TY - GEN

T1 - Perfect tilings of binary spaces

AU - Cohen, Gerard

AU - Litsyn, Simon

AU - Vardy, Alexander

AU - Zemor, Gilles

PY - 1993

Y1 - 1993

N2 - We study partitions of the space F2n 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 F2n 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 F2n. 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.

AB - We study partitions of the space F2n 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 F2n 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 F2n. 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.

UR - http://www.scopus.com/inward/record.url?scp=0027307807&partnerID=8YFLogxK

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AN - SCOPUS:0027307807

SN - 0780308786

T3 - Proceedings of the 1993 IEEE International Symposium on Information Theory

SP - 370

BT - Proceedings of the 1993 IEEE International Symposium on Information Theory

PB - Publ by IEEE

T2 - Proceedings of the 1993 IEEE International Symposium on Information Theory

Y2 - 17 January 1993 through 22 January 1993

ER -