Regular Article: The Diametric Theorem in Hamming Spaces驴Optimal Anticodes

  • Authors:

  • Rudolf Ahlswede;Levon H. Khachatrian

  • Affiliations:

  • Mathematik, Universitaat Bielefeld, Bielefeld, 33501, Germany;Mathematik, Universitaat Bielefeld, Bielefeld, 33501, Germany

  • Venue:
  • Advances in Applied Mathematics
  • Year:
  • 1998

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Abstract

For a Hamming space (X^n"@a,d"H), the set ofn-length words over the alphabet X"@a={0,1,...,@a-1} endowed with the distanced"H, which for two wordsx^n=(x"1,...,x"n),y^n=(y"1,...,y"n)@?X^n"@acounts the number of different components, we determine the maximal cardinality of subsets with a prescribed diameterdor, in another language, anticodes with distanced. We refer to the result as the diametric theorem. In a sense anticodes are dual to codes, which have a prescribedlowerbound 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 @a=2) in the restricted case, where alln-length words considered have exactlykones.