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

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.

**1.**Keith Ball,*A lower bound for the optimal density of lattice packings*, Internat. Math. Res. Notices**10**(1992), 217–221. MR**1191572**, 10.1155/S1073792892000242**2.**J. W. S. Cassels,*An introduction to the geometry of numbers*, Springer-Verlag, Berlin-New York, 1971. Second printing, corrected; Die Grundlehren der mathematischen Wissenschaften, Band 99. MR**0306130****3.**J. H. Conway and N. J. A. Sloane,*Sphere packings, lattices and groups*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1993. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR**1194619****4.**P. M. Gruber and C. G. Lekkerkerker,*Geometry of numbers*, 2nd ed., North-Holland Mathematical Library, vol. 37, North-Holland Publishing Co., Amsterdam, 1987. MR**893813****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 Tracts in Mathematics and Mathematical Physics, No. 54, Cambridge University Press, New York, 1964. MR**0172183****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