High-accuracy semidefinite programming bounds for kissing numbers

Hans Mittelmann; Frank Vallentin

doi:10.1080/10586458.2010.10129070

High-accuracy semidefinite programming bounds for kissing numbers

Hans Mittelmann, Frank Vallentin

Mathematical and Statistical Sciences, School of (SoMSS)

Research output: Contribution to journal › Article › peer-review

28 Scopus citations

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 ≤ 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 + 7680q³ + 4320q⁴ + 276480q⁵ + 61440q⁶ + ・・・.

Original language	English (US)
Pages (from-to)	175-179
Number of pages	5
Journal	Experimental Mathematics
Volume	19
Issue number	2
DOIs	https://doi.org/10.1080/10586458.2010.10129070
State	Published - 2010

Keywords

Average theta series
Extremal modular form
Kissing number
Semidefinite programming

ASJC Scopus subject areas

Mathematics(all)

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10.1080/10586458.2010.10129070

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title = "High-accuracy semidefinite programming bounds for kissing numbers",

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 ≤ 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 + 7680q3 + 4320q4 + 276480q5 + 61440q6 + ・・・.",

keywords = "Average theta series, Extremal modular form, Kissing number, Semidefinite programming",

author = "Hans Mittelmann and Frank Vallentin",

note = "Funding Information: The second author was partially supported by the Deutsche Forschungsgemeinschaft (DFG) under grant SCHU 1503/4.",

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doi = "10.1080/10586458.2010.10129070",

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AU - Mittelmann, Hans

AU - Vallentin, Frank

N1 - Funding Information: The second author was partially supported by the Deutsche Forschungsgemeinschaft (DFG) under grant SCHU 1503/4.

PY - 2010

Y1 - 2010

N2 - 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 series 1 + 7680q3 + 4320q4 + 276480q5 + 61440q6 + ・・・.

AB - 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 series 1 + 7680q3 + 4320q4 + 276480q5 + 61440q6 + ・・・.

KW - Average theta series

KW - Extremal modular form

KW - Kissing number

KW - Semidefinite programming

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High-accuracy semidefinite programming bounds for kissing numbers

Abstract

Keywords

ASJC Scopus subject areas

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