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A Gilbert-Varshamov type bound for Euclidean packings
Author(s):
Gabriele
Nebe;
Chaoping
Xing.
Journal:
Math. Comp.
77
(2008),
2339-2344.
MSC (2000):
Primary 11H31, 52C17, 11H71, 11H06
Posted:
April 28, 2008
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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:
-
- 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:
10.1090/S0025-5718-08-02113-3
PII:
S 0025-5718(08)02113-3
Received by editor(s):
July 3, 2007
Received by editor(s) in revised form:
October 12, 2007
Posted:
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
Copyright of article:
Copyright
2008,
American Mathematical Society
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