Dpto. Álgebra, Análisis Matemático, Geometría y Topología 96 http://uvadoc.uva.es/handle/10324/1129 2021-04-14T17:54:20Z 2021-04-14T17:54:20Z Accelerated Processing for Maximum Distance Separable Codes using Composite Extension Fields Ruano Benito, Diego Lucani, Daniel E. Geil, Olav http://uvadoc.uva.es/handle/10324/40152 2020-05-15T10:22:13Z 2019-01-01T00:00:00Z This paper describes a new design of Reed-Solomon (RS) codes when using composite extension fields. Our ultimate goal is to provide codes that remain Maximum Distance Separable (MDS), but that can be processed at higher speeds in the encoder and decoder. This is possible by using coefficients in the generator matrix that belong to smaller (and faster) finite fields of the composite extension and limiting the use of the larger (and slower) finite fields to a minimum. We provide formulae and an algorithm to generate such constructions starting from a Vandermonde RS generator matrix and show that even the simplest constructions, e.g., using only processing in two finite fields, can speed up processing by as much as two-fold compared to a Vandermonde RS and Cauchy RS while using the same decoding algorithm, and more than two-fold compared to other RS Cauchy and FFT-based RS. 2019-01-01T00:00:00Z Squares of matrix-product codes Cascudo, Ignacio Gundersen, Jaron Skovsted Ruano Benito, Diego http://uvadoc.uva.es/handle/10324/40143 2020-05-15T10:22:12Z 2020-01-01T00:00:00Z The component-wise or Schur product \$C*C'\$ of two linear error-correcting codes \$C\$ and \$C'\$ over certain finite field is the linear code spanned by all component-wise products of a codeword in \$C\$ with a codeword in \$C'\$. When \$C=C'\$, we call the product the square of \$C\$ and denote it \$C^{*2}\$. Motivated by several applications of squares of linear codes in the area of cryptography, in this paper we study squares of so-called matrix-product codes, a general construction that allows to obtain new longer codes from several ``constituent'' codes. We show that in many cases we can relate the square of a matrix-product code to the squares and products of their constituent codes, which allow us to give bounds or even determine its minimum distance. We consider the well-known \$(u,u+v)\$-construction, or Plotkin sum (which is a special case of a matrix-product code) and determine which parameters we can obtain when the constituent codes are certain cyclic codes. In addition, we use the same techniques to study the squares of other matrix-product codes, for example when the defining matrix is Vandermonde (where the minimum distance is in a certain sense maximal with respect to matrix-product codes). 2020-01-01T00:00:00Z Improved Bounds on the Threshold Gap in Ramp Secret Sharing Cascudo, Ignacio Gundersen, Jaron Skovsted Ruano Benito, Diego http://uvadoc.uva.es/handle/10324/40136 2020-05-15T10:22:12Z 2019-01-01T00:00:00Z In this paper, we consider linear secret sharing schemes over a finite field F q , where the secret is a vector in Fℓ q and each of the n shares is a single element of F q . We obtain lower bounds on the so-called threshold gap g of such schemes, defined as the quantity r-t where r is the smallest number such that any subset of r shares uniquely determines the secret and t is the largest number such that any subset of t shares provides no information about the secret. Our main result establishes a family of bounds which are tighter than previously known bounds for ℓ ≳ 2 . Furthermore, we also provide bounds, in terms of n and q , on the partial reconstruction and privacy thresholds, a more fine-grained notion that considers the amount of information about the secret that can be contained in a set of shares of a given size. Finally, we compare our lower bounds with known upper bounds in the asymptotic setting. 2019-01-01T00:00:00Z Entanglement-assisted quantum error-correcting codes over arbitrary finite fields Galindo, Carlos Hernando, Fernando Matsumoto, Ryutaroh Ruano Benito, Diego http://uvadoc.uva.es/handle/10324/40134 2021-03-03T21:45:44Z 2019-01-01T00:00:00Z We prove that the known formulae for computing the optimal number of maximally entangled pairs required for entanglement-assisted quantum error-correcting codes (EAQECCs) over the binary field hold for codes over arbitrary finite fields as well. We also give a Gilbert-Varshamov bound for EAQECCs and constructions of EAQECCs coming from punctured self-orthogonal linear codes which are valid for any finite field. 