We investigate perfect codes in $mathbb{Z}^n$ under the $ell_p$ metric. Upper bounds for the packing radius $r$ of a linear perfect code, in terms of the metric parameter $p$ and the dimension $n$ are derived. For $p = 2$ and $n = 2, 3$, we determine
all radii for which there are linear perfect codes. The non-existence results for codes in $mathbb{Z}^n$ presented here imply non-existence results for codes over finite alphabets $mathbb{Z}_q$, when the alphabet size is large enough, and has implications on some recent constructions of spherical codes.
In this paper we consider the problem of transmitting a continuous alphabet discrete-time source over an AWGN channel. We propose a constructive scheme based on a set of curves on the surface of a N-dimensional sphere. Our approach shows that the des
ign of good codes for this communication problem is related to geometrical properties of spherical codes and projections of N-dimensional rectangular lattices. Theoretical comparisons with some previous works in terms of the mean square error as a function of the channel SNR as well as simulations are provided.
