Johnson Selmer Martin Johnson (21 May 1916 – 26 June 1996) was an American mathematician, a researcher at the RAND Corporation. Biography Johnson was born on May 21, 1916, in Buhl, Minnesota. He earned a B.A. and then an M.A. in mathematics from the Uni ...

, is also known as the triangular

association scheme The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. In algebraic combinatorics, association schemes ...

. It consists of the set of all binary vectors ''X'' of length ''ℓ'' and weight ''n'', such that

v=\left, X\=\binom

.F. J. MacWilliams and N. J. A. Sloane, ''The Theory of Error-Correcting Codes'', Elsevier, New York, 1978. Two vectors ''x'', ''y'' ∈ ''X'' are called ''i''th associates if dist(''x'', ''y'') = 2''i'' for ''i'' = 0, 1, ..., ''n''. The

eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

are given by :

p_\left(k\right)=E_\left(k\right),

q_\left(i\right)=\fracE_\left(k\right),

where :

\mu_=\frac\binom,

and ''E''_''k''(''x'') is an Eberlein polynomial defined by :

E_\left(x\right)=\sum_^(-1)^\binom \binom\binom,\qquad k=0,\ldots,n.

References

{{reflist Combinatorics