The Diametric Theorem in Hamming Spaces-Optimal Anticodes

نویسندگان

  • Rudolf Ahlswede
  • Levon H. Khachatrian
چکیده

Ž n . For a Hamming space X , d , the set of n-length words over the alphabet a H 4 n X s 0, 1, . . . , a y 1 endowed with the distance d , which for two words x s a H Ž . n Ž . n x , . . . , x , y s y , . . . , y g X counts the number of different components, 1 n 1 n a 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. Ž . 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 Ž . diametric theorem for a s 2 in the restricted case, where all n-length words considered have exactly k ones. Q 1998 Academic Press

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تاریخ انتشار 1998