Constructive spherical codes on layers of flat tori

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.Comment: 9 pages, 5 figures, submitted to IEEE Transactions on Information Theor

Asymptotic bounds for spherical codes

The set of all error-correcting codes C over a fixed finite alphabet F of cardinality q determines the set of code points in the unit square with coordinates (R(C), delta (C)):= (relative transmission rate, relative minimal distance). The central problem of the theory of such codes consists in maximizing simultaneously the transmission rate of the code and the relative minimum Hamming distance between two different code words. The classical approach to this problem explored in vast literature consists in the inventing explicit constructions of "good codes" and comparing new classes of codes with earlier ones. Less classical approach studies the geometry of the whole set of code points (R,delta) (with q fixed), at first independently of its computability properties, and only afterwords turning to the problems of computability, analogies with statistical physics etc. The main purpose of this article consists in extending this latter strategy to domain of spherical codes.Comment: 34 pages amstex, 3 figure

Space Frequency Codes from Spherical Codes

A new design method for high rate, fully diverse ('spherical') space frequency codes for MIMO-OFDM systems is proposed, which works for arbitrary numbers of antennas and subcarriers. The construction exploits a differential geometric connection between spherical codes and space time codes. The former are well studied e.g. in the context of optimal sequence design in CDMA systems, while the latter serve as basic building blocks for space frequency codes. In addition a decoding algorithm with moderate complexity is presented. This is achieved by a lattice based construction of spherical codes, which permits lattice decoding algorithms and thus offers a substantial reduction of complexity.Comment: 5 pages. Final version for the 2005 IEEE International Symposium on Information Theor

Asymptotic bounds for spherical codes

The set of all error-correcting codes C over a fixed finite alphabet F of cardinality q determines the set of code points in the unit square with coordinates (R(C), delta (C)):= (relative transmission rate, relative minimal distance). The central problem of the theory of such codes consists in maximizing simultaneously the transmission rate of the code and the relative minimum Hamming distance between two different code words. The classical approach to this problem explored in vast literature consists in the inventing explicit constructions of "good codes" and comparing new classes of codes with earlier ones. Less classical approach studies the geometry of the whole set of code points (R,delta) (with q fixed), at first independently of its computability properties, and only afterwords turning to the problems of computability, analogies with statistical physics etc. The main purpose of this article consists in extending this latter strategy to domain of spherical codes

Curves on torus layers and coding for continuous alphabet sources

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.Comment: 5 pages, 4 figures. Accepted for presentation at 2012 IEEE International Symposium on Information Theory (ISIT). 2th version: typos corrected. 3rd version: some typos corrected, a footnote added in Section III B, a comment added in the beggining of Section V and Theorem I adde

Computability questions in the sphere packing problem

We consider the sets of dimensions for which there is an optimal sphere packing with special regularity properties (respectively, a lattice, or a periodic set with a given bound on the number of translations, or an arbitrary periodic set). We show that all these sets are oracle-computable, given an oracle that orders an associated set of spherical codes by increasing Kolmogorov complexity.Comment: 24 pages, LaTe

Three-dimensional Magnetohydrodynamic Simulations of Buoyant Bubbles in Galaxy Clusters

We report results of 3D MHD simulations of the dynamics of buoyant bubbles in magnetized galaxy cluster media. The simulations are three dimensional extensions of two dimensional calculations reported by Jones & De Young (2005). Initially spherical bubbles and briefly inflated spherical bubbles all with radii a few times smaller than the intracluster medium (ICM) scale height were followed as they rose through several ICM scale heights. Such bubbles quickly evolve into a toroidal form that, in the absence of magnetic influences, is stable against fragmentation in our simulations. This ring formation results from (commonly used) initial conditions that cause ICM material below the bubbles to drive upwards through the bubble, creating a vortex ring; that is, hydrostatic bubbles develop into "smoke rings", if they are initially not very much smaller or very much larger than the ICM scale height. Even modest ICM magnetic fields with beta = P_gas/P_mag ~ 10^3 can influence the dynamics of the bubbles, provided the fields are not tangled on scales comparable to or smaller than the size of the bubbles. Quasi-uniform, horizontal fields with initial beta ~ 10^2 bifurcated our bubbles before they rose more than about a scale height of the ICM, and substantially weaker fields produced clear distortions. On the other hand, tangled magnetic fields with similar, modest strengths are generally less easily amplified by the bubble motions and are thus less influential in bubble evolution. Inclusion of a comparably strong, tangled magnetic field inside the initial bubbles had little effect on our bubble evolution, since those fields were quickly diminished through expansion of the bubble and reconnection of the initial field.Comment: 20 pages, 12 figures. Accepted for publication in The Astrophysical Journa

The global stability of M33: still a puzzle

The inner disc of the local group galaxy M33 appears to be in settled rotational balance, and near IR images reveal a mild, large-scale, two-arm spiral pattern with no strong bar. We have constructed N-body models that match all the extensive observational data on the kinematics and surface density of stars and gas in the inner part of M33. We find that currently favoured models are unstable to the formation of a strong bar of semi-major axis 2 < a_B < 3 kpc within 1 Gyr, which changes the dynamical properties of the models to become inconsistent with the current, apparently well-settled, state. The formation of a bar is unaffected by how the gas component is modelled, by increasing the mass of the nuclear star cluster, or by making the dark matter halo counter-rotate, but it can be prevented by either reducing the mass-to-light ratio of the stars to Upsilon_V ~ 0.6 or Upsilon_K ~ 0.23 in solar units or by increasing the random motions of the stars. Also a shorter and weaker bar results when the halo is rigid and unresponsive. However, all three near-stable models support multi-arm spirals, and not the observed large-scale bi-symmetric spiral. A two-arm spiral pattern could perhaps be tidally induced, but such a model would require an implausibly low mass disc to avoid a bar and there is no visible culprit. Thus the survival of the current state of this exceptionally well-studied galaxy is not yet understood. We also suspect that many other unbarred galaxies present a similar puzzle.Comment: 15 pages, 8 figures, to appear in MNRAS. A nymber of revisions from v