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 $v = | X | = (\binom{ℓ}{n})$ .^[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} (k) = E_{i} (k),

q_{k} (i) = \frac{μ_{k}}{v_{i}} E_{i} (k),

where

μ_{i} = \frac{ℓ - 2 i + 1}{ℓ - i + 1} (\binom{ℓ}{i}),

and E_k(x) is an Eberlein polynomial defined by

E_{k} (x) = \sum_{j = 0}^{k} (- 1)^{j} (\binom{x}{j}) (\binom{n - x}{k - j}) (\binom{ℓ - n - x}{k - j}), k = 0, \dots, n .

References

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↑ 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.

[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.

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Johnson scheme

References

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