The optimal isodual lattice quantizer in three dimensions

J. H. Conway; N. J. A. Sloane

doi:10.3934/amc.2007.1.257

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2007, Volume 1, Issue 2: 257-260. Doi: 10.3934/amc.2007.1.257

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The optimal isodual lattice quantizer in three dimensions

J. H. Conway^1, and
N. J. A. Sloane^2,

1.
Mathematics Department, Princeton University, Princeton, NJ 08544
2.
AT&T Shannon Labs, 180 Park Avenue, Florham Park, NJ 07932-0971

Received: December 31, 2006

Revised: December 31, 2006

Published: June 2007

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Abstract

The mean-centered cuboidal (or m.c.c.) lattice is known to be the optimal packing and covering among all isodual three-dimensional lattices. In this note we show that it is also the best quantizer. It thus joins the isodual lattices $\mathbb Z$, $A_2$ and (presumably) $D_4, E_8$ and the Leech lattice in being simultaneously optimal with respect to all three criteria.

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Mathematics Subject Classification: Primary: 52C07; Secondary: 11H55, 94A29.

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