This work presents an extension of the Construction πAlattices proposed by Huang and Narayanan, to Hurwitz quaternion integers. This construction is provided by using an isomorphism from a version of the Chinese remainder theorem applied to maximal orders, in contrast to natural orders in prior works. Exploiting this map, we analyse the performance of the resulting multilevel lattice codes and highlight via computer simulations their notably reduced computational complexity provided by the multistage decoding. Moreover, it is shown that there is a sequence of Construction πAlattices that attain with high probability the Poltyrev-limit.

In this paper, we present a scheme for obtaining Voronoi constellations of Generalized Construction D’ lattice obtained from q-ary codes. For this, we first extend the Generalized Construction D’ to a chain of linear q-ary codes, establish a connection with Construction A and derive a generator matrix, expressions for the minimum distance and the volume of this lattice. This connection also allowed to extend an efficient previous scheme for obtaining Voronoi constellations of Construction A to Generalized Construction D’ lattices.

In this work we extend the Construction$\pi_{A}$lattices proposed in [1], to Hurwitz quaternion integers. This construction is provided by using an isomorphism from a version of the Chinese remainder theorem applied to maximal orders in contrast to natural orders in prior works. Exploiting this map, we analyze the performance of the resulting multilevel lattice codes and show via computer simulations their notably reduced computational complexity provided by the multistage decoding.

We present a communication-efficient distributed protocol for computing the Babai point, an approximate nearest point for a random vector${\mathbf{X}}\in \mathbb {R}^{n}$in a given lattice. We show that the protocol is optimal in the sense that it minimizes the sum rate when the components of$\boldsymbol {X}$are mutually independent. We then investigate the error probability, i.e. the probability that the Babai point does not coincide with the nearest lattice point, motivated by the fact that for some cases, a distributed algorithm for finding the Babai point is sufficient for finding the nearest lattice point itself. Two different probability models for$\boldsymbol {X}$are considered—uniform and Gaussian. For the uniform model, in dimensions two and three, the error probability is seen to grow with the packing density, and we demonstrate that the densest lattice in dimension two presents the worst error probability. For higher dimensions, we develop probabilistic concentration bounds as well as bounds based on geometric arguments for the error probability. The probabilistic bounds lead to the conclusion that for lattices which generate suitably thin coverings of$\mathbb {R}^{n}$(which includes lattices that meet Rogers’ bound on the covering radius), the error probability goes to unity as$n$grows. Probabilistic and geometric bounds are also used to estimate the error probability under the uniform model for various lattices including the$A_{n}$family and the Leech lattice,$\Lambda _{24}$. On the other hand, for the Gaussian model, the error probability goes to zero as the lattice dimension tends to infinity, provided the noise variance is sufficiently small.

We present a new systematic approach to constructing spherical codes in dimensions$2^{k}$, based on Hopf foliations. Using the fact that a sphere$S^{2n-1}$is foliated by manifolds$S_{\cos \eta }^{n-1} \times S_{\sin \eta }^{n-1}$,$\eta \in [0,\pi /2]$, we distribute points in dimension$2^{k}$via a recursive algorithm from a basic construction in$\mathbb {R}^{4}$. Our procedure outperforms some current constructive methods in several small-distance regimes and constitutes a compromise between achieving a large number of codewords for a minimum given distance and effective constructiveness with low encoding computational cost. Bounds for the asymptotic density are derived and compared with other constructions. The encoding process has storage complexity$O(n)$and time complexity$O(n \log n)$. We also propose a sub-optimal decoding procedure, which does not require storing the codebook and has time complexity$O(n \log n)$.

Construction C* was recently introduced as a generalization of the multilevel Construction C (or Forney's code-formula), such that the coded levels may be dependent. Both constructions do not produce a lattice in general, hence the central idea of this paper is to present a 3-level lattice Construction C* scheme that admits an efficient nearest-neighborhood decoding. In order to achieve this objective, we choose coupled codes for levels 1 and 3, and set the second level code C2as an independent linear binary self-dual code, which is known to have a rich mathematical structure among families of linear codes. Our main result states a necessary and sufficient condition for this construction to generate a lattice. We then present examples of efficient lattices and also non-lattice constellations with good packing properties.

We present a new approach to construct spherical codes in 2kdimensions, based on Hopf foliations. Using the fact that a sphere S2n-1is foliated by manifolds Scos ηn-1× Ssin ηn-1, η ∈ [0, π/2], we distribute points in dimension 2kvia a recursive algorithm from a basic construction in R4. Our procedure outperforms some current constructive methods in several small-distance regimes and constitutes a compromise between optimality and computational effort.

