Constructions of algebraic lattices

In this work we present constructions of algebraic lattices in Euclidean space with optimal center density in dimensions 2, 3, 4, 6, 8 and 12, which are rotated versions of the lattices Λn, for n = 2,3,4,6,8 and K12. These algebraic lattices are constructed through twisted canonical homomorphism via ideals of a ring of algebraic integers. Mathematical subject classification: 18B35, 94A15, 20H10.49350

Constructions of algebraic lattices

In this work we present constructions of algebraic lattices in Euclidean space with optimal center density in dimensions 2, 3, 4, 6, 8 and 12, which are rotated versions of the lattices Λn, for n = 2,3,4,6,8 and K12. These algebraic lattices are constructed through twisted canonical homomorphism via ideals of a ring of algebraic integers. Mathematical subject classification: 18B35, 94A15, 20H10

Curves On A Sphere, Shift-map Dynamics, And Error Control For Continuous Alphabet Sources

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 ṡ = As, with A a square skew-symmetric matrix. The second code is generated by the shift map sn = bnsn-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 ZN 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 (mse) 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.49716581672Andersson, H., Loeliger, H.-A., Codes from iterated maps Proc. 1995 IEEE Int. Symp. Information Theory, Whistler, BC, Canada, Sept. 17-22, 1995, p. 309Battail, G., On signal-to-noise ratios in communication systems (1966) IEEE Trans. Commun., COM-14, pp. 71-72. , FebBerger, T., Tufts, D.W., Optimum pulse amplitude modulation. Part I: Transmitter-receiver design and bounds from information theory (1967) IEEE Trans. Inform. Theory, IT-13, pp. 196-208. , AprChen, B., Wornell, G.W., Analog error correcting codes based on chaotic dynamical systems (1998) IEEE Trans. Commun., 46, pp. 881-890. , JulyCohen, H., (1993) A Course in Computational Algebraic Number Theory, , Berlin, Germany: Springer-VerlagCosta, S.I.R., Slepian-type codes on the flat torus Proc. IEEE Int. Symp. Information Theory, Sorrento, Italy, July 2000Costa, S.I.R., On closed twisted curves (1990) Proc. Amer. Math. Soc., 109 (1), pp. 205-214Costa, S.I.R., Muniz, M., Augustini, E., Pallazzo R., Jr., Graphs, tesselations and perfect codes on flat tori IEEE Trans. Inform. Theory, , submitted for publicationCurtis, M.L., (1987) Matrix Groups, , New York: Springer-VerlagDo Carmo, M.P., (1976) Differential Geometry of Curves and Surfaces, , Englewood Cliffs, NJ: Prentice-HallGoblick, T.J., Theoretical limitation on the transmission of data from analog sources (1965) IEEE Trans. Inform. Theory, IT-12, pp. 558-566. , OctHayes, S., Grebogi, C., Ott, E., Communicating with chaos (1993) Phys. Rev. Lett., 70, pp. 3031-3034. , MayProakis, J.G., (1983) Digital Communications, , New York: McGraw-HillSakrison, D.J., (1970) Notes on Analog Communication, , New York: Van Nostrand ReinholdSeidman, L.P., An upper bound on average estimation error in nonlinear systems (1968) IEEE Trans. Inform. Theory, IT-14, pp. 243-250. , MarShannon, C.E., A mathematical theory of communication (1948) Bell Syst. Tech. J., 27, pp. 379-423. , JulyShannon, C.E., Coding theorems for a discrete source with a fidelity criterion (1959) IRE Nat. Conv. Rec., Pt. 4, pp. 142-163. , MarTimor, U., Design of signals for analog communication (1970) IEEE Trans. Inform. Theory, IT-16, pp. 581-587. , SeptTimor, U., An upper bound on the estimation error in the threshold region (1970) IEEE Trans. Inform. Theory, IT-16, pp. 692-699. , NovVaishampayan, V.A., High quality A/D conversion using single-bit quantizers: A novel approach based on generalized frequency modulation (1998) Proc. 1998 URSI Int. Symp. Signals, Systems, and Electronics, 29, pp. 438-443. , Sept./OctVaishampayan, V.A., Method and apparatus for converting an analog signal to a digital signal (2000), U.S. Patent 6 160 505, DecVaishampayan, V.A., Sloane, N.J.A., Costa, S.I.R., Dynamical systems, curves and coding for continuous alphabet sources Proc. 2002 Information Theory Workshop, Bangalore, India, Oct. 20-25, 2002, , http://www.research.att.com/Wozencraft, J.M., Jacobs, I.M., (1965) Principles of Communication Engineering, , New York: WileyZiv, J., The behavior of analog communication systems (1970) IEEE Trans. Inform. Theory, IT-16, pp. 587-594. , SeptZiv, J., Zakai, M., Some lower bounds on signal parameter estimation (1969) IEEE Trans. Inform. Theory, IT-15, pp. 386-391. , Ma

