Erez Recently and Zamir have shown that a lattice code with
lattice decoding can achieve the capacity of the additive white Gaussian
noise (AWGN) channel. This was achieved by using a minimum mean-square
error (MMSE) scaling and dithering to transform the AWGN channel in to a
modulo-lattice additive noise (mod-$\Lambda$) channel.
Further, Lui et. al. have shown that lattice decoding can achieve the
error
exponent for the AWGN channel using a scaling other than the
MMSE scaling at rates above between the critical rate and capacity.
We present a simple geometric derivation the of optimal scaling and show
that the optimal parameters have a natural relation to the optimizing
parameters of the Gallager exponent and can be interpreted through an
``inflated spherical code". We show using this interpretation, that for
spherical codes, {\em threshold decoding} of an inflated codebook yields
the
maximum...