Tables of sphere packings and spherical codes

@article{Sloane1981TablesOS,
  title={Tables of sphere packings and spherical codes},
  author={N. J. A. Sloane},
  journal={IEEE Trans. Inf. Theory},
  year={1981},
  volume={27},
  pages={327-338}
}
  • N. Sloane
  • Published 1 May 1981
  • Mathematics
  • IEEE Trans. Inf. Theory
The theta function of a sphere packing gives the number of centers at each distance from the origin. The theta functions of a number of important packings ( A_{n},D_{n},E_{n} , the Leech lattice, and others) and tables of the first fifty or so of their coefficients are given in this paper. 

Figures and Tables from this paper

Spherical codes generated by binary partitions of symmetric pointsets
TLDR
Several constructions are presented by which spherical codes are generated from groups of binary codes, the main ideas are code concatenation and set partitioning.
Nearest neighbor algorithm for spherical codes from the Leech lattice
TLDR
A nearest-neighbor algorithm that works on this is developed to determine the point in the code closet to some arbitrary vector in R/sup 24/.
Asymptotically dense spherical codes - Part h Wrapped spherical codes
TLDR
It is shown that the asymptotically maximum spherical coding density is achieved by wrapped spherical codes whenever /spl Lambda/ is the densest possible sphere packing.
Packing of regular tetrahedral quartets of circles on a sphere
  • T. Tarnai, P. W. Fowler, S. Kabai
  • Mathematics
    Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
  • 2003
How must 4N non‐overlapping equal circles forming N quartets be packed on a sphere so that the angular diameter of the circles will be as large as possible under the constraint that, within each
Constructing Spherical Codes by Global Optimization Methods
TLDR
Most of the 4-and 5-dimensional codes in this work are better than the previously known codes and it is shown how to reduce the number of optimization variables by restricting the solutions to have certain symmetries.
New Results in the Theory of Packing and Covering
Let J be a system of sets. If all members of J are contained in a given set C and each point of C belongs to at most one member of J then J is said to be a packing into C. If, on the other hand, each
The Coxeter–Todd lattice, the Mitchell group, and related sphere packings
This paper studies the Coxeter-Todd lattice Ag, its automorphism group (which is Mitchell's reflection group 6-P$C7(4,3)-2), and the associated 12-dimensional real lattice K12. We give several
A Note on the Leech Lattice as a Code for the Gaussian Channel
Voronoi regions of lattices, second moments of polytopes, and quantization
TLDR
The answers to the squared distance questions and a description of the Voronoi (or nearest neighbor) regions of these lattices have applications to quantization and to the design of signals for the Gaussian channel.
...
...

References

SHOWING 1-10 OF 70 REFERENCES
Hamming Association Schemes and Codes on Spheres
TLDR
A linear programming bound on the size of a minimum distance code on $S^n $ generalizes the bound obtained by N. J. A. Sloane in the discrete case.
Sphere Packings and Error-Correcting Codes
Error-correcting codes are used in several constructions for packings of equal spheres in n-dimensional Euclidean spaces En. These include a systematic derivation of many of the best sphere packings
Sphere packings constructed from BCH and Justesen codes
TLDR
Bose-Chaudhuri-Hocquenghem and Justesen codes are used to pack equa spheres in n –dimensional Euclidean space with density Δ satisfying for all sufficiently large n of the form m 2 m, appearing to be the densest packings yet constructed in high dimensional space.
Packing and Covering
Introduction 1. Packaging and covering densities 2. The existence of reasonably dense packings 3. The existence of reasonably economical coverings 4. The existence of reasonably dense lattice
Uniqueness of Certain Spherical Codes
TLDR
This paper shows that there is essentially only one way of arranging 240 nonoverlapping unit spheres in R 8 (resp. R 24) so that they all touch another unit sphere, and only one method of arranging 56(resp. 4600) spheres inR 8 so thatthey all touch two further, touching spheres.
Distance Properties of Group Codes for the Gaussian Channel
TLDR
A theorem on the representation of a doubly transitive permutation group is used to solve the optimal vector problem for the irreducible representation of dimension $n - 1$ of the symmetric group of degree n.
The number of lattice points in a $k$-dimensional hypersphere
Thus for fc ̂ 4 8k = fc/2 1. The value of fc which has received the greatest attention is fc = 2, the number of lattice points in a circle. Wilton [2] gives an account of the early work in this
Codes over GF ( 4 ) and Complex Lattices
The connections between binary and ternary error-correcting codes on the one hand, and lattices and sphere-packings in W on the other have been studied by several authors [6,7,
...
...