The diametric theorem in Hamming spaces-optimal anticodes

For a Hamming space (𝒳_αⁿ,d_H ), the set of n-length words over the alphabet 𝒳_α ={0,1,...,α-1} endowed with the distance d_H, which for two words xⁿ=(x₁,...,xⁿ), y ⁿ=(y₁,...,y_n)∈𝒳_α 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 diametric theorem