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Finite Fields by International Conference On Finite Fields%3a Theory%2c Applications%2c And Algorithms (4th 1997 University Of Waterloo)
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1Finite Dimensional Systems With Random External Fields And Neutrino Propagation In Fluctuating Media
By E. Torrente-Lujan
We develop the general formalism for the study of neutrino propagation in presence of stochastic media. This formalism allows the systematic derivation of evolution equations for averaged quantities as survival probabilities and higher order distribution moments. The formalism applies equally to any finite dimensional Schroedinger equation in presence of a stochastic external force. New integro-differential equations valid for finite correlated processes are obtained for the first time. For the particular case of exponentially correlated processes a second order ordinary equation is obtained. As a consequence, the Redfield equation valid for Gaussian delta-correlated noise is rederived in a simple way. The formalism, together with the quantum correlation theorem is applied to the computation of higher moments and correlation functions of practical interest in forthcoming high precision neutrino experiments. It is shown that equal and not equal time correlators follow similar differential equations.
“Finite Dimensional Systems With Random External Fields And Neutrino Propagation In Fluctuating Media” Metadata:
- Title: ➤ Finite Dimensional Systems With Random External Fields And Neutrino Propagation In Fluctuating Media
- Author: E. Torrente-Lujan
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-hep-ph9807361
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2Multiplication In Cyclotomic Rings And Its Application To Finite Fields
By Francisco Arguello
A representation of finite fields that has proved useful when implementing finite field arithmetic in hardware is based on an isomorphism between subrings and fields. In this paper, we present an unified formulation for multiplication in cyclotomic rings and cyclotomic fields in that most arithmetic operations are done on vectors. From this formulation we can generate optimized algorithms for multiplication. For example, one of the proposed algorithms requires approximately half the number of coordinate-level multiplications at the expense of extra coordinate-level additions. Our method is then applied to the finite fields GF(q^m) to further reduce the number of operations. We then present optimized algorithms for multiplication in finite fields with type-I and type-II optimal normal bases.
“Multiplication In Cyclotomic Rings And Its Application To Finite Fields” Metadata:
- Title: ➤ Multiplication In Cyclotomic Rings And Its Application To Finite Fields
- Author: Francisco Arguello
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-0807.3699
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3Motivic Complexes Over Finite Fields And The Ring Of Correspondences At The Generic Point
By James S. Milne and Niranjan Ramachandran
Already in the 1960s Grothendieck understood that one could obtain an almost entirely satisfactory theory of motives over a finite field when one assumes the full Tate conjecture. In this note we prove a similar result for motivic complexes. In particular Beilinson's Q-algebra of "correspondences at the generic point" is then defined for all connected varieties. We compute this for all smooth projective varieties (hence also for varieties birational to such a variety).
“Motivic Complexes Over Finite Fields And The Ring Of Correspondences At The Generic Point” Metadata:
- Title: ➤ Motivic Complexes Over Finite Fields And The Ring Of Correspondences At The Generic Point
- Authors: James S. MilneNiranjan Ramachandran
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-math0607483
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4Finite Temperature Quark Confinement Via Chromomagnetic Fields
By Michael Ogilvie
A natural mechanism for finite temperature quark confinement arises via the coupling of the adjoint Polyakov loop to the chromomagnetic field. Lattice simulations and analytical results both support this hypothesis. Finite temperature SU(3) lattice simulations show that a large external coupling to the chromomagnetic field restores confinement at temperatures above the normal deconfining temperature. A one-loop calculation of the effective potential for SU(2) gluons in a background field shows that a constant chromomagnetic field can drive the Polyakov loop to confining behavior, and the Polyakov loop can in turn remove the well-known tachyonic mode associated with gluons in an external chromomagnetic field. For abelian background fields, tachyonic modes are necessary for confinement at one loop.
“Finite Temperature Quark Confinement Via Chromomagnetic Fields” Metadata:
- Title: ➤ Finite Temperature Quark Confinement Via Chromomagnetic Fields
- Author: Michael Ogilvie
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-hep-lat9709127
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5Chiral Phase Transitions In Strong Chromomagnetic Fields At Finite Temperature And Dimensional Reduction
A natural mechanism for finite temperature quark confinement arises via the coupling of the adjoint Polyakov loop to the chromomagnetic field. Lattice simulations and analytical results both support this hypothesis. Finite temperature SU(3) lattice simulations show that a large external coupling to the chromomagnetic field restores confinement at temperatures above the normal deconfining temperature. A one-loop calculation of the effective potential for SU(2) gluons in a background field shows that a constant chromomagnetic field can drive the Polyakov loop to confining behavior, and the Polyakov loop can in turn remove the well-known tachyonic mode associated with gluons in an external chromomagnetic field. For abelian background fields, tachyonic modes are necessary for confinement at one loop.
“Chiral Phase Transitions In Strong Chromomagnetic Fields At Finite Temperature And Dimensional Reduction” Metadata:
- Title: ➤ Chiral Phase Transitions In Strong Chromomagnetic Fields At Finite Temperature And Dimensional Reduction
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-hep-ph9701323
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6First-Principles Perturbative Computation Of Phonon Properties Of Insulators In Finite Electric Fields
By Xinjie Wang and David Vanderbilt
We present a perturbative method for calculating phonon properties of an insulator in the presence of a finite electric field. The starting point is a variational total-energy functional with a field-coupling term that represents the effect of the electric field. This total-energy functional is expanded in small atomic displacements within the framework of density-functional perturbation theory. The linear response of field-polarized Bloch functions to atomic displacements is obtained by minimizing the second-order derivatives of the total-energy functional. In the general case of nonzero phonon wavevector, there is a subtle interplay between the couplings between neighboring k-points introduced by the presence of the electric field in the reference state, and further-neighbor k-point couplings determined by the wavevector of the phonon perturbation. As a result, terms arise in the perturbation expansion that take the form of four-sided loops in k-space. We implement the method in the {\tt ABINIT} code and perform illustrative calculations of the field-dependent phonon frequencies for III-V semiconductors.
