# American Institute of Mathematical Sciences

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

## The optimal isodual lattice quantizer in three dimensions

 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.
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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