Kissing number

Kissing number

The last chapter

Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016

The Newton number or kissing number problem is a local variant of the sphere packing problem. Fix a sphere of radius a in Euclidean n-space. The Newton number τn is the maximal number of spheres of radius a which touch the given sphere and do not penetrate each other. Because of scaling, this does not depend on the radius a. Consider an n-dimensional integer lattice. Each point has the same number of closest neighbors. This number is then a lower bound on τn. The E8 lattice and the Leech lattice are exceptional even in this respect. In fact, we saw that each point in the E8 lattice has 240 neighbors. It can be proved that higher kissing numbers are impossible: τ8 = 240.

A primal–dual penalty method via rounded weighted-ℓ₁ Lagrangian duality

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Journal Information

Published in Optimization, 2022

R. S. Burachik, C. Y. Kaya, C. J. Price

A kissing number is defined as the maximum number of spheres of radius r that touch another given sphere of radius r, without overlapping. The kissing number problem seeks to find the maximum number of non-overlapping spheres of radius r in that can simultaneously touch (kiss) a central sphere of the same radius. We present and formulate the mathematical model for this problem as in [25].

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