“First-Principles Perturbative Computation Of Phonon Properties Of Insulators In Finite Electric Fields” Metadata:
- Title: ➤ First-Principles Perturbative Computation Of Phonon Properties Of Insulators In Finite Electric Fields
- Authors: Xinjie WangDavid Vanderbilt
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-cond-mat0605435
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7A Constructive Approach To The Finite Wavelet Frames Over Prime Fields
By Asghar Rahimi and Niloufar Seddighi
In this article, we present a constructive method for computing the frame coefficients of finite wavelet frames over prime fields using tools from computational harmonic analysis and group theory.
“A Constructive Approach To The Finite Wavelet Frames Over Prime Fields” Metadata:
- Title: ➤ A Constructive Approach To The Finite Wavelet Frames Over Prime Fields
- Authors: Asghar RahimiNiloufar Seddighi
“A Constructive Approach To The Finite Wavelet Frames Over Prime Fields” Subjects and Themes:
- Subjects: Functional Analysis - Mathematics
Edition Identifiers:
- Internet Archive ID: arxiv-1703.05012
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8On The Difference Between Permutation Polynomials Over Finite Fields
By Nurdagül Anbar, Almasa Oduzak, Vandita Patel, Luciane Quoos, Anna Somoza and Alev Topuzoğlu
The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that if $p>(d^2-3d+4)^2$, then there is no complete mapping polynomial $f$ in $\Fp[x]$ of degree $d\ge 2$. For arbitrary finite fields $\Fq$, a similar non-existence result is obtained recently by I\c s\i k, Topuzo\u glu and Winterhof in terms of the Carlitz rank of $f$. Cohen, Mullen and Shiue generalized the Chowla-Zassenhaus-Cohen Theorem significantly in 1995, by considering differences of permutation polynomials. More precisely, they showed that if $f$ and $f+g$ are both permutation polynomials of degree $d\ge 2$ over $\Fp$, with $p>(d^2-3d+4)^2$, then the degree $k$ of $g$ satisfies $k \geq 3d/5$, unless $g$ is constant. In this article, assuming $f$ and $f+g$ are permutation polynomials in $\Fq[x]$, we give lower bounds for $k %=\mathrm{deg(h)} $ in terms of the Carlitz rank of $f$ and $q$. Our results generalize the above mentioned result of I\c s\i k et al. We also show for a special class of polynomials $f$ of Carlitz rank $n \geq 1$ that if $f+x^k$ is a permutation of $\Fq$, with $\gcd(k+1, q-1)=1$, then $k\geq (q-n)/(n+3)$.
“On The Difference Between Permutation Polynomials Over Finite Fields” Metadata:
- Title: ➤ On The Difference Between Permutation Polynomials Over Finite Fields
- Authors: ➤ Nurdagül AnbarAlmasa OduzakVandita PatelLuciane QuoosAnna SomozaAlev Topuzoğlu
“On The Difference Between Permutation Polynomials Over Finite Fields” Subjects and Themes:
- Subjects: Algebraic Geometry - Mathematics
Edition Identifiers:
- Internet Archive ID: arxiv-1703.08086
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9Encoding Points On Hyperelliptic Curves Over Finite Fields In Deterministic Polynomial Time
By Jean-Gabriel Kammerer, Reynald Lercier and Guénaël Renault
We present families of (hyper)elliptic curve which admit an efficient deterministic encoding function.
“Encoding Points On Hyperelliptic Curves Over Finite Fields In Deterministic Polynomial Time” Metadata:
- Title: ➤ Encoding Points On Hyperelliptic Curves Over Finite Fields In Deterministic Polynomial Time
- Authors: Jean-Gabriel KammererReynald LercierGuénaël Renault
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-1005.1454
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10On The Characteristic Polynomial Of Frobenius Of Supersingular Abelian Varieties Of Dimension Up To 7 Over Finite Fields
By Vijaykumar Singh, Alexey Zatysev and Gary McGuire
In this article, we derive the list of the characteristic polynomials of the Frobenius endomorphism of simple supersingular abelian varieties of dimension $1,~2,~3,~4,~5,~6,~7$ over $\mathbb{F}_q$ where $q=p^n$, $n$ odd.
“On The Characteristic Polynomial Of Frobenius Of Supersingular Abelian Varieties Of Dimension Up To 7 Over Finite Fields” Metadata:
- Title: ➤ On The Characteristic Polynomial Of Frobenius Of Supersingular Abelian Varieties Of Dimension Up To 7 Over Finite Fields
- Authors: Vijaykumar SinghAlexey ZatysevGary McGuire
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-1011.2257
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11Singularity Of Random Matrices Over Finite Fields
By Kenneth Maples
Let $A$ be an $n \times n$ random matrix with iid entries over a finite field of order $q$. Suppose that the entries do not take values in any additive coset of the field with probability greater than $1 - \alpha$ for some fixed $0 < \alpha < 1$. We show that the singularity probability converges to the uniform limit with an exponentially small error depending only on $\alpha$. We also show that the distribution of the determinant of $A$ converges to its limiting distribution at an exponential rate.
“Singularity Of Random Matrices Over Finite Fields” Metadata:
- Title: ➤ Singularity Of Random Matrices Over Finite Fields
- Author: Kenneth Maples
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-1012.2372
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12Weak Version Of Restriction Estimates For Spheres And Paraboloids In Finite Fields
By Hunseok Kang and Doowon Koh
We study L^p-L^r restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension $d$ is even, then it is conjectured that the L^{(2d+2)/(d+3)}-L^2 Stein-Tomas restriction result can be improved to the L^{(2d+4)/(d+4)}-L^2 estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured L^p-L^2 restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in $d$ dimensions and those for homogeneous varieties in (d+1)dimensions.
