Kostka polynomial: Difference between revisions

Revision as of 17:03, 8 April 2013

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. Inform. 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. 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.

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+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&nbsp;''n'', such that <math>v=\left|X\right|=\binom{\ell}{n}</math>.<ref>P. Delsarte and V. I. Levenshtein, “Association schemes and coding theory,“  ''IEEE Trans. Inform. Theory'', vol. 44, no. 6, pp. 2477&ndash;2504, 1998.</ref><ref>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.</ref><ref>F. J. MacWilliams and N. J. A. Sloane, ''The Theory of Error-Correcting Codes'', Elsevier, New York, 1978.</ref>  Two vectors&nbsp;''x'',&nbsp;''y''&nbsp;&isin;&nbsp;''X'' are called ''i''th associates if dist(''x'',&nbsp;''y'')&nbsp;=&nbsp;2''i'' for ''i''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;...,&nbsp;''n''. The [[eigenvalues]] are given by
+: <math>
+p_{i}\left(k\right)=E_{i}\left(k\right),
+</math>
+: <math>
+q_{k}\left(i\right)=\frac{\mu_{k}}{v_{i}}E_{i}\left(k\right),
+</math>
+where
+: <math>
+\mu_{i}=\frac{\ell-2i+1}{\ell-i+1}\binom{\ell}{i},
+</math>
+and ''E''<sub>''k''</sub>(''x'') is an '''Eberlein polynomial''' defined by
+: <math>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.</math>
+==References==
+{{reflist}}
+[[Category:Combinatorics]]

Kostka polynomial: Difference between revisions

Revision as of 17:03, 8 April 2013

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