2019-01-01T00:00:00Z An Analytic Study of the Reversal of Hartmann Flows by Rotating Magnetic Fields Núñez Jiménez, Manuel http://uvadoc.uva.es/handle/10324/39948 2020-05-15T10:22:12Z 2012-01-01T00:00:00Z The effects of a background uniform rotating magnetic field acting in a conducting fluid with a parallel flow are studied analytically. The stationary version with a transversal magnetic field is well known as generating Hartmann boundary layers. The Lorentz force includes now one term depending on the rotation speed and the distance to the boundary wall. As one intuitively expects, the rotation of magnetic field lines pushes backwards or forwards the flow. One consequence is that near the wall the flow will eventually reverse its direction, provided the rate of rotation and/or the magnetic field are large enough. The configuration could also describe a fixed magnetic field and a rotating flow. 2012-01-01T00:00:00Z Resonances and oscillatory behavior near multi-species plasma equilibria Núñez Jiménez, Manuel http://uvadoc.uva.es/handle/10324/39945 2020-05-15T10:22:12Z 2014-01-01T00:00:00Z We consider dynamic multi-species plasma equilibria whose variables depend on a single spatial coordinate and linear perturbations of these. The linearized system may be reduced to a second-order one satisfied by the respective fluid streamfunctions. For the two-species case, the electron mass is a parameter small enough for a WKB asymptotic analysis to be justified. It turns out that the points where either the ion or electron equilibrium velocity equals the ratio between the temporal and transversal frequencies of the perturbation are turning or singular points of the system, connecting exponentially increasing or decreasing solutions to oscillatory ones. The crucial role of singular points in the balance between the different contributions to the electron kinetic energy is explored. 2014-01-01T00:00:00Z On the asymptotic balance between electric and magnetic energies for hydromagnetic relativistic flows Núñez Jiménez, Manuel http://uvadoc.uva.es/handle/10324/39943 2020-05-15T10:22:11Z 2013-01-01T00:00:00Z In the equations of classical magnetohydrodynamics, the displacement current is considered vanishingly small due to low plasma velocities. For velocities comparable to the speed of light, the full relativistic electromagnetic equations must be used. In the absence of gravitational forcings and with an isotropic Ohm’s law, it is proved that for poloidal magnetic field and velocity and toroidal electric field, the electric and magnetic energies tend to be equivalent in average for large times. This represents a partial extension of Cowling’s theorem for axisymmetric fields. 2013-01-01T00:00:00Z Blowup of certain analytic solutions of the Hall magnetohydrodynamic equations Núñez Jiménez, Manuel Álvarez López, Jorge Rojo García, Jesús http://uvadoc.uva.es/handle/10324/39942 2020-05-15T10:22:11Z 2008-01-01T00:00:00Z A recent analytic solution of the Hall magnetohydrodynamics equations is analyzed. It is shown that its evolution in time depends upon a certain set of inequalities upon the initial values of the velocity and the magnetic field. For most of the cases, both magnitudes will blow up in a finite time. This shows that for keeping the original structure of the solution, energy must be introduced into the system until eventually it cannot hold any longer. 2008-01-01T00:00:00Z Dynamo effect of spacetime curvature in force-free magnetospheres Núñez Jiménez, Manuel http://uvadoc.uva.es/handle/10324/39941 2020-05-15T10:22:11Z 2012-01-01T00:00:00Z We study the possibility of growth of the electric and magnetic fields in a force-free plasma due strictly to the gravitational curvature of the spacetime domain where those fields lie. To this end, we identify a total energy by analogy with the results of classical magnetohydrodynamics. After obtaining the general evolution equation for the total energy, we apply to it to the fiducial observers in a number of classical metrics: Schwarzschild, Boyer-Lindquist, Kerr-Schild, Robertson-Walker, and post-Newtonian approximation. As a rule the shift velocity plays the role of minus the fluid velocity in Newtonian MHD, but the details are often highly intricate. 2012-01-01T00:00:00Z Uniform estimates on the velocity in Rayleigh–Bénard convection Núñez Jiménez, Manuel http://uvadoc.uva.es/handle/10324/39596 2020-05-15T10:22:11Z 2005-01-01T00:00:00Z The kinetic energy of a fluid located between two plates at different temperatures is easily bounded by classical inequalities. However, experiments and numerical simulations indicate that when the convection is turbulent, the volume of the domains in which the speed is large, is rather small. This could imply that the maximum of the speed, in contrast with its quadratic mean, does not admit an a priori upper bound. It is proved that, provided the pressure remains bounded, a uniform estimate for the speed maximum does indeed exist, and that it depends on the maxima of certain ratios between temperature, pressure, and velocity. 