“Weak Version Of Restriction Estimates For Spheres And Paraboloids In Finite Fields” Metadata:
- Title: ➤ Weak Version Of Restriction Estimates For Spheres And Paraboloids In Finite Fields
- Authors: Hunseok KangDoowon Koh
“Weak Version Of Restriction Estimates For Spheres And Paraboloids In Finite Fields” Subjects and Themes:
- Subjects: Mathematics - Classical Analysis and ODEs
Edition Identifiers:
- Internet Archive ID: arxiv-1401.6997
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13The K($\pi$,1)-property For Marked Curves Over Finite Fields
By Philippe Lebacque and Alexander Schmidt
We investigate the K($\pi$,1)-property for p of smooth, marked curves (X,T) defined over finite fields of characteristic p. We prove that (X,T) has the K($\pi$,1)-property if X is affine and give positive and negative examples in the proper case. We also consider the unmarked proper case over a finite field of characteristic different to p.
“The K($\pi$,1)-property For Marked Curves Over Finite Fields” Metadata:
- Title: ➤ The K($\pi$,1)-property For Marked Curves Over Finite Fields
- Authors: Philippe LebacqueAlexander Schmidt
“The K($\pi$,1)-property For Marked Curves Over Finite Fields” Subjects and Themes:
- Subjects: Mathematics - Number Theory - Algebraic Geometry
Edition Identifiers:
- Internet Archive ID: arxiv-1401.1979
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The book is available for download in "texts" format, the size of the file-s is: 0.18 Mbs, the file-s for this book were downloaded 24 times, the file-s went public at Sat Jun 30 2018.
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14On K-simplexes In (2k-1)-dimensional Vector Spaces Over Finite Fields
By Le Anh Vinh
We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence.
“On K-simplexes In (2k-1)-dimensional Vector Spaces Over Finite Fields” Metadata:
- Title: ➤ On K-simplexes In (2k-1)-dimensional Vector Spaces Over Finite Fields
- Author: Le Anh Vinh
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-0903.2506
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15DTIC ADA086813: Discrete And Continuous Spectra Of Finite-Width, Parallel-Plate Simulator Fields.
By Defense Technical Information Center
The discrete (referred as modes) and continuous spectra of finite-width, parallel-plate simulator fields are investigated. Formulas suitable for numerical calculation are obtained for both the transverse-magnetic (TM) and transverse-electric (TE) fields. Numerical results for the propagation constants and the field distributions of the higher order TM modes are presented. It is found that in the working volume of the parallel-plate simulator some of the higher-order TM modes resemble those of a closed waveguide, and the field of the continuous spectrum decays at least as z-2 for waves propagating along the z-direction.
“DTIC ADA086813: Discrete And Continuous Spectra Of Finite-Width, Parallel-Plate Simulator Fields.” Metadata:
- Title: ➤ DTIC ADA086813: Discrete And Continuous Spectra Of Finite-Width, Parallel-Plate Simulator Fields.
- Author: ➤ Defense Technical Information Center
- Language: English
“DTIC ADA086813: Discrete And Continuous Spectra Of Finite-Width, Parallel-Plate Simulator Fields.” Subjects and Themes:
- Subjects: ➤ DTIC Archive - Yang,F C - DIKEWOOD INDUSTRIES INC ALBUQUERQUE NM - *PLATES - *PARALLEL ORIENTATION - *ELECTROMAGNETIC PULSES - *ELECTROMAGNETIC PULSE SIMULATORS - SIMULATORS - MAGNETIC FIELDS - ELECTRIC FIELDS - TRANSVERSE WAVES - INTEGRAL EQUATIONS - CONTINUOUS SPECTRA
Edition Identifiers:
- Internet Archive ID: DTIC_ADA086813
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16Linearized Polynomial Maps Over Finite Fields
By Joost Berson
We consider polynomial maps described by so-called "(multivariate) linearized polynomials". These polynomials are defined using a fixed prime power, say q. Linearized polynomials have no mixed terms. Considering invertible polynomial maps without mixed terms over a characteristic zero field, we will only obtain (up to a linear transformation of the variables) triangular maps, which are the most basic examples of polynomial automorphisms. However, over the finite field F_q automorphisms defined by linearized polynomials have (in general) an entirely different structure. Namely, we will show that the linearized polynomial maps over F_q are in one-to-one correspondence with matrices having coefficients in a univariate polynomial ring over F_q. Furthermore, composition of polynomial maps translates to matrix multiplication, implying that invertible linearized polynomial maps correspond to invertible matrices. This alternate description of the linearized polynomial automorphism subgroup leads to the solution of many famous conjectures (most notably, the Jacobian Conjecture) for this kind of polynomials and polynomial maps.
“Linearized Polynomial Maps Over Finite Fields” Metadata:
- Title: ➤ Linearized Polynomial Maps Over Finite Fields
- Author: Joost Berson
- Language: English
Edition Identifiers:
- Internet Archive ID: arxiv-1201.1137
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17Number Of Solutions Of Systems Of Homogeneous Polynomial Equations Over Finite Fields
By Mrinmoy Datta and Sudhir R. Ghorpade
We consider the problem of determining the maximum number of common zeros in a projective space over a finite field for a system of linearly independent multivariate homogeneous polynomials defined over that field. There is an elaborate conjecture of Tsfasman and Boguslavsky that predicts the maximum value when the homogeneous polynomials have the same degree that is not too large in comparison to the size of the finite field. We show that this conjecture holds in the affirmative if the number of polynomials does not exceed the total number of variables. This extends the results of Serre (1991) and Boguslavsky (1997) for the case of one and two polynomials, respectively. Moreover, it complements our recent result that the conjecture is false, in general, if the number of polynomials exceeds the total number of variables.
