LatNet Builder Manual  2.0.1-11
Software Package for Constructing Highly Uniform Point Sets
LatticeTester::NormaPalpha< Int, RedDbl > Class Template Reference

This class implements theoretical bounds on the values of \(P_{\alpha}\) for a lattice (see class Palpha). More...

#include <NormaPalpha.h>

Inherits LatticeTester::Normalizer< RedDbl >.

Public Member Functions

 NormaPalpha (const Int &m, int alpha, int s, NormType norm=L2NORM)
 Constructor for the bounds \(B_{\alpha}(s)\) obtained for lattices, in all dimensions \(\le s\), where \(\alpha= {}\)alpha. More...
 
double calcBound (int alpha, int s)
 Computes and returns the bound \(B_{\alpha}(s)\) given in [33]  (p. More...
 
void init (int alpha)
 Initializes the bounds for the Palpha normalization.
 
int getAlpha () const
 Returns the value of \(\alpha\).
 
- Public Member Functions inherited from LatticeTester::Normalizer< RedDbl >
 Normalizer (RedDbl &logDensity, int t, std::string Name, NormType norm=L2NORM, double beta=1)
 Constructor for the bounds. More...
 
 Normalizer (int t, std::string Name, NormType norm=L2NORM, double beta=1)
 Constructor only used by the NormaPalpha class. More...
 
virtual ~Normalizer ()
 Destructor.
 
virtual void init (RedDbl &logDensity, double beta)
 Initializes the bounds on the length of the shortest vector. More...
 
std::string ToString () const
 Returns this object as a string.
 
NormType getNorm () const
 Returns the norm associated with this object.
 
void setLogDensity (RedDbl logDensity)
 Sets the log-density associated with this object to logDensity.
 
RedDbl getLogDensity () const
 Returns the logDensity associated with this object.
 
void setNorm (NormType norm)
 Sets the norm associated with this object to norm.
 
int getDim () const
 Returns the maximal dimension for this object.
 
double getPreComputedBound (int j) const
 Returns the bound on the length of the shortest nonzero vector in dimension \(j\) as computed in Normalizer::init.
 
double getBound (int j) const
 Calculates and returns the bound on the length of the shortest nonzero vector in dimension \(j\).
 
virtual double getGamma (int j) const
 Returns the value of the lattice constant \(\gamma_j\) in dimension \(j\). More...
 

Additional Inherited Members

- Static Public Attributes inherited from LatticeTester::Normalizer< RedDbl >
static const int MAX_DIM = 48
 
- Protected Attributes inherited from LatticeTester::Normalizer< RedDbl >
std::string m_name
 Name of the normalizer.
 
NormType m_norm
 Norm associated with this object.
 
RedDbl m_logDensity
 log of the density, ie log of the number of points of the lattice per unit of volume.
 
int m_maxDim
 Only elements 1 to m_maxDim (inclusive) of arrays are defined.
 
double m_beta
 Beta factor.
 
double * m_bounds
 Contains the bounds on the length of the shortest nonzero vector in the lattice in each dimension.
 

Detailed Description

template<typename Int, typename RedDbl>
class LatticeTester::NormaPalpha< Int, RedDbl >

This class implements theoretical bounds on the values of \(P_{\alpha}\) for a lattice (see class Palpha).

Constructor & Destructor Documentation

◆ NormaPalpha()

template<typename Int , typename RedDbl >
LatticeTester::NormaPalpha< Int, RedDbl >::NormaPalpha ( const Int &  m,
int  alpha,
int  s,
NormType  norm = L2NORM 
)

Constructor for the bounds \(B_{\alpha}(s)\) obtained for lattices, in all dimensions \(\le s\), where \(\alpha= {}\)alpha.

The lattices have rank \(1\), with \(m\) points per unit volume. Restriction: \(2 \le s \le48\), \(\alpha\ge2\), and \(m\) prime.

Member Function Documentation

◆ calcBound()

template<typename Int , typename RedDbl >
double LatticeTester::NormaPalpha< Int, RedDbl >::calcBound ( int  alpha,
int  s 
)

Computes and returns the bound \(B_{\alpha}(s)\) given in [33]  (p.

83, Theorem 4.4). Given \(s > 1\), \(\alpha> 1\), \(m\) prime, and \(m > e^{\alpha s/(\alpha-1)}\), then there exists an integer vector \(\mathbf{a} \in\mathbb Z^s\) such that

\[ P_{\alpha}(s, \mathbf{a}) \le B_{\alpha}(s) = \frac{e}{s}^{\alpha s} \frac{(2\ln m + s)^{\alpha s}}{m^{\alpha}}. \]

If the conditions for the existence of the bound are not satisfied, the function returns \(-1\).

References NTL::conv(), and LatticeTester::IntFactor< Int >::isPrime().


The documentation for this class was generated from the following file: