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Discrete & Computational Geometry

, Volume 8, Issue 2, pp 109–130 | Cite as

On the covering multiplicity of lattices

  • J. H. Conway
  • N. J. A. Sloane
Article

Abstract

Let the lattice Λ have covering radiusR, so that closed balls of radiusR around the lattice points just cover the space. The covering multiplicityCM(Λ) is the maximal number of times the interiors of these balls overlap. We show that the least possible covering multiplicity for ann-dimensional lattice isn ifn≤8, and conjecture that it exceedsn in all other cases. We determine the covering multiplicity of the Leech lattice and of the latticesIn, An, Dn, En and their duals for small values ofn. Although it appears thatCM(In)=2n−1 ifn≤33, asn → ∞ we haveCM(In)∼2.089... n . The results have application to numerical integration.

Keywords

Open Ball Closed Ball Deep Hole Theta Series Dual Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • J. H. Conway
    • 1
  • N. J. A. Sloane
    • 2
  1. 1.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  2. 2.Mathematical Sciences Research CenterAT&T Bell LaboratoriesMurray HillUSA

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