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A Gilbert-Varshamov type bound for Euclidean packings


Authors: Gabriele Nebe and Chaoping Xing
Journal: Math. Comp. 77 (2008), 2339-2344
MSC (2000): Primary 11H31, 52C17, 11H71, 11H06
DOI: https://doi.org/10.1090/S0025-5718-08-02113-3
Published electronically: April 28, 2008
MathSciNet review: 2429888
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper develops a method to obtain a Gilbert-Varshamov type bound for dense packings in the Euclidean spaces using suitable lattices. For the Leech lattice the obtained bounds are quite reasonable for large dimensions, better than the Minkowski-Hlawka bound, but not as good as the lower bound given by Keith Ball in 1992.


References [Enhancements On Off] (What's this?)

  • 1. Keith Ball, A lower bound for the optimal density of lattice packings. Internat. Math. Res. Notices, 1992, no. 10, 217-221 MR 1191572 (93k:11061)
  • 2. J. W. S. Cassels, ``An Introduction to the Geometry of Numbers,'' Springer-Verlag, New York, 1971. MR 0306130 (46:5257)
  • 3. J. H. Conway and N. J. A. Sloane, ``Sphere Packings, Lattices and Groups,'' Second Edition, Springer-Verlag, New York, 1992. MR 1194619 (93h:11069)
  • 4. P. M. Gruber, C. G. Lekkerkerker, ``Geometry of Numbers.'' Second edition, North-Holland Publishing Co., Amsterdam, 1987. MR 893813 (88j:11034)
  • 5. J. Cannon et al., The Magma Computational Algebra System for Algebra, Number Theory and Geometry, published electronically at http://magma.maths.usyd.edu.au/magma/.
  • 6. Gabriele Nebe's homepage, http://www.math.rwth-aachen.de/homes/Gabriele.Nebe/.
  • 7. C. A. Rogers, ``Packing and Covering,'' Cambridge Univ. Press, 1964. MR 0172183 (30:2405)
  • 8. N.J.A. Sloane, Table of Densest Packings Presently Known, see the website: http://www.research.att.com/~njas/lattices/density.html.

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Additional Information

Gabriele Nebe
Affiliation: Lehrstuhl D für Mathematik, RWTH Aachen, Germany
Email: nebe@math.rwth-aachen.de

Chaoping Xing
Affiliation: Division of Mathematical Science, School of Physical & Mathematical Sciences, Nanyang Technological University, Singapore 637616
Email: xingcp@ntu.edu.sg

DOI: https://doi.org/10.1090/S0025-5718-08-02113-3
Received by editor(s): July 3, 2007
Received by editor(s) in revised form: October 12, 2007
Published electronically: April 28, 2008
Additional Notes: The research of the second author was partially supported by the Singapore MoE Tier 1 grant RG60/07 and the National Scientific Research Project 973 of China 2004CB318000
The second author is the corresponding author
Article copyright: © Copyright 2008 American Mathematical Society

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