A Geometric Proof of the Gap Theorem

Hollmann, Körner, and Litsyn used generalized Steiner systems to prove that it is impossible to partition ann-cube intokHamming spheres if 2<k<n+2. Furthermore, ifk=n+2, they showed the only partition of then-cube consists of a single sphere of radiusn−2 andn+1 spheres of radius 0. We give a geometric proof that this is the only nontrivial partition of ann-cube into fewer than 2p+2 spheres, wherepis the largest prime withp⩽n. We also show thatk=8 is the only value ofkbetween 4 and 11 such that it is possible to partition a cube other than the (k−2)-cube intokspheres.