Curves on Flat Tori and Analog Source-Channel Codes

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.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.59106646665

Fisher Information Matrix And Hyperbolic Geometry

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. © 2005 IEEE.3436Agustini, E., Costa, S.I.R., An Upper Bound for Signal Transmission Error Probability in Hyperbolic Spaces, , Submitted paperAmari, S., (1986) Differential Geometrical Methods in Statistics, 28. , Lecture Notes in Statistics, Springer-Verlag, HeidelbergAmari, S., Nagaoka, N., (2000) Methods of Information Geometry, 141. , Translations of Mathematical Monographs, Am. Math. SocAtkinson, C., Mitchell, A.F.S., Raos distance measure (1981) Samkhyã - The Indian Journal of Statistics, 43, pp. 345-365Banuelos, R., Carroll, T., Extremal problems for conditioned brownian motion and the hyperbolic metric (2000) Ann. Inst. Fourier, 50 (5), pp. 1507-1532Beardon, A., (1983) The Geometry of Discrete Groups, , Springer-Verlag, NewYorkBurbea, J., Rao, C.R., Entropy differential metric,distance and divergence measures in probability spaces: A unified approach (1982) Journal of Multivariate Analysis, 12, pp. 575-590Costa, M.H.M., Cover, T.M., On the similarity of the entropy power inequality and the Brunn- Minkowski inequality (1984) IEEE Trans. Inform. Theory, 30 (6), pp. 837-839. , NovCosta, S.I.R., Santos, S.A., The Hyperbolic Model of the Mean X Standard Deviation Plane, , Pre-print in preparationCosta, S.I.R., Muniz, M., Agustini, E., Palazzo, Graphs, tessellations, and perfect codes on flat tori (2004) IEEE Trans. Inform. Theory, 50 (10), pp. 2363-2378Cover, T.M., Thomas, J.A., (1999) Elements of Information Theory, , Wiley-Interscience Publication, New YorkDembo, A., Cover, T.M., Informationtheoretic inequalities (1991) IEEE Trans. Inform. Theory, 37 (6), pp. 1501-1518. , NovForney, D., Geometrically uniform codes (1991) IEEE Trans. Inform. Theory, 37 (5), pp. 1241-1260Huiling, L., Kume, A., The Fréchet mean shape and the shape of the means (2000) Adv. in Appl. Probab., 32 (1), pp. 101-113Rao, C.R., Information and accuracy attainable in the estimation of statistical parameters (1945) Bull. Calcutta Math. Soc., 37, pp. 81-9

Construção e análise de códigos esféricos com boas taxas binárias

In this paper we deals with the construction of spherical codes for the Gaussian channel. We present families of structured spherical codes, designed in layers of flat tori, that can be constructed in linear time for even dimensions and improves, for certain parameters, the lower bound for the binary rate given by Shannon.Neste trabalho consideramos a distribuição de um conjunto discreto de pontos sobre a superfície de uma esfera euclidiana unitária, com o propósito de construir códigos esféricos para o canal Gaussiano. Apresentamos famílias de códigos esféricos estruturados, que podem ser construídas em tempo linear para dimensões pares e superam, para alguns parâmetros, o limitante inferior de Shannon para a taxa binária de informação.132

