scholarly journals Constructions of Dense Lattices of Full Diversity

2020 ◽  
Vol 21 (2) ◽  
pp. 299
Author(s):  
A. A. Andrade ◽  
A. J. Ferrari ◽  
J. C. Interlando ◽  
R. R. Araujo
Keyword(s):  
Real Number ◽  
Fading Channels ◽  
Packing Density ◽  
Signal Transmission ◽  
Cyclotomic Field ◽  
Sphere Packing ◽  
Number Fields ◽  
Trace Form ◽  
Full Diversity ◽  
Totally Real

A lattice construction using Z-submodules of rings of integers of number fields is presented. The construction yields rotated versions of the laminated lattices A_n for n = 2,3,4,5,6, which are the densest lattices in their respective dimensions. The sphere packing density of a lattice is a function of its packing radius, which in turn can be directly calculated from the minimum squared Euclidean norm of the lattice. Norms in a lattice that is realized by a totally real number field can be calculated by the trace form of the field restricted to its ring of integers. Thus, in the present work, we also present the trace form of the maximal real subfield of a cyclotomic field. Our focus is on totally real number fields since their associated lattices have full diversity. Along with high packing density, the full diversity feature is desirable in lattices that are used for signal transmission over both Gaussian and Rayleigh fading channels.

Constructions of rotated lattice constellations in dimensions power of 3

2018 ◽  
Vol 17 (09) ◽  
pp. 1850175 ◽  
Author(s):  
Agnaldo José Ferrari ◽  
Antonio Aparecido de Andrade
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Fading Channels ◽  
Lower Bounds ◽  
Rayleigh Fading ◽  
Signal Transmission ◽  
Number Fields ◽  
Rayleigh Fading Channels ◽  
Full Diversity ◽  
Totally Real

In this paper, we present the constructions of rotated [Formula: see text]-lattices, where [Formula: see text] is a positive integer, via [Formula: see text]-modules of the ring of the integers [Formula: see text]. Our focus is on totally real number fields since the associated lattices have full diversity and then may be suitable for signal transmission over both Gaussian and Rayleigh fading channels. Lower bounds for the minimum product distances of such construction are also presented.

Algebraic constructions of rotated unimodular lattices and direct sum of Barnes–Wall lattices

2020 ◽  
pp. 2150029
Author(s):  
João Eloir Strapasson ◽  
Agnaldo José Ferrari ◽  
Grasiele Cristiane Jorge ◽  
Sueli Irene Rodrigues Costa
Keyword(s):  
Real Number ◽  
Fading Channels ◽  
Rayleigh Fading ◽  
Signal Transmission ◽  
Number Fields ◽  
Rayleigh Fading Channels ◽  
Direct Sum ◽  
Full Diversity ◽  
Algebraic Constructions ◽  
Totally Real

In this paper, we construct some families of rotated unimodular lattices and rotated direct sum of Barnes–Wall lattices [Formula: see text] for [Formula: see text] and [Formula: see text] via ideals of the ring of the integers [Formula: see text] for [Formula: see text] and [Formula: see text]. We also construct rotated [Formula: see text] and [Formula: see text]-lattices via [Formula: see text]-submodules of [Formula: see text]. Our focus is on totally real number fields since the associated lattices have full diversity and then may be suitable for signal transmission over both Gaussian and Rayleigh fading channels. The minimum product distances of such constructions are also presented here.

scholarly journals ON NUMBER FIELDS WITH EQUIVALENT INTEGRAL TRACE FORMS

2012 ◽  
Vol 08 (07) ◽  
pp. 1569-1580 ◽  
Author(s):  
GUILLERMO MANTILLA-SOLER
Keyword(s):  
Quadratic Form ◽  
Real Number ◽  
Number Field ◽  
Number Fields ◽  
Trace Form ◽  
Integral Quadratic Form ◽  
Trace Forms ◽  
Fundamental Discriminant ◽  
Totally Real

