On the Voronoi Regions of Certain Lattices

The Voronoi region of a lattice $L_n \subseteq \mathbb{R}^n $ is the convex polytope consisting of all points of $\mathbb{R}^{\text{n}} $ that are closer to the origin than to any other point of $L_n $. In this paper we calculate the second moments of the Voronoi regions of the lattices $E_6^ * ,E_7^ * ,K_{12} ,\Lambda _{16} $ and $\Lambda _{24} $. The results show that these lattices are the best quantizers presently known in dimensions 6, 7,12,16 and 24. The calculations are performed by Monte Carlo integration, and make use of fast algorithms for finding the closest lattice point to an arbitrary point of the space. We also establish two general theorems concerning the number of faces of the Voronoi region of a lattice.

[1]  H. Coxeter,et al.  Extreme Forms , 1951, Canadian Journal of Mathematics.

[2]  L. Tóth.  Lagerungen in der Ebene auf der Kugel und im Raum , 1953 .

[3]  M. H. Quenouille.  NOTES ON BIAS IN ESTIMATION , 1956 .

[4]  E. S. Barnes,et al.  Some extreme forms defined in terms of Abelian groups , 1959, Journal of the Australian Mathematical Society.

[5]  J. Leech.  Notes on Sphere Packings , 1967, Canadian Journal of Mathematics.

[6]  John H. Conway,et al.  A characterisation of Leech's lattice , 1969 .

[7]  J. Halton.  A Retrospective and Prospective Survey of the Monte Carlo Method , 1970 .

[8]  Hans Niemeier.  Definite quadratische formen der dimension 24 und diskriminante 1 , 1973 .

[9]  Walter Feit,et al.  Codes over GF ( 4 ) and Complex Lattices , 1978 .

[10]  Sergeĭ Sergeevich Ryshkov,et al.  C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra : with application to the theory of coverings , 1978 .

[11]  Allen Gersho,et al.  Asymptotically optimal block quantization , 1979, IEEE Trans. Inf. Theory.

[12]  D. Ray-Chaudhuri.  Relations between combinatorics and other parts of mathematics , 1979 .

[13]  N. J. A. Sloane,et al.  Tables of sphere packings and spherical codes , 1981, IEEE Trans. Inf. Theory.

[14]  N. J. A. Sloane,et al.  Fast quantizing and decoding and algorithms for lattice quantizers and codes , 1982, IEEE Trans. Inf. Theory.

[15]  N. J. A. Sloane,et al.  Voronoi regions of lattices, second moments of polytopes, and quantization , 1982, IEEE Trans. Inf. Theory.

[16]  N. J. A. Sloane,et al.  On a problem of Ryskov concerning lattice coverings , 1982 .

[17]  Allen Gersho,et al.  On the structure of vector quantizers , 1982, IEEE Trans. Inf. Theory.

[18]  Paul L. Zador,et al.  Asymptotic quantization error of continuous signals and the quantization dimension , 1982, IEEE Trans. Inf. Theory.

[19]  N. Sloane,et al.  The Optimal Lattice Quantizer in Three Dimensions , 1983 .

[20]  Neil J. A. Sloane,et al.  Complex and integral laminated lattices , 1983 .

[21]  N. J. A. Sloane,et al.  The Coxeter–Todd lattice, the Mitchell group, and related sphere packings , 1983, Mathematical Proceedings of the Cambridge Philosophical Society.