“Number Of Solutions Of Systems Of Homogeneous Polynomial Equations Over Finite Fields” Metadata:
- Title: ➤ Number Of Solutions Of Systems Of Homogeneous Polynomial Equations Over Finite Fields
- Authors: Mrinmoy DattaSudhir R. Ghorpade
- Language: English
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- Subjects: Mathematics - Algebraic Geometry
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- Internet Archive ID: arxiv-1507.03029
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18Proper Toric Maps Over Finite Fields
By Mark Andrea de Cataldo
We determine a strong form of the decomposition theorem for proper toric maps over finite fields.
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- Title: ➤ Proper Toric Maps Over Finite Fields
- Author: Mark Andrea de Cataldo
- Language: English
“Proper Toric Maps Over Finite Fields” Subjects and Themes:
- Subjects: Algebraic Geometry - Mathematics
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- Internet Archive ID: arxiv-1503.02679
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19Discrete Painleve II Equation Over Finite Fields
We determine a strong form of the decomposition theorem for proper toric maps over finite fields.
“Discrete Painleve II Equation Over Finite Fields” Metadata:
- Title: ➤ Discrete Painleve II Equation Over Finite Fields
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- Internet Archive ID: arxiv-1206.4456
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20How To Construct Curves Over Finite Fields With Many Points
By Gerard van der Geer and Marcel van der Vlugt
In this paper we give several methods to construct curves over finite fields with many points and illustrate this with examples of the results.
“How To Construct Curves Over Finite Fields With Many Points” Metadata:
- Title: ➤ How To Construct Curves Over Finite Fields With Many Points
- Authors: Gerard van der GeerMarcel van der Vlugt
- Language: English
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- Internet Archive ID: arxiv-alg-geom9511005
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21Solitons With Nested Structure Over Finite Fields
By Fumitaka Yura
We propose a solitonic dynamical system over finite fields that may be regarded as an analogue of the box-ball systems. The one-soliton solutions of the system, which have nested structures similar to fractals, are also proved. The solitonic system in this paper is described by polynomials, which seems to be novel. Furthermore, in spite of such complex internal structures, numerical simulations exhibit stable propagations before and after collisions among multiple solitons with preserving their patterns.
“Solitons With Nested Structure Over Finite Fields” Metadata:
- Title: ➤ Solitons With Nested Structure Over Finite Fields
- Author: Fumitaka Yura
“Solitons With Nested Structure Over Finite Fields” Subjects and Themes:
- Subjects: ➤ Nonlinear Sciences - Mathematics - Exactly Solvable and Integrable Systems - Cellular Automata and Lattice Gases - Pattern Formation and Solitons - Mathematical Physics
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- Internet Archive ID: arxiv-1405.2018
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22Jacobi-Trudi Determinants Over Finite Fields
By Ben Anzis, Shuli Chen, Yibo Gao, Jesse Kim, Zhaoqi Li and Rebecca Patrias
In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over the integers to a finite field $\mathbb{F}_q$, what is the probability that the Schur function $s_\lambda$ maps to zero? We show that this probability is always at least $1/q$ and is asymptotically $1/q$. Moreover, we give a complete classification of all shapes that can achieve probability $1/q$. In addition, we identify certain families of shapes where the corresponding Schur functions being sent to zero are independent events, and we look into the probability that a Schur functions is mapped to nonzero values in $\mathbb{F}_q$.
“Jacobi-Trudi Determinants Over Finite Fields” Metadata:
- Title: ➤ Jacobi-Trudi Determinants Over Finite Fields
- Authors: ➤ Ben AnzisShuli ChenYibo GaoJesse KimZhaoqi LiRebecca Patrias
“Jacobi-Trudi Determinants Over Finite Fields” Subjects and Themes:
- Subjects: Combinatorics - Mathematics
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- Internet Archive ID: arxiv-1611.00216
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23Nonlinear Roth Type Theorems In Finite Fields
By Jean Bourgain and Mei-Chu Chang
We obtain smoothing estimates for certain nonlinear convolution operators on prime fields, leading to quantitative nonlinear Roth type theorems. Compared with the usual linear setting (i.e. arithmetic progressions), the nonlinear nature of the operators leads to different phenomena, both qualitatively and quantitatively.
“Nonlinear Roth Type Theorems In Finite Fields” Metadata:
- Title: ➤ Nonlinear Roth Type Theorems In Finite Fields
- Authors: Jean BourgainMei-Chu Chang
“Nonlinear Roth Type Theorems In Finite Fields” Subjects and Themes:
- Subjects: Number Theory - Mathematics
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- Internet Archive ID: arxiv-1608.05448
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24On Coordinatising Planes Of Prime Power Order Using Finite Fields
By Robert S. Coulter
We revisit the coordinatisation method for projective planes. First, we discuss how the behaviour of the additive and multiplicative loops can be described in terms of its action on the "vertical" line, and how this means one can coordinatise certain planes in an optimal sense. We then move to consider projective planes of prime power order only. Specifically, we consider how coordinatising planes of prime power order using finite fields as the underlying labelling set leads to some general restrictions on the form of the resulting planar ternary ring (PTR) when viewed as a trivariate polynomial over the field. We also consider the Lenz-Barlotti type of the plane being coordinatised, deriving further restrictions on the form of the PTR polynomial.
“On Coordinatising Planes Of Prime Power Order Using Finite Fields” Metadata:
- Title: ➤ On Coordinatising Planes Of Prime Power Order Using Finite Fields
- Author: Robert S. Coulter
“On Coordinatising Planes Of Prime Power Order Using Finite Fields” Subjects and Themes:
- Subjects: Combinatorics - Mathematics
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- Internet Archive ID: arxiv-1609.01337
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25Reversed Dickson Polynomials Of The (k+1)-th Kind Over Finite Fields
By Neranga Fernando
We discuss the properties and the permutation behaviour of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ over finite fields. The results in this paper unify and generalize several recently discovered results on reversed Dickson polynomials over finite fields.
