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May  2007, 1(2): 257-260. doi: 10.3934/amc.2007.1.257

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, United States
2. 

AT&T Shannon Labs, 180 Park Avenue, Florham Park, NJ 07932-0971, United States

Received  January 2007 Revised  January 2007 Published  May 2007

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.
Keywords: f.c.c. lattice, self-dual lattice, m.c.c. lattice., b.c.c. lattice, Quantizing, isodual lattice.
Mathematics Subject Classification: Primary: 52C07; Secondary: 11H55, 94A2.
Citation: J. H. Conway, N. J. A. Sloane. The optimal isodual lattice quantizer in three dimensions. Advances in Mathematics of Communications, 2007, 1 (2) : 257-260. doi: 10.3934/amc.2007.1.257
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