THE DIAMETRIC THEOREM IN HAMMING-SPACES - OPTIMAL ANTICODES

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 di ameter 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 f ind it remarkable that the diametric theorem (for arbitrary a) can be derived from our recent complete intersection theorem, which can be vi ewed as a diametric theorem (for alpha = 2) in the restricted case, wh ere all n-length words considered have exactly k ones. (C) 1998 Academ ic Press.