“Reversed Dickson Polynomials Of The (k+1)-th Kind Over Finite Fields” Metadata:
- Title: ➤ Reversed Dickson Polynomials Of The (k+1)-th Kind Over Finite Fields
- Author: Neranga Fernando
“Reversed Dickson Polynomials Of The (k+1)-th Kind Over Finite Fields” Subjects and Themes:
- Subjects: Number Theory - Mathematics
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- Internet Archive ID: arxiv-1605.04338
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26New MDS Euclidean And Hermitian Self-dual Codes Over Finite Fields
By Hongxi Tong and Xiaoqing Wang
In this paper, we construct MDS Euclidean self-dual codes which are extended cyclic duadic codes. And we obtain many new MDS Euclidean self-dual codes. We also construct MDS Hermitian self-dual codes from generalized Reed-Solomon codes and constacyclic codes. And we give some results on Hermitian self-dual codes, which are the extended cyclic duadic codes.
“New MDS Euclidean And Hermitian Self-dual Codes Over Finite Fields” Metadata:
- Title: ➤ New MDS Euclidean And Hermitian Self-dual Codes Over Finite Fields
- Authors: Hongxi TongXiaoqing Wang
“New MDS Euclidean And Hermitian Self-dual Codes Over Finite Fields” Subjects and Themes:
- Subjects: Information Theory - Computing Research Repository - Mathematics
Edition Identifiers:
- Internet Archive ID: arxiv-1606.07161
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27The Reversible Negacyclic Codes Over Finite Fields
By Shixin Zhu, Binbin Pang and Zhonghua Sun
In this paper, by investigating the factor of the $x^n+1$, we deduce that the structure of the reversible negacyclic code over the finite field $\mathbb{F}_{q}$, where $q$ is an odd prime power. Though studying $q-$cyclotomic cosets modulo $2n$, we obtain the parameters of negacyclic BCH code of length $n=\frac{q^\ell+1}{2}$ , $n=\frac{q^m-1}{2(q-1)}$ and $n=\frac{q^{t\cdot2^\tau}-1}{2(q^t+1)}$. Some optimal linear codes from negacyclic codes are given. Finally, we discuss a class of MDS LCD negacyclic codes.
“The Reversible Negacyclic Codes Over Finite Fields” Metadata:
- Title: ➤ The Reversible Negacyclic Codes Over Finite Fields
- Authors: Shixin ZhuBinbin PangZhonghua Sun
“The Reversible Negacyclic Codes Over Finite Fields” Subjects and Themes:
- Subjects: Information Theory - Computing Research Repository - Mathematics
Edition Identifiers:
- Internet Archive ID: arxiv-1610.08206
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28Del Pezzo Surfaces Over Finite Fields And Their Frobenius Traces
By Barinder Banwait, Francesc Fité and Daniel Loughran
Let $S$ be a smooth cubic surface over a finite field $\mathbb F_q$. It is known that $\#S(\mathbb F_q) = 1 + aq + q^2$ for some $a \in \{-2,-1,0,1,2,3,4,5,7\}$. Serre has asked which values of a can arise for a given $q$. Building on special cases treated by Swinnerton-Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton-Dyer's tables on cubic surfaces over finite fields.
“Del Pezzo Surfaces Over Finite Fields And Their Frobenius Traces” Metadata:
- Title: ➤ Del Pezzo Surfaces Over Finite Fields And Their Frobenius Traces
- Authors: Barinder BanwaitFrancesc FitéDaniel Loughran
“Del Pezzo Surfaces Over Finite Fields And Their Frobenius Traces” Subjects and Themes:
- Subjects: Number Theory - Mathematics
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- Internet Archive ID: arxiv-1606.00300
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29Modular Characteristic Classes For Representations Over Finite Fields
By Anssi Lahtinen and David Sprehn
The cohomology of the degree-$n$ general linear group over a finite field of characteristic $p$, with coefficients also in characteristic $p$, remains poorly understood. For example, the lowest degree previously known to contain nontrivial elements is exponential in $n$. In this paper, we introduce a new system of characteristic classes for representations over finite fields, and use it to construct a wealth of explicit nontrivial elements in these cohomology groups. In particular we obtain nontrivial elements in degrees linear in $n$. We also construct nontrivial elements in the mod $p$ homology and cohomology of the automorphism groups of free groups, and the general linear groups over the integers. These elements reside in the unstable range where the homology and cohomology remain poorly understood.
“Modular Characteristic Classes For Representations Over Finite Fields” Metadata:
- Title: ➤ Modular Characteristic Classes For Representations Over Finite Fields
- Authors: Anssi LahtinenDavid Sprehn
“Modular Characteristic Classes For Representations Over Finite Fields” Subjects and Themes:
- Subjects: Representation Theory - Algebraic Topology - Mathematics
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- Internet Archive ID: arxiv-1607.01052
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30Tensor Products Of Unipotent Characters Of General Linear Groups Over Finite Fields
By Emmanuel Letellier
We study multiplicities of unipotent characters in tensor products of unipotent characters of GL(n,q). We prove that these multiplicities are polynomials in q with non-negative integer coefficients. We study the degree of these polynomials and give a necessary and sufficient condition in terms of the representation theory of symmetric groups for these polynomials to be non-zero.
“Tensor Products Of Unipotent Characters Of General Linear Groups Over Finite Fields” Metadata:
- Title: ➤ Tensor Products Of Unipotent Characters Of General Linear Groups Over Finite Fields
- Author: Emmanuel Letellier
- Language: English
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- Internet Archive ID: arxiv-1204.2690
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31On A Class Of Diagonal Equations Over Finite Fields
By Ioulia N. Baoulina
Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form $x_1^{2^m}+\dots+x_n^{2^m}=0$ over a finite field of characteristic $p\equiv\pm 3\pmod{8}$. All of the evaluations are effected in terms of parameters occurring in quadratic partitions of some powers of $p$.
