# 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} }

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 $n = 9, \ldots, 23$.

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## 17 Citations

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## References

SHOWING 1-10 OF 25 REFERENCES

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

- MathematicsExp. 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 ScienceDiscret. 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

- Mathematics
- 2003

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…

### Symmetry groups, semidefinite programs, and sums of squares

- Mathematics, Computer Science
- 2004

### Better Bounds for Planar Sets Avoiding Unit Distances

- MathematicsDiscret. 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 ScienceCanadian Journal of Mathematics
- 1981

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.

### New Bounds on the Number of Unit Spheres That Can Touch a Unit Sphere in n Dimensions

- Computer ScienceJ. Comb. Theory, Ser. A
- 1979

### Invariant Semidefinite Programs

- Mathematics
- 2012

The basic theory is given and it is illustrated in applications from coding theory, combinatorics, geometry, and polynomial optimization.