High-Accuracy Semidefinite Programming Bounds for Kissing Numbers

@article{Mittelmann2010HighAccuracySP,
  title={High-Accuracy Semidefinite Programming Bounds for Kissing Numbers},
  author={Hans D. Mittelmann and Frank Vallentin},
  journal={Experimental Mathematics},
  year={2010},
  volume={19},
  pages={175 - 179}
}
The kissing number in n-dimensional Euclidean space is the maximal number of nonoverlapping unit spheres that simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing number based on semidefinite programming. This paper is a report on high-accuracy calculations of these upper bounds for n ≤ 24. The bound for n = 16 implies a conjecture of Conway and Sloane: there is no 16-dimensional periodic sphere packing with average theta… 

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References

SHOWING 1-10 OF 32 REFERENCES

New upper bounds for kissing numbers from semidefinite programming

TLDR
This paper applies semidefinite programming to codes on the unit sphere and compute new upper bounds for the kissing number in several dimensions.

The kissing number in four dimensions

The kissing number problem asks for the maximal number k(n) of equal size nonoverlapping spheres in n-dimensional space that can touch another sphere of the same size. This problem in dimension three

Kissing numbers, sphere packings, and some unexpected proofs

The ikissing number problemi asks for the maximal number of white spheres that can touch a black sphere of the same size in n-dimensional space. The answers in dimensions one, two and three are

Sphere packings, I

  • T. Hales
  • Physics, Mathematics
    Discret. Comput. Geom.
  • 1997
TLDR
A program to prove the Kepler conjecture on sphere packings is described and it is shown that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.

Sphere Packings, Lattices and Groups

The second edition of this book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also continue to

Lattices , Linear Codes , and Invariants , Part I

TLDR
The present article is a two-part series devoted to lattices, linear codes, and their relations with other branches of mathematics, focusing on certain invariants attached to lattice and codes.

Extremal Lattices

This paper deals with discrete subgroups of euclidean vector spaces, equivalently finitely generated free abelian groups (isomorphic to Z for some n ∈ N) together with a positive definite quadratic

The D 4 Root System Is Not Universally Optimal

TLDR
It is proved that the D 4 root system (equivalently, the set of vertices of the regular 24-cell) is not a universally optimal spherical code, and it is conjectured that every 5-design consisting of 24 points in S 3 is in a 3-parameter family.