“On A Class Of Diagonal Equations Over Finite Fields” Metadata:
- Title: ➤ On A Class Of Diagonal Equations Over Finite Fields
- Author: Ioulia N. Baoulina
“On A Class Of Diagonal Equations Over Finite Fields” Subjects and Themes:
- Subjects: Number Theory - Mathematics
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- Internet Archive ID: arxiv-1602.01807
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32On Superspecial Abelian Varieties Over Finite Fields
By Jiangwei Xue, Tse-Chung Yang and Chia-Fu Yu
In this paper we establish a new lattice description for superspecial abelian varieties over a finite field $\mathbb {F}_q$ of $q=p^a$ elements. Our description depends on the parity of the exponent $a$ of $q$. When $q$ is an odd power of the prime $p$, we give an explicit formula for the number of superspecial abelian surfaces over $\mathbb F_q$.
“On Superspecial Abelian Varieties Over Finite Fields” Metadata:
- Title: ➤ On Superspecial Abelian Varieties Over Finite Fields
- Authors: Jiangwei XueTse-Chung YangChia-Fu Yu
“On Superspecial Abelian Varieties Over Finite Fields” Subjects and Themes:
- Subjects: Number Theory - Mathematics
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- Internet Archive ID: arxiv-1602.02541
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33Zero-nonzero Patterns For Nilpotent Matrices Over Finite Fields
By Kevin N. Vander Meulen and Adam Van Tuyl
Fix a field F. A zero-nonzero pattern A is said to be potentially nilpotent over F if there exists a matrix with entries in F with zero-nonzero pattern A that allows nilpotence. In this paper we initiate an investigation into which zero-nonzero patterns are potentially nilpotent over F, with a special emphasis on the case that F = Z_p is a finite field. As part of this investigation, we develop methods, using the tools of algebraic geometry and commutative algebra, to eliminate zero-nonzero patterns A as being potentially nilpotent over any field F. We then use these techniques to classify all irreducible zero-nonzero patterns of order two and three that are potentially nilpotent over Z_p for each prime p.
“Zero-nonzero Patterns For Nilpotent Matrices Over Finite Fields” Metadata:
- Title: ➤ Zero-nonzero Patterns For Nilpotent Matrices Over Finite Fields
- Authors: Kevin N. Vander MeulenAdam Van Tuyl
- Language: English
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- Internet Archive ID: arxiv-0812.0527
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34On Point Sets In Vector Spaces Over Finite Fields That Determine Only Acute Angle Triangles
By Igor E. Shparlinski
For three points $\vec{u}$,$\vec{v}$ and $\vec{w}$ in the $n$-dimensional space $\F_q^n$ over the finite field $\F_q$ of $q$ elements we give a natural interpretation of an acute angle triangle defined by this points. We obtain an upper bound on the size of a set $\cZ$ such that all triples of distinct points $\vec{u}, \vec{v}, \vec{w} \in \cZ$ define acute angle triangles. A similar question in the real space $\cR^n$ dates back to P. Erd{\H o}s and has been studied by several authors.
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- Title: ➤ On Point Sets In Vector Spaces Over Finite Fields That Determine Only Acute Angle Triangles
- Author: Igor E. Shparlinski
- Language: English
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- Internet Archive ID: arxiv-0903.2520
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35Optimal Curves Of Low Genus Over Finite Fields
By Alexey Zaytsev
The Hasse-Weil-Serre bound is improved for curves of low genera over finite fields with discriminant in {-3,-4,-7,-8,-11,-19} by studying optimal curves.
“Optimal Curves Of Low Genus Over Finite Fields” Metadata:
- Title: ➤ Optimal Curves Of Low Genus Over Finite Fields
- Author: Alexey Zaytsev
- Language: English
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- Internet Archive ID: arxiv-0706.4203
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36Gauss-Green Theorem For Weakly Differentiable Vector Fields, Sets Of Finite Perimeter, And Balance Laws
By G. Q. Chen, M. Torres and W. Ziemer
We analyze a class of weakly differentiable vector fields (\FF \colon \rn \to \rn) with the property that (\FF\in L^{\infty}) and (\div \FF) is a Radon measure. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence-measure field $\FF$ over the boundary of an arbitrary set of finite perimeter, which ensures the validity of the Gauss-Green theorem. To achieve this, we establish a fundamental approximation theorem which states that, given a Radon measure $\mu$ that is absolutely continuous with respect to $\mathcal{H}^{N-1}$ on $\rn$, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure-theoretic interior of the set with respect to the measure $\|\mu\|$. With this approximation theorem, we derive the normal trace of $\FF$ on the boundary of any set of finite perimeter, (E), as the limit of the normal traces of $\FF$ on the boundaries of the approximate sets with smooth boundary, so that the Gauss-Green theorem for $\FF$ holds on (E). With these results, we analyze the Cauchy fluxes that are bounded by a Radon measure over any oriented surface (i.e. an $(N-1)$-dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure-valued source terms from the formulation of balance law. This framework also allows the recovery of Cauchy entropy fluxes through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws.
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- Authors: G. Q. ChenM. TorresW. Ziemer
- Language: English
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37Arithmetic Of Rationaly Connected Quintic 3-folds Over Finite And Function Fields
By Marc Perret
We prove that both classical Chevalley-Warning-Ax and Tsen theorems hold for the blowing up of a quintic 3-fold along a line of multiplicity 3. Both proofs, which are of the same spirit than the original ones, involve the description of this blowing-up as a subvariety of a toric variety.
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- Author: Marc Perret
- Language: English
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38Factoring Polynomials Over Finite Fields Using Balance Test
By Chandan Saha
We study the problem of factoring univariate polynomials over finite fields. Under the assumption of the Extended Riemann Hypothesis (ERH), (Gao, 2001) designed a polynomial time algorithm that fails to factor only if the input polynomial satisfies a strong symmetry property, namely square balance. In this paper, we propose an extension of Gao's algorithm that fails only under an even stronger symmetry property. We also show that our property can be used to improve the time complexity of best deterministic algorithms on most input polynomials. The property also yields a new randomized polynomial time algorithm.