Upper Bounds For A Commutative Group Code

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. © 2006 IEEE.275277Ericson, T., Zinoviev, V., (2001) Codes on Euclidian Spheres, , North-Holland Mathematical Libray, Elsevier Science Pub CoBöröczky, K., Packing of spheres in spaces of constant curvature (1978) Acta Mathematica Academia Scientiarum Hungariacae, 32, pp. 243-261Costa, S.I.R., Vaishampayan, V., Curves on a sphere, shift-map dynamics, and error control for continuos alphabet sources (2003) IEEE Transaction on Information Theory, 49 (7). , JulyCosta, S.I.R., Muniz, M., Agustini, E., Palazzo, R., Graphs, tessellations, and perfect codes on flat tori (2004) IEEE Transaction on Information Theory, 50 (10). , OctoberCoxeter, H.S.M., An upper bound for the number of equal nonoverlapping spheres that can touch another of the same size (1963) Proc. Symp. in Pure Mathematics, 7, pp. 53-72Coxeter, H.S.M., (1999) The Beauty of Geometry - Twelve Essays, , Reprinted in, Dover Publications, INCIngermasson, I., Commutative group codes for the Gaussian channel (1973) IEEE Transaction on Information Theory, IT-19, pp. 215-219. , MarSlepian, D., Group codes for the Gaussian channel (1968) The Bell System Technical Journal, 4 (7), pp. 575-602. , AprilToth, L.F., Über die abschatzung des kürzesten abstandes zweier punkte eines auf einer kugelfläche liegenden punktsystemes (1943) Jber. Deut. Math Verein, 53, pp. 66-68Sloane, N.J.A., Conway, J.H., (1991) Sphere Packings, Lattices and Groups, , Springer-Verlag 3 ed

Rotated D N-lattices

Based on algebraic number theory we construct some families of rotated D n-lattices with full diversity which can be good for signal transmission over both Gaussian and Rayleigh fading channels. Closed-form expressions for the minimum product distance of those lattices are obtained through algebraic properties. © 2012 Elsevier Inc.1321123972406Andrade, A.A., Alves, C., Carlos, T.B., Rotated lattices via the cyclotomic field Q(ζ2r) (2006) Int. J. Appl. Math., 19 (3), pp. 321-331Bayer-Fluckiger, E., Lattices and number fields (1999) Contemp. Math., 241, pp. 69-84Bayer-Fluckiger, E., Ideal lattices (2002) Proceedings of the Conference Number Theory and Diophantine Geometry, pp. 168-184. , Cambridge Univ. PressBayer-Fluckiger, E., Upper bounds for Euclidean minima of algebraic number fields (2006) J. Number Theory, 121 (2), pp. 305-323Bayer-Fluckiger, E., Nebe, G., On the Euclidean minimum of some real number fields (2005) J. Theor. Nombres Bordeaux, 17 (2), pp. 437-454Bayer-Fluckiger, E., Oggier, F., Viterbo, E., New algebraic constructions of rotated Zn-lattice constellations for the Rayleigh fading channel (2004) IEEE Trans. Inform. Theory, 50 (4), pp. 702-714Bayer-Fluckiger, E., Suarez, I., Ideal lattices over totally real number fields and Euclidean minima (2006) Arch. Math., 86 (3), pp. 217-225Boutros, J., Viterbo, E., Rastello, C., Belfiori, J.C., Good lattice constellations for both Rayleigh fading and Gaussian channels (1996) IEEE Trans. Inform. Theory, 42 (2), pp. 502-517Conway, J.H., Sloane, N.J.A., (1988) Sphere Packings, Lattices and Groups, , Springer-VerlagLopes, J.O.D., Discriminants of subfields of Q(ζ2r) (2003) J. Algebra Appl., 2, pp. 463-469Micciancio, D., Goldwasser, S., Complexity of Lattice Problems: A Cryptographic Perspective (2002) Kluwer Internat. Ser. Engrg. Comput. Sci., 671. , Kluwer Academic PublishersSamuel, P., (1970) Algebraic Theory of Numbers, , Hermann, ParisStewart, I.N., Tall, D.O., (1987) Algebraic Number Theory, , Chapman & Hall, LondonWashington, L.C., (1982) Introduction to Ciclotomic Fields, , Springer-Verlag, New Yor

