# 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…

## 54 Citations

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

- Computer ScienceExp. Math.
- 2018

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

- Computer Science
- 2018

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

- Mathematics
- 2017

The kissing number in K dimensions is the maximum number of unit balls arranged around a central unit ball in such a way that the intersection of the interiors of any pair of balls in the configuration is empty.

### Kissing numbers – a survey

- Computer Science
- 2015

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

### Kissing number in hyperbolic space

- Mathematics
- 2019

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…

### Delsarte method in the problem on kissing numbers in high-dimensional spaces

- Mathematics
- 2014

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…

### Kissing Number Cells and Integral Conjecture

- Mathematics
- 2013

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

- Mathematics, Computer Science
- 2018

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

- MathematicsForum of Mathematics, Sigma
- 2014

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

- Computer Science, Mathematics
- 2022

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.

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