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.
This paper is a strongly geometrical approach to the Fisher distance, which is a measure of dissimilarity between two probability distribution functions. The Fisher distance, as well as other divergence measures, are also used in many applications to
establish a proper data average. The main purpose is to widen the range of possible interpretations and relations of the Fisher distance and its associated geometry for the prospective applications. It focuses on statistical models of the normal probability distribution functions and takes advantage of the connection with the classical hyperbolic geometry to derive closed forms for the Fisher distance in several cases. Connections with the well-known Kullback-Leibler divergence measure are also devised.
A new class of spherical codes is constructed by selecting a finite subset of flat tori from a foliation of the unit sphere S^{2L-1} of R^{2L} and designing a structured codebook on each torus layer. The resulting spherical code can be the image of a
lattice restricted to a specific hyperbox in R^L in each layer. Group structure and homogeneity, useful for efficient storage and decoding, are inherited from the underlying lattice codebook. A systematic method for constructing such codes are presented and, as an example, the Leech lattice is used to construct a spherical code in R^{48}. Upper and lower bounds on the performance, the asymptotic packing density and a method for decoding are derived.
A method for finding an optimum $n$-dimensional commutative group code of a given order $M$ is presented. The approach explores the structure of lattices related to these codes and provides a significant reduction in the number of non-isometric cases
to be analyzed. The classical factorization of matrices into Hermite and Smith normal forms and also basis reduction of lattices are used to characterize isometric commutative group codes. Several examples of optimum commutative group codes are also presented.
