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

@article{Machado2018ImprovingTS,
title={Improving the Semidefinite Programming Bound for the Kissing Number by Exploiting Polynomial Symmetry},
author={Fabr{\'i}cio Caluza Machado and Fernando M{\'a}rio de Oliveira Filho},
journal={Exp. Math.},
year={2018},
volume={27},
pages={362-369}
}
• Published 16 September 2016
• Computer Science
• Exp. Math.
The kissing number of $\mathbb{R}^n$ is the maximum number of pairwise-nonoverlapping unit spheres that can simultaneously touch a central unit sphere. Mittelmann and Vallentin (2010), based on the semidefinite programming bound of Bachoc and Vallentin (2008), computed the best known upper bounds for the kissing number for several values of $n \leq 23$. In this paper, we exploit the symmetry present in the semidefinite programming bound to provide improved upper bounds for $n = 9, \ldots, 23$.

## Tables from this paper

### 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

### 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.

### Exact Semidefinite Programming Bounds for Packing Problems

• Computer Science, Mathematics
SIAM J. Optim.
• 2021
An algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of therationals, and sharp bounds are applied to get sharp bounds for packing problems, which prove certain optimal packing configurations are unique up to rotations.

### 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.

### Kissing number in non-Euclidean spaces

• Mathematics
• 2020
This paper provides upper and lower bounds on the kissing number of congruent radius $r > 0$ spheres in hyperbolic $\mathbb{H}^n$ and spherical $\mathbb{S}^n$ spaces, for $n\geq 2$. For that purpose,

### Dimension reduction for semidefinite programs via Jordan algebras

• Mathematics
Math. Program.
• 2020
This work shows if an orthogonal projection map satisfies certain invariance conditions, restricting to its range yields an equivalent primal–dual pair over a lower-dimensional symmetric cone—namely, the cone-of-squares of a Jordan subalgebra of symmetric matrices and presents a simple algorithm for minimizing the rank of this projection and hence the dimension of this subal algebra.

### High-dimensional sphere packing and the modular bootstrap

• Computer Science
Journal of High Energy Physics
• 2020
A more detailed picture of the behavior for finite c is given than was previously available, and an exponential improvement for sphere packing density bounds in high dimen- sions is indicated.

### Upper bounds for energies of spherical codes of given cardinality and separation

• Computer Science
Des. Codes Cryptogr.
• 2020
A linear programming framework for obtaining upper bounds for the potential energy of spherical codes of fixed cardinality and minimum distance is introduced using Hermite interpolation and polynomials are constructed.

### Conic Linear Optimization for Computer-Assisted Proofs (hybrid meeting)

• Mathematics
• 2022
From a mathematical perspective, optimization is the science of proving inequalities. In this sense, computational optimization is a method for computer-assisted proofs. Conic (linear) optimization

### An extension the semidefinite programming bound for spherical codes

• O. Musin
• Computer Science, Mathematics
• 2019
In this paper we present an extension of known semidefinite and linear programming upper bounds for spherical codes and consider a version of this bound for distance graphs. We apply the main result

## References

SHOWING 1-10 OF 25 REFERENCES

### High-Accuracy Semidefinite Programming Bounds for Kissing Numbers

• Mathematics
Exp. Math.
• 2010
High-accuracy calculations of upper bounds for the kissing number based on semidefinite programming are reported on, finding that there is no 16-dimensional periodic sphere packing with average theta series.

### New upper bounds for kissing numbers from semidefinite programming

• Computer Science
• 2006
This paper applies semidefinite programming to codes on the unit sphere and compute new upper bounds for the kissing number in several dimensions.

### New Upper Bounds for the Density of Translative Packings of Three-Dimensional Convex Bodies with Tetrahedral Symmetry

• Mathematics, Computer Science
Discret. Comput. Geom.
• 2017
New upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $$l^p_3$$l3p-norm) and of Platonic and Archimedean solids having tetrahedral symmetry are determined.

### 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

### Better Bounds for Planar Sets Avoiding Unit Distances

• Mathematics
Discret. Comput. Geom.
• 2016
It is shown that any 1-avoiding set in $$\mathbb {R}^n (n\ge 2)$$Rn(n≥2) that displays block structure that is made up of blocks such that the distance between any two points from the same block is less than 1 and points from distinct blocks lie farther than 1 unit of distance apart from each other has density strictly less than1/2n.

### Uniqueness of Certain Spherical Codes

• Computer Science