Experimental Mathematics

High-Accuracy Semidefinite Programming Bounds for Kissing Numbers

Hans D. Mittelmann and Frank Vallentin

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Abstract

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 $n \leq 24$. The bound for $n = 16$ implies a conjecture of Conway and Sloane: there is no $16$-dimensional periodic sphere packing with average theta series $1 + 7680q^3 + 4320q^4 + 276480q^5 + 61440q^6 + \cdots.$

Article information

Source
Experiment. Math., Volume 19, Issue 2 (2010), 174-178.

Dates
First available in Project Euclid: 17 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.em/1276784788

Mathematical Reviews number (MathSciNet)
MR2676746

Zentralblatt MATH identifier
1279.11070

Subjects
Primary: 11F11: Holomorphic modular forms of integral weight 52C17: Packing and covering in $n$ dimensions [See also 05B40, 11H31] 90C10: Integer programming

Keywords
Kissing number semidefinite programming average theta series extremal modular form

Citation

Mittelmann, Hans D.; Vallentin, Frank. High-Accuracy Semidefinite Programming Bounds for Kissing Numbers. Experiment. Math. 19 (2010), no. 2, 174--178. https://projecteuclid.org/euclid.em/1276784788


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