Geometric Contacts Of Surfaces Immersed In Rn, N ≥ 5

We study the extrinsic geometry of surfaces immersed in Rn, n ≥ 5, by analyzing their contacts with different standard geometrical models, such as hyperplanes and hyperspheres. We investigate the relation between different types of contact and the properties of the curvature ellipses at each point. In particular, we focalize our attention on the hyperspheres having contacts of corank two with the surface. This leads in a natural way to the concept of umbilical focus and umbilic curvature. © 2008 Elsevier B.V. All rights reserved.273442454Bruce, J.W., The duals of generic hypersurfaces (1981) Math. Scand., 49, pp. 36-39Chen, B.Y., Yano, K., Integral formulas for submanifolds and their applications (1971) J. Differential Geometry, 5, pp. 467-477Feldman, E.A., Geometry of immersions I (1965) Trans. Amer. Math. Soc., 120, pp. 185-224Giblin, P.J., Sapiro, G., Affine-invariant distances, envelopes and symmetry sets (1998) Geom. Dedicata, 71 (3), pp. 237-261Golubitsky, M., Guillemin, V., Stable Mappings and their Singularities (1973) GTM, 14. , Springer-Verlag, BerlinIzumiya, S., Sano, T., Generic affine differential geometry of space curves (1998) Proc. Edinburgh Math. Soc., 128 A (2), pp. 301-314Izumiya, S., Pei, D.-H., Sano, T., Singularities of hyperbolic Gauss maps (2003) Proc. London Math. Soc., 86, pp. 485-512Izumiya, S., Pei, D., Romero Fuster, M.C., The lightcone Gauss map of a spacelike surface in Minkowski 4-space (2004) Asian J. Math., 8, pp. 511-530Izumiya, S., Pei, D.-H., Romero Fuster, M.C., Takahashi, M., The horospherical geometry of submanifolds in hyperbolic space (2005) J. London Math. Soc. (2), 71, pp. 779-800Little, J., On singularities of submanifolds of higher dimensional Euclidean space (1969) Annali Mat. Pura Appl. 4A, 83, pp. 261-336Looijenga, E.J.N., (1974) Structural stability of smooth families of C∞-functions, , PhD Thesis, University of AmsterdamMochida, D.K.H., Romero-Fuster, M.C., Ruas, M.A.S., The geometry of surfaces in 4-space from a contact viewpoint (1995) Geom. Dedicata, 54, pp. 323-332Mochida, D.K.H., Romero-Fuster, M.C., Ruas, M.A.S., Osculating hyperplanes and asymptotic directions of codimension 2 submanifolds of Euclidean spaces (1999) Geom. Dedicata, 77, pp. 305-315Mochida, D.K.H., Romero-Fuster, M.C., Ruas, M.A.S., Inflection points and nonsingular embeddings of surfaces in R5 (2003) Rocky Mountain J. Math., 33 (3), pp. 995-1010Montaldi, J.A., (1983) Contact with application to submanifolds, , PhD Thesis, University of LiverpoolMontaldi, J.A., On contact between submanifolds (1986) Michigan Math. J., 33, pp. 195-199Montaldi, J.A., On generic composites of maps (1991) Bull. London Math. Soc., 23, pp. 81-85Moore, C.L.E., Wilson, E.B., Differential geometry of two-dimensional surfaces in hyperspaces (1916) Proc. Amer. Acad. Arts Sci., 52, pp. 267-368Moraes, S.M., Romero-Fuster, M.C., Semiumbilics and 2-regular immersions of surfaces in Euclidean spaces (2005) Rocky Mountain J. Math., 35 (4), pp. 1327-1346Palais, R.S., Terng, C., Critical Points Theory and Submanifold Geometry (1988) Lectures Notes in Mathematics, 1353. , Springer-VerlagPorteous, I.R., The normal singularities of a submanifolds (1971) J. Differential Geometry, 5, pp. 543-564Romero-Fuster, M.C., Sphere stratifications and the Gauss map (1983) Proc. Edinburgh Math. Soc., 95 A, pp. 115-136Romero-Fuster, M.C., Ruas, M.A.S., Tari, F., Asymptotic curves on surfaces in R5 (2008) Comm. Contemp. Math., 10 (3), pp. 1-27Romero-Fuster, M.C., Sanabria-Codesal, E., On the flat ridges of submanifolds of codimension 2 in Rn (2002) Proc. Edinburgh Math. Soc., 132 A (4), pp. 975-984Romero-Fuster, M.C., Sánchez-Bringas, F., Isometric reduction of the codimension and 2-regular immersions of submanifolds PreprintSpivak, M., (1979) A Comprehensive Introduction to Differential Geometry, vols. 3 and 4, , Publish or Perish, Bosto