Let K be a number field. The integral trace form is the integral quadratic form given by tr k/ℚ(x2)|OK. In this article we study the existence of non-conjugated number fields with equivalent integral trace forms. As a corollary of one of the main results of this paper, we show that any two non-totally real number fields with the same signature and same prime discriminant have equivalent integral trace forms. Additionally, based on previous results obtained by the author and the evidence presented here, we conjecture that any two totally real quartic fields of fundamental discriminant have equivalent trace zero forms if and only if they are conjugated.

scholarly journals Weak Arithmetic Equivalence

2015 ◽  
Vol 58 (1) ◽  
pp. 115-127 ◽  
Author(s):  
Guillermo Mantilla-Soler
Keyword(s):  
Zeta Function ◽  
Real Number ◽  
Number Field ◽  
Number Fields ◽  
Trace Form ◽  
Root Numbers ◽  
Local Root ◽  
Arithmetic Equivalence ◽  
Totally Real ◽  
Local Root Numbers

AbstractInspired by the invariant of a number field given by its zeta function, we define the notion of weak arithmetic equivalence and show that under certain ramification hypotheses this equivalence determines the local root numbers of the number field. This is analogous to a result of Rohrlich on the local root numbers of a rational elliptic curve. Additionally, we prove that for tame non-totally real number fields, the integral trace form is invariant under arithmetic equivalence

scholarly journals A Construction of Rotated Lattices via Totally Real Subfields of the Cyclotomic Field Q(z_p)

2019 ◽  
Vol 20 (3) ◽  
pp. 561
Author(s):  
Antonio A. Andrade ◽  
Everton L. Oliveira ◽  
José C. Interlando
Keyword(s):  
Fading Channel ◽  
Rayleigh Fading ◽  
Cyclotomic Field ◽  
Number Fields ◽  
Rayleigh Fading Channel ◽  
Ring Of Integers ◽  
Full Diversity ◽  
Alternative Approach ◽  
Totally Real ◽  
Modulation Diversity

The theory of lattices have shown to be useful in information theory and rotated lattices with high modulations diversity have been extensively studied as an alternative approach for transmission over a Rayleigh-fading channel, where the performance of this modulation schemes essentially depends of the modulation diversity and of the minimum product distance to achieve substantial coding gains. The maximum diversity of a rotated lattice is guaranteed when we use totally real number fields and the minimum product distance is optimized by considering fields with minimum discriminant. In this paper, we present a construction of rotated lattice for the Rayleigh fading channel in Euclidean spaces with full diversity, where this construction is through a totally real subfield K of the cyclotomic field Q(z_p), where p is an odd prime, obtained by endowing their ring of integers.

REPRESENTING Ω(l∞) FOR REAL ABELIAN FIELDS

2003 ◽  
Vol 02 (03) ◽  
pp. 237-276 ◽  
Author(s):  
JÜRGEN RITTER ◽  
ALFRED WEISS
Keyword(s):  
Real Number ◽  
Prime Number ◽  
Cyclotomic Field ◽  
Number Fields ◽  
Iwasawa Theory ◽  
Root Number ◽  
Abelian Extensions ◽  
Main Conjecture ◽  
Abelian Fields ◽  
Totally Real

For real subfields K of a cyclotomic field ℚ(ς) we remove the tameness assumption at a given odd prime number l, which was needed in [11] in order to establish the equivalence of the Lifted Root Number Conjecture at l and an equivariant main conjecture of Iwasawa theory for abelian extensions of totally real number fields K/k.

On linear relations for special L-values over certain totally real number fields

2021 ◽  
pp. 1-18
Author(s):  
Seiji Kuga
Keyword(s):  
Real Number ◽  
Fourier Coefficients ◽  
Number Fields ◽  
Arithmetic Functions ◽  
Hilbert Modular Form ◽  
Linear Relations ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular ◽  
Totally Real

In this paper, we give linear relations between the Fourier coefficients of a special Hilbert modular form of half integral weight and some arithmetic functions. As a result, we have linear relations for the special [Formula: see text]-values over certain totally real number fields.

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