Tilings with Generalized Lee Spheres

We discuss tilings of ℝ n with bodies which are generalizations of the well known Lee spheres. It is shown that if n=2 then there exists a tiling of ℝ n with any generalized Lee spheres of order n. Tilings of ℝ n with generalized Lee spheres with radius one are also shown. All the tilings we consider are based on lattices.

[1]  S. Golomb,et al.  Perfect Codes in the Lee Metric and the Packing of Polyominoes , 1970 .

[2]  Steven Galovich,et al.  Splittings of Abelian groups by integers , 1981 .

[3]  Jaakko Astola.  An Elias-type bound for Lee codes over large alphabets and its application to perfect codes , 1982, IEEE Trans. Inf. Theory.

[4]  N. J. A. Sloane,et al.  Sphere Packings, Lattices and Groups , 1987, Grundlehren der mathematischen Wissenschaften.

[5]  Karel A. Post.  Nonexistence Theorems on Perfect Lee Codes over Large Alphabets , 1975, Inf. Control..

[6]  Solomon W. Golomb.  A general formulation of error matrices (Corresp.) , 1969, IEEE Trans. Inf. Theory.

[7]  Sherman K. Stein.  A symmetric star body that tiles but not as a lattice , 1972 .

[8]  William Hamaker.  Factoring groups and tiling space , 1973 .

[9]  W.A.A. Nuij.  Solution to Problem E2422 [1973, 691] - Tiling with incomparable rectangles , 1974 .

[10]  M. Birkner,et al.  Blow-up of semilinear PDE's at the critical dimension. A probabilistic approach , 2002 .

[11]  Hugh Everett,et al.  Packing and covering by translates of certain nonconvex bodies , 1979 .

[12]  Alexander Vardy,et al.  Interleaving Schemes for Multidimensional Cluster Errors , 1998, IEEE Trans. Inf. Theory.

[13]  Sherman Stein.  Packings of Rn by certain error spheres , 1984, IEEE Trans. Inf. Theory.

[14]  Rudolf Ahlswede,et al.  On Perfect Codes and Related Concepts , 2001, Des. Codes Cryptogr..

[15]  Dean R. Hickerson.  Splittings of finite groups , 1983 .

[16]  Sherman K. Stein.  Factoring by subsets , 1967 .

[17]  Sherman Stein,et al.  Combinatorial packings of R3 by certain error spheres , 1984, IEEE Trans. Inf. Theory.