Perfect Tilings of Binary Spaces

We study partitions of the space F/sup n//sub 2// of all the binary n-tuples into disjoint sets, such that each set is an additive coset of a given set V. Such a partition is called a perfect tiling of F/sup n/sub 2// and denoted (V, A), where A is the set of coset representatives. A sufficient condition for a set V to be a tile is given in terms of the cardinality of V+V. A perfect tiling (V, A) is said to be proper if V generates F/sup n/sub 2//. We show that the classification of perfect tilings can be reduced to the study of proper perfect tilings. We then prove that each proper perfect tiling is uniquely associated with a perfect binary code. A construction of proper perfect tilings from perfect binary codes is presented. Furthermore, we introduce a class of perfect tilings obtained by iterating a simple recursive construction. Finally, we generalize the well-known Lloyd theorem, originally stated for tilings by spheres, for the case of arbitrary perfect tilings.