High-Accuracy Semidefinite Programming Bounds for Kissing Numbers

  title={High-Accuracy Semidefinite Programming Bounds for Kissing Numbers},
  author={Hans D. Mittelmann and Frank Vallentin},
  journal={Experimental Mathematics},
  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… 

Improving the Semidefinite Programming Bound for the Kissing Number by Exploiting Polynomial Symmetry

The symmetry present in the semidefinite programming bound of Bachoc and Vallentin (2008) is exploited to provide improved upper bounds for the kissing number for values of n = 9, \ldots, 23.

Improving the Semidefinite Programming Bound for the Kissing Number by Exploiting Polynomial Symmetry

This article exploits the symmetry present in the semidefinite programming bound to provide improved upper bounds for the kissing number for n = 9, …, 23.

Mathematical Programming Bounds for Kissing Numbers

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Kissing numbers – a survey

This work surveys old and recent results on the kissing numbers keeping the generality of spherical codes.

Kissing number in hyperbolic space

This paper provides upper and lower bounds on the kissing number of congruent radius $r > 0$ spheres in $\mathbb{H}^n$, for $n\geq 2$. For that purpose, the kissing number is replaced by the kissing

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We consider extremal problems for continuous functions that are nonpositive on a closed interval and can be represented as series in Gegenbauer polynomials with nonnegative coefficients. These

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Kissing Numbers (1) appear to be the product of dimension number and the dimension’s simplex vertex number for 0-3 Euclidean spatial dimensions, but depart from the linear product of dimension and

Positive semidefinite approximations to the cone of copositive kernels

Two convergent hierarchies of subsets of copositive kernels, in terms of non-negative and positive definite kernels are proposed, which results in fast-to-compute upper bounds on the kissing number that lie between the currently existing LP and SDP bounds.

Upper bounds for packings of spheres of several radii

Abstract We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space.

Solving clustered low-rank semidefinite programs arising from polynomial optimization

A primal-dual interior point method specialized to clustered low-rank semidefinite programs, which arise from multivariate polynomial (matrix) programs through sums-of-squares characterizations and sampling, which allows for the computation of improved kissing number bounds in dimensions 11 through 23.



New upper bounds for kissing numbers from semidefinite programming

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
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

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

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.