Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-28T10:10:58.005Z Has data issue: false hasContentIssue false
  • English
  • Français

The dual lattice of an extreme six-dimensional lattice

Part of: Geometry of numbers

Published online by Cambridge University Press:  09 April 2009

David Coulson
David Coulson
Affiliation:
Monash University Clayton, Victoria 3168, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The best lattice quantizers seem to be duals of extreme lattices. The quantizing constant associated with the dual lattice of Barnes's senary form φ6 is found, together with a new type of quantizing technique. The quantizing constant is better than expected in the sense that it is better than D*6 even though D6 provides a denser packing. This is the smallest dimension for which this occurs.

MSC classification

Secondary: 11H06: Lattices and convex bodies 11H55: Quadratic forms (reduction theory, extreme forms, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 3 , June 1991 , pp. 373 - 383
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Barnes, E. S., ‘The perfect and extreme senary forms’, Canad. J. Math. 9 (1957), 235–242.Google Scholar
[2]Barnes, E. S., ‘The complete enumeration of extreme senary forms’, Philos. Trans. Roy. Soc. London Ser. A 249 (1957), 461–506.Google Scholar
[3]Barnes, E. S., ‘The optimal lattice quantizer in three dimensions’, SIAM Discrete Mathematics 4 (1983), 117–130.Google Scholar
[4]Conway, J. H. and Sloane, N. J. A., ‘Voronoi regions of lattices, second moments of polytopes and quantisation’, IEEE Trans. Inform. Theory 28 (1982), 211–226.Google Scholar
[5]Conway, J. H. and Sloane, N. J. A., ‘Fast quantising and decoding algorithms for lattice quantisers and codes’, IEEE Trans. Inform. Theory 28 (1982), 227–232.CrossRefGoogle Scholar
[6]Conway, J. H. and Sloane, N. J. A., ‘Complex and integral laminated lattices’, Trans. Amer. Math. Soc. 280 (1983), 463–490.CrossRefGoogle Scholar
[7]Conway, J. H. and Sloane, N. J. A., ‘Voronoi regions of certain lattices’, SIAD 5 (1984), 294–305.Google Scholar
[8]Conway, J. H. and Sloane, N. J. A., Sphere packings, lattices and groups, (Springer-Verlag, Chapter 2, 1988).Google Scholar
[9]Conway, J. H. and Sloane, N. J. A., ‘Low dimensional lattices.2 subgroups of GL(N, Z)’, Proc. Roy. Soc. London Ser. A 419 (1988), 29–68.Google Scholar
[10]Coxeter, H. S. M., ‘Extreme forms’, Canad. J. Math. 3 (1951), 391–441.Google Scholar
[11]Craig, M., ‘Extreme forms and cyclotomy’, Mathematika 25 (1978), 44–56.Google Scholar
[12]Worley, R. T., ‘The Voronoi region of E*6’, J. Austral. Math. Soc. 43 (1987), 268–278.Google Scholar
[13]Worley, R. T., ‘The Voronoi region of E*7’, SIAD 9 (1988), 134–141.Google Scholar