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- Author: Chandan Saha
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39Complete Padovan Sequences In Finite Fields
By Juan B. Gil, Michael D. Weiner and Catalin Zara
Given a prime $p\ge 5$, and given $1
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- Authors: Juan B. GilMichael D. WeinerCatalin Zara
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40Finite Fields
By Lidl, Rudolf
Given a prime $p\ge 5$, and given $1
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- Author: Lidl, Rudolf
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41Improved Upper Bounds For The Number Of Points On Curves Over Finite Fields
By Everett W. Howe and Kristin E. Lauter
We give new arguments that improve the known upper bounds on the maximal number N_q(g) of rational points of a curve of genus g over a finite field F_q for a number of pairs (q,g). Given a pair (q,g) and an integer N, we determine the possible zeta functions of genus-g curves over F_q with N points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus-g curve over F_q with N points must have a low-degree map to another curve over F_q, and often this is enough to give us a contradiction. In particular, we able to provide eight previously unknown values of N_q(g), namely: N_4(5) = 17, N_4(10) = 27, N_8(9) = 45, N_{16}(4) = 45, N_{128}(4) = 215, N_3(6) = 14, N_9(10) = 54, and N_{27}(4) = 64. Our arguments also allow us to give a non-computer-intensive proof of the recent result of Savitt that there are no genus-4 curves over F_8 having exactly 27 rational points. Furthermore, we show that there is an infinite sequence of q's such that for every g with 0 < g < log_2 q, the difference between the Weil-Serre bound on N_q(g) and the actual value of N_q(g) is at least g/2.
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- Title: ➤ Improved Upper Bounds For The Number Of Points On Curves Over Finite Fields
- Authors: Everett W. HoweKristin E. Lauter
- Language: English
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42Locally Decodable Codes From Nice Subsets Of Finite Fields And Prime Factors Of Mersenne Numbers
By Kiran S. Kedlaya and Sergey Yekhanin
A k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that one can probabilistically recover any bit x_i of the message by querying only k bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. The major goal of LDC related research is to establish the optimal trade-off between length and query complexity of such codes. Recently [Y] introduced a novel technique for constructing locally decodable codes and vastly improved the upper bounds for code length. The technique is based on Mersenne primes. In this paper we extend the work of [Y] and argue that further progress via these methods is tied to progress on an old number theory question regarding the size of the largest prime factors of Mersenne numbers. Specifically, we show that every Mersenne number m=2^t-1 that has a prime factor p>m^\gamma yields a family of k(\gamma)-query locally decodable codes of length Exp(n^{1/t}). Conversely, if for some fixed k and all \epsilon > 0 one can use the technique of [Y] to obtain a family of k-query LDCs of length Exp(n^\epsilon); then infinitely many Mersenne numbers have prime factors arger than known currently.
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- Title: ➤ Locally Decodable Codes From Nice Subsets Of Finite Fields And Prime Factors Of Mersenne Numbers
- Authors: Kiran S. KedlayaSergey Yekhanin
- Language: English
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- Internet Archive ID: arxiv-0704.1694
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43Averages Over Hyperplanes, Sum-product Theory In Vector Spaces Over Finite Fields And The Erdos-Falconer Distance Conjecture
By Derrick Hart, Alex Iosevich, Doowon Koh and Misha Rudnev
We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${\mathbb F}_q$, the finite field with q elements, by $A \cdot A+... +A \cdot A$, where A is a subset ${\mathbb F}_q$ of sufficiently large size. We also use the incidence machinery we develope and arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting due to the influence of the arithmetic. On the positive side, we obtain good exponents for the Erdos -Falconer distance problem for subsets of the unit sphere in $\mathbb F_q^d$ and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher dimensional vector spaces over general finite fields.
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- Title: ➤ Averages Over Hyperplanes, Sum-product Theory In Vector Spaces Over Finite Fields And The Erdos-Falconer Distance Conjecture
- Authors: Derrick HartAlex IosevichDoowon KohMisha Rudnev
- Language: English
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- Internet Archive ID: arxiv-0707.3473
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44The Riemann Hypothesis For Function Fields Over A Finite Field
By Machiel van Frankenhuijsen
We discuss Enrico Bombieri's proof of the Riemann hypothesis for curves over a finite field. Reformulated, it states that the number of points on a curve $\C$ defined over the finite field $\F_q$ is of the order $q+O(\sqrt{q})$. The first proof was given by Andr\'e Weil in 1942. This proof uses the intersection of divisors on $\C\times\C$, making the application to the original Riemann hypothesis so far unsuccessful, because $\spec\Z\times\spec\Z=\spec\Z$ is one-dimensional. A new method of proof was found in 1969 by S. A. Stepanov. This method was greatly simplified and generalized by Bombieri in 1973. Bombieri's method uses functions on $\C\times\C$, again precluding a direct translation to a proof of the original Riemann hypothesis. However, the two coordinates on $\C\times\C$ have different roles, one coordinate playing the geometric role of the variable of a polynomial, and the other coordinate the arithmetic role of the coefficients of this polynomial. The Frobenius automorphism of $\C$ acts on the geometric coordinate of $\C\times\C$. In the last section, we make some suggestions how Nevanlinna theory could provide a model of $\spec\Z\times\spec\Z$ that is two-dimensional and carries an action of Frobenius on the geometric coordinate.
