Sphere Packing and Coding Theory

Authored by: G.A. Kabatiansky , J.A. Rush

Discrete and Computational Geometry

Print publication date:  April  2004
Online publication date:  April  2004

Print ISBN: 9781584883012
eBook ISBN: 9781420035315
Adobe ISBN:

10.1201/9781420035315.ch61

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Abstract

Consider a metric space equipped with some measure (natural in all examples) in which all balls of the same radius have the same “volume” (measure). A set of metric balls of the same radius is called a sphere packing if the intersection of any two balls has measure zero. In the case where the space has finite measure the density of a sphere packing is defined as the ratio of the measure of the union of the balls to the measure of the whole space. In the other case one can consider some natural “large subspace” of the space, such as a ball of large radius, and define the density of a sphere packing as the limit of the ratio of the corresponding volumes. The most famous instance of this problem is sphere packing in Euclidean n-dimensional space, where one asks how densely it is possible to fill ? n with nonoverlapping balls of a fixed (and by homogeneity, irrelevant) radius.

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