PUB - Publikationen an der Universität Bielefeld

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.