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The dual lattice of an extreme six-dimensional lattice

Published online by Cambridge University Press:  09 April 2009

David Coulson
Affiliation:
Monash University Clayton, Victoria 3168, Australia
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Abstract

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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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Barnes, E. S., ‘The perfect and extreme senary forms’, Canad. J. Math. 9 (1957), 235242.Google Scholar
[2]Barnes, E. S., ‘The complete enumeration of extreme senary forms’, Philos. Trans. Roy. Soc. London Ser. A 249 (1957), 461506.Google Scholar
[3]Barnes, E. S., ‘The optimal lattice quantizer in three dimensions’, SIAM Discrete Mathematics 4 (1983), 117130.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), 211226.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), 227232.CrossRefGoogle Scholar
[6]Conway, J. H. and Sloane, N. J. A., ‘Complex and integral laminated lattices’, Trans. Amer. Math. Soc. 280 (1983), 463490.CrossRefGoogle Scholar
[7]Conway, J. H. and Sloane, N. J. A., ‘Voronoi regions of certain lattices’, SIAD 5 (1984), 294305.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), 2968.Google Scholar
[10]Coxeter, H. S. M., ‘Extreme forms’, Canad. J. Math. 3 (1951), 391441.Google Scholar
[11]Craig, M., ‘Extreme forms and cyclotomy’, Mathematika 25 (1978), 4456.Google Scholar
[12]Worley, R. T., ‘The Voronoi region of E*6’, J. Austral. Math. Soc. 43 (1987), 268278.Google Scholar
[13]Worley, R. T., ‘The Voronoi region of E*7’, SIAD 9 (1988), 134141.Google Scholar