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- Title: ➤ The Riemann Hypothesis For Function Fields Over A Finite Field
- Author: Machiel van Frankenhuijsen
- Language: English
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45The Weight Hierarchies And Chain Condition Of A Class Of Codes From Varieties Over Finite Fields
By Wu, Xinen, Feng, Gui-Liang and Rao, T. R. N
The generalized Hamming weights of linear codes were first introduced by Wei. These are fundamental parameters related to the minimal overlap structures of the subcodes and very useful in several fields. It was found that the chain condition of a linear code is convenient in studying the generalized Hamming weights of the product codes. In this paper we consider a class of codes defined over some varieties in projective spaces over finite fields, whose generalized Hamming weights can be determined by studying the orbits of subspaces of the projective spaces under the actions of classical groups over finite fields, i.e., the symplectic groups, the unitary groups and orthogonal groups. We give the weight hierarchies and generalized weight spectra of the codes from Hermitian varieties and prove that the codes satisfy the chain condition.
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- Title: ➤ The Weight Hierarchies And Chain Condition Of A Class Of Codes From Varieties Over Finite Fields
- Authors: Wu, XinenFeng, Gui-LiangRao, T. R. N
- Language: English
“The Weight Hierarchies And Chain Condition Of A Class Of Codes From Varieties Over Finite Fields” Subjects and Themes:
- Subjects: ➤ PYROELECTRICITY - ELECTRIC DIPOLES - ELECTRIC POTENTIAL - PIEZOELECTRICITY - SUBSTRATES - THERMAL STABILITY - ELECTRODES - DIELECTRIC POLARIZATION - POLYMERIC FILMS - PATENT APPLICATIONS - DIELECTRICS
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- Internet Archive ID: nasa_techdoc_19960041244
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46Quantization Of Symplectic Vector Spaces Over Finite Fields
By Shamgar Gurevich and Ronny Hadani
In this paper, we construct a quantization functor, associating a complex vector space H(V) to a finite dimensional symplectic vector space V over a finite field of odd characteristic. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp(V). The main new technical result is a proof of a stronger form of the Stone-von Neumann property for the Heisenberg group. Our result answers, for the case of the Heisenberg group, a question of Kazhdan about the possible existence of a canonical vector space attached to a coadjoint orbit of a general unipotent group over finite field.
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- Title: ➤ Quantization Of Symplectic Vector Spaces Over Finite Fields
- Authors: Shamgar GurevichRonny Hadani
- Language: English
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47Separability And Entanglement In Finite Dimer-type Chains In General Transverse Fields
By Norma Canosa, Raul Rossignoli and Juan Mauricio Matera
We determine the conditions under which general dimer-type spin chains with $XYZ$ couplings of arbitrary range in a general transverse field will exhibit an exactly separable parity-breaking eigenstate. We also provide sufficient conditions which ensure that it will be a ground state. We then examine the exact side limits at separability of the entanglement between any two spins in a finite chain, showing that in the vicinity of separability, the system will loose all signatures of dimerization, with pairwise entanglement approaching infinite range and becoming independent of separation and interaction range. The possibility of a non-uniform exactly separable ground state induced by an alternating field is also shown. As illustration, we examine the behavior of the pairwise entanglement in a finite $XY$ dimer chain under a uniform as well as alternating field. Related aspects of the magnetization are also discussed.
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- Title: ➤ Separability And Entanglement In Finite Dimer-type Chains In General Transverse Fields
- Authors: Norma CanosaRaul RossignoliJuan Mauricio Matera
- Language: English
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- Internet Archive ID: arxiv-1008.4412
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48Topologically Protected Landau Levels In Bilayer Graphene In Finite Electric Fields
By Tohru Kawarabayashi, Yasuhiro Hatsugai and Hideo Aoki
The zero-energy Landau level of bilayer graphene is shown to be anomalously sharp (delta-function like) against bond disorder as long as the disorder is correlated over a few lattice constants.The robustness of the zero-mode anomaly can be attributed to the preserved chiral symmetry. Unexpectedly, even when we apply a finite potential difference (i.e., an electric field) between the top and the bottom layers, the valley-split $n=0$ Landau levels remain anomalously sharp although they are now shifted away from the zero energy, while the $n=1$ Landau levels exhibit the usual behavior.
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- Authors: Tohru KawarabayashiYasuhiro HatsugaiHideo Aoki
- Language: English
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- Internet Archive ID: arxiv-1111.5894
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49On Ritt's Decomposition Theorem In The Case Of Finite Fields
By Jaime Gutierrez and David Sevilla
A classical theorem by Ritt states that all the complete decomposition chains of a univariate polynomial satisfying a certain tameness condition have the same length. In this paper we present our conclusions about the generalization of these theorem in the case of finite coefficient fields when the tameness condition is dropped.
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- Title: ➤ On Ritt's Decomposition Theorem In The Case Of Finite Fields
- Authors: Jaime GutierrezDavid Sevilla
- Language: English
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50Counting Quiver Representations Over Finite Fields Via Graph Enumeration
By Geir T. Helleloid and Fernando Rodriguez-Villegas
Let $\Gamma$ be a quiver on n vertices $v_1, v_2, ..., v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\alpha \in \N^n$. Hua gave a formula for $A_{\Gamma}(\alpha, q)$, the number of isomorphism classes of absolutely indecomposable representations of $\Gamma$ over the finite field $\F_q$ with dimension vector $\alpha$. Kac showed that $A_{\Gamma}(\bm{\alpha}, q)$ is a polynomial in q with integer coefficients. Using Hua's formula, we show that for each non-negative integer s, the s-th derivative of $A_{\Gamma}(\alpha,q)$ with respect to q, when evaluated at q = 1, is a polynomial in the variables $g_{ij}$, and we compute the highest degree terms in this polynomial. Our formulas for these coefficients depend on the enumeration of certain families of connected graphs.
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- Title: ➤ Counting Quiver Representations Over Finite Fields Via Graph Enumeration
- Authors: Geir T. HelleloidFernando Rodriguez-Villegas
- Language: English
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- Internet Archive ID: arxiv-0810.2127
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