2005-01-01T00:00:00Z Invariant subspaces of the periodic Navier–Stokes and magnetohydrodynamics equations: Symmetries and inverse cascades Núñez Jiménez, Manuel http://uvadoc.uva.es/handle/10324/39594 2020-05-15T10:22:11Z 2000-01-01T00:00:00Z It is shown that when the initial condition and the forcing term of the periodic Navier–Stokes or magnetohydrodynamics equations have Fourier coefficients which vanish outside a certain semigroup of frequencies, the same happens to the solutions for all time. Subgroups of frequencies correspond to solutions possessing certain symmetries. By taking as a semigroup the frequencies whose Fourier components are non-negative integers, we get a class of solutions for which the higher modes do not influence the evolution of the lower ones; therefore, the phenomenon of inverse cascading cannot occur for them. 2000-01-01T00:00:00Z Existence theorems for two-fluid magnetohydrodynamics Núñez Jiménez, Manuel http://uvadoc.uva.es/handle/10324/39593 2020-05-15T10:22:11Z 2005-01-01T00:00:00Z The description of a plasma as composed by two types of fluids, formed by ions and electrons, is more complete than the classical one-fluid magnetohydrodynamics MHD model and it has proved necessary to explain the phenomena of fast magnetic reconnection. We prove a finite-time theorem of existence and uniqueness of solutions for this system for either Dirichlet or periodic boundary conditions in dimension three. It turns out that the regularity estimates for the magnetic field are finer than the MHD ones. 2005-01-01T00:00:00Z Plasma velocity in hydromagnetic dynamos Núñez Jiménez, Manuel http://uvadoc.uva.es/handle/10324/39589 2020-05-15T10:22:11Z 2002-01-01T00:00:00Z Hydromagnetic dynamos are plasma configurations generating for some time an exponentially increasing magnetic field. By using a number of functional inequalities, we estimate the rate of increase of magnetic energy in terms of the plasma resistivity and diferent norms on the plasma velocity. Our bounds are proved to be optimal as far as the powers of the relevant magnitudes are concerned. 2002-01-01T00:00:00Z On the gravitational potential of modified Newtonian dynamics Núñez Jiménez, Manuel http://uvadoc.uva.es/handle/10324/38736 2020-05-15T10:22:11Z 2013-01-01T00:00:00Z The mathematical structure of the Poisson equation of Modified Newtonian Dynamics (MOND) is studied. The appropriate setting turns out to be an Orlicz-Sobolev space whose Orlicz function is related to Milgrom’s μ-function, where there exists existence and uniqueness of weak solutions. Since these do not have in principle much regularity, a further study is performed where the gravitational field is not too large, where MOND is most relevant. In that case the field turns out to be H¨older continuous. Quasilinear MOND is also analyzed. 2013-01-01T00:00:00Z Basic results on the equations of magnetohydrodynamics of partially ionized inviscid plasmas Núñez Jiménez, Manuel http://uvadoc.uva.es/handle/10324/38696 2020-05-15T10:22:11Z 2009-01-01T00:00:00Z The equations of evolution of partially ionized plasmas have been far more studied in one of their many simplifications than in its original form. They present a relation between the velocity of each species, plus the magnetic and electric fields, which yield as an analog of Ohm’s law a certain elliptic equation. Therefore, the equations represent a functional evolution system, not a classical one. Nonetheless, a priori estimates and theorems of existence may be obtained in appropriate Sobolev spaces. 2009-01-01T00:00:00Z Transport of energy in dissipative advection phenomena Núñez Jiménez, Manuel http://uvadoc.uva.es/handle/10324/38692 2020-05-15T10:22:10Z 2003-01-01T00:00:00Z A study of the distribution of energy among the different scales is performed for several systems in fluid mechanics, including the Navier–Stokes, magnetohydrodynamics and active scalars equations. It is found that all these systems possess a common structure which enables us to deduce how the energy introduced by the forcing is transferred to the scales present in the flow. It is also shown that in special cases an energy cascade will occur. The limits of this method are also considered. 2003-01-01T00:00:00Z