In the present paper, we investigate the existence of lattice perfect codes when considered as sublattices of other lattices under the Euclidean metric. We generalize bounds on the radius of perfect codes in a generic lattice, previously known for the cubic lattice. The new bounds are based on covering density, and covering radius of the ambient lattices, and, along with algebraic methods, allow to characterise all perfect codes in small dimension for a given ambient lattice. We provide case studies for some well known ambient lattices, such as the hexagonal lattice, and the checkerboard lattices. In contrast to the cubic lattice, these case studies show that, by changing the ambient lattice, one can find rich sets of perfect codes.

Lattice and special nonlattice multilevel constellations constructed from binary codes, such as Constructions A, C, and D, have relevant applications in Mathematics (sphere packing) and in Communication (multi-stage decoding and efficient vector quantization). In this work, we explore some properties of Construction C, in particular its geometric uniformity. We then propose a new multilevel construction, inspired by bit interleaved coded modulation (BICM), that we call Construction$C^\star $. We investigate the geometric uniformity, latticeness, and minimum distance properties of Construction$C^\star $, and discuss its superior packing efficiency when compared to Construction C.

The Golomb-Welch conjecture (1968) on the non-existence of perfect Lee codes in Znwith radius e ≥ 2 and dimensions n ≥ 3, widely believed to be true, has been up to now only proved for large radius in any dimension, for small dimensions, and for some small radii and specific n. The main result of this paper is that for radius e = 2, there are no perfect Lee linear codes in Znfor infinitely many values of n.

We study the problem of characterizing when two memoryless binary asymmetric channels, described by their transition probabilities (p, q) and (p', q'), are equivalent from the point of view of maximum likelihood decoding when restricted to n-block binary codes. This equivalence of channels induces a partition (depending on n) on the space of parameters (p, q) into regions associated with the equivalence classes. Explicit expressions for describing these regions, their number and areas are derived. Some perspectives of applications of our results to decoding problems are also presented.

We consider the closest lattice point problem in a distributed network setting and study the communication cost and the error probability for computing an approximate nearest lattice point, using the nearest-plane algorithm, due to Babai. Two distinct communication models, centralized and interactive, are considered. The importance of proper basis selection is addressed. Assuming a reduced basis for a two-dimensional lattice, we determine the approximation error of the nearest plane algorithm. The communication cost for determining the Babai point, or equivalently, for constructing the rectangular nearest-plane partition, is calculated in the interactive setting. For the centralized model, an algorithm is presented for reducing the communication cost of the nearest plane algorithm in an arbitrary number of dimensions.

In information geometry a Riemannian manifold of probability distributions is considered with the metric given by Fisher information matrix. In this paper we approach explicit forms for the Fisher-Rao distance in spaces composed by multivariate normal distributions of particular forms. Upper bounds for the distance between two general normal distributions are presented and their tightness are discussed in specific cases.

We address the problem of constructing explicit mappings from a k-dimensional continuous alphabet source to an n-dimensional Gaussian channel. The source is assumed to be uniformly distributed on the unit cube [0, 1)k. The scheme considered is based on a family of piecewise linear mappings and its performance is shown to be related to specific projected lattices of Zn. We study sufficient conditions for the mean squared error of such mappings to scale optimally with the signal-to-noise ratio of the channel and present an explicit construction for the case k = n-1. However, in some other cases our scheme requires the source to be uniformly distributed over a fundamental region of a specific lattice, that may be not congruent to [0, 1)k. A dissection technique is presented in order to overcome the source support mismatch and the MSE degradation of such a transformation is analyzed. An example construction of a 2 : n expansion mapping using the dissection technique is presented and is shown to exhibit optimal scaling of the MSE with the channel SNR.

In this paper, we consider the problem of transmitting a continuous alphabet discrete-time source over an additive white Gaussian noise channel in the bandwidth expansion case. We propose a constructive scheme based on a set of curves on the surface of a 2N-dimensional sphere. Our approach shows that the design of good codes for this communication problem relies on geometrical properties of spherical codes and projections of N-dimensional rectangular lattices. Theoretical comparisons with some previous works in terms of the mean squared error as a function of the channel SNR, as well as simulations, are provided.

A new class of spherical codes is constructed by selecting a finite subset of flat tori from a foliation of the unit sphere S2L-1⊂ R2Land designing a structured codebook on each torus layer. The resulting spherical code can be the image of a lattice restricted to a specific box in RLin 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 as well some examples of constructions. Upper and lower bounds on the performance, the asymptotic packing density and a method for decoding are derived.

