The diametric theorem in Hamming spaces - Optimal anticodes

Ahlswede R, Khachatrian LH (1998)
ADVANCES IN APPLIED MATHEMATICS 20(4): 429-449.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
For a Hamming space (H-alpha(n), d(H)), the set of n-length words over the alphabet H-alpha = {0,1,...,alpha-1} endowed with the distance d(H), which for two words x(n) = (x(1),...,x(n)), y(n) = (y(1),...,y(n)) is an element of H-alpha(n) counts the number of different components, we determine the maximal cardinality of subsets with a prescribed diameter d or, in another language, anticodes with distance d. We refer to the result as the diametric theorem. In a sense anticodes are dual to codes, which have a prescribed lower bound on the pairwise distance. It is a hopeless task to determine their maximal sizes exactly. We find it remarkable that the diametric theorem (for arbitrary a) can be derived from our recent complete intersection theorem, which can be viewed as a diametric theorem (for alpha = 2) in the restricted case, where all n-length words considered have exactly k ones. (C) 1998 Academic Press.
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Zeitschriftentitel
ADVANCES IN APPLIED MATHEMATICS
Band
20
Zeitschriftennummer
4
Seite
429-449
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Ahlswede R, Khachatrian LH. The diametric theorem in Hamming spaces - Optimal anticodes. ADVANCES IN APPLIED MATHEMATICS. 1998;20(4):429-449.
Ahlswede, R., & Khachatrian, L. H. (1998). The diametric theorem in Hamming spaces - Optimal anticodes. ADVANCES IN APPLIED MATHEMATICS, 20(4), 429-449. doi:10.1006/aama.1998.0588
Ahlswede, R., and Khachatrian, L. H. (1998). The diametric theorem in Hamming spaces - Optimal anticodes. ADVANCES IN APPLIED MATHEMATICS 20, 429-449.
Ahlswede, R., & Khachatrian, L.H., 1998. The diametric theorem in Hamming spaces - Optimal anticodes. ADVANCES IN APPLIED MATHEMATICS, 20(4), p 429-449.
R. Ahlswede and L.H. Khachatrian, “The diametric theorem in Hamming spaces - Optimal anticodes”, ADVANCES IN APPLIED MATHEMATICS, vol. 20, 1998, pp. 429-449.
Ahlswede, R., Khachatrian, L.H.: The diametric theorem in Hamming spaces - Optimal anticodes. ADVANCES IN APPLIED MATHEMATICS. 20, 429-449 (1998).
Ahlswede, Rudolf, and Khachatrian, LH. “The diametric theorem in Hamming spaces - Optimal anticodes”. ADVANCES IN APPLIED MATHEMATICS 20.4 (1998): 429-449.