Inequalities for Code Pairs

Our main discovery is the inequality: If A, B ⊂ {1,..., α}m; m ∈ ℕ; satisfies for the Hamming distance dd(a,b)−d(a,b′)+d(a′,b′)−d(a′,b)≠1,2foralla,a′∈Aandb,b′∈B then |A||B| ⩽ d*m, where d*={αforα=2,3,4,⌊α2Ȱ⌈α2Ȱforα≥4, and the bound is best.It is much more general than its predecessors ([1], [2]) and has a perspicuous combinatorial proof