In this paper we consider the problem of transmitting a continuous alphabet discrete-time source over an AWGN channel. The design of good curves for this purpose relies on geometrical properties of spherical codes and projections of N-dimensional lattices. We propose a constructive scheme based on a set of curves on the surface of a 2N-dimensional sphere and present comparisons with some previous works.

Q-ary lattices can be obtained from q-ary codes using the so-called Construction A. We investigate these lattices in the Lee metric and show how their decoding process can be related to the associated codes. For prime q we derive a Lee sphere decoding algorithm for q-ary lattices, present a brief discussion on its complexity and some comparisons with the classic sphere decoding.

A cylinder anchored at two distinct points of the lattice Znis called a strut if its interior does not contain a lattice point. We address the problem of constructing struts of maximal radius in Zn. Our main result is a general construction technique, which we call the lifting construction, which produces a sequence of struts that are optimal in the limit. We also tighten a previous result of ours - an achievable lower bound on the volume of a strut. The problem is motivated by a nonlinear analog communication problem. We demonstrate, through simulation, improvements in performance that are obtained using our construction.

A new class of spherical codes is constructed by selecting a finite subset of flat tori that foliate the unit sphere S2L-1sub R2Land constructing a structured codebook on each torus in the finite subset. The codebook on each torus is the image of a lattice restricted to a specific hyperbox in RL. Group structure and homogeneity, useful for efficient decoding, are inherited from the underlying lattice codebook. Upper and lower bounds on performance are derived and a systematic search algorithm is presented for constructing optimal codebooks. The torus layer spherical codes presented here exhibit good performance when compared to the well known apple-peeling, wrapped and laminated codes.

Good spherical codes have large minimum squared distance. An important quota in the theory of spherical codes is the maximum number of points M(n, ρ) displayed on the sphere Sn-1, having a minimum squared distance ρ. The aim of this work is to study this problem within the class of group codes. We establish a bound for the number of points of a commutative group code in dimension even.

We introduce and discuss the concept of Gaussian probability density function (pdf) for the n-dimensional hyperbolic space which has been proposed as an environment for coding and decoding signals. An upper bound for the error probability of signal transmission associated with the hyperbolic distance is established. The pdf and the upper bound were developed using Poincare models for the hyperbolic spaces

The Fisher information matrix induces a metric on parametric spaces of families of probability density functions. We analyse here the family of normal distributions showing how hyperbolic geometry arises naturally from the Fisher information metric.

Quadrature amplitude modulation (QAM)-like signal sets are considered in this paper as coset constellations placed on regular graphs on surfaces known as flat tori. Such signal sets can be related to spherical, block, and trellis codes and may be viewed as geometrically uniform (GU) in the graph metric in a sense that extends the concept introduced by Forney . Homogeneous signal sets of any order can then be labeled by a cyclic group, induced by translations on the Euclidean plane. We construct classes of perfect codes on square graphs including Lee spaces, and on hexagonal and triangular graphs, all on flat tori. Extension of this approach to higher dimensions is also considered.

We consider two codes based on dynamical systems, for transmitting information from a continuous alphabet, discrete-time source over a Gaussian channel. The first code, a homogeneous spherical code, is generated by the linear dynamical system s/spl dot/=As, with A a square skew-symmetric matrix. The second code is generated by the shift map s/sub n/=b/sub n/s/sub n-1/(mod 1). The performance of each of these codes is determined by the geometry of its locus or signal set, specifically, its arc length and minimum distance, suitably defined. We show that the performance analyses for these systems are closely related, and derive exact expressions and bounds for relevant geometric parameters. We also observe that the lattice /spl Zopf//sup N/ underlies both modulation systems and we develop a fast decoding algorithm that relies on this observation. Analytic results show that for fixed bandwidth expansion, good scaling behavior of the mean squared error is obtained relative to the channel signal-to-noise ratio (SNR). Particularly interesting is the resulting observation that sampled, exponentially chirped modulation codes are good bandwidth expansion codes.

Good codes for transmitting a continuous-alphabet source over an AWGN channel can be constructed using simple dynamical systems. The trajectories of the dynamical systems that we consider are curves in /spl Ropf//sup N/, and we use these curves as signal sets for a modulation system. In this paper we consider the problem of choosing the parameters of the dynamical system such that the length of its trajectory is maximized subject to a constraint on the minimum distance between its "folds". We provide some general results on the construction of such curves and show how to select the parameters optimally in the case N=6. This is done by reducing the problem to one of choosing a vector (1, a, b) in /spl Zopf//sup 3/ for which a high packing density is obtained for the lattice /spl Lambda//sub p/ obtained by projecting /spl Zopf//sup 3/ into the plane orthogonal to (1, a, b). Two approaches are used to prove the central result of the paper.