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.
double	calcBound (int alpha, int s)
	Computes and returns the bound \(B_{\alpha}(s)\) given in [31]  (p.
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)
	Complete constructor for a Normalizer.
	Normalizer (int t, std::string Name, NormType norm=L2NORM, double beta=1)
	Constructor that does not take the density as an argument.
virtual	~Normalizer ()
	Destructor.
virtual void	init (RedDbl &logDensity, double beta)
	This is a method that will initialize the bounds this normalizer can return.
std::string	ToString () const
	Returns a string that describes this object.
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 for dimension j as computed in Normalizer::init().
virtual RedDbl	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 a lattice constant \(\gamma\) in dimension \(j\).

Additional Inherited Members
Static Public Attributes inherited from LatticeTester::Normalizer< RedDbl >
static const int	MAX_DIM = 48
	The maximum dimension of the lattices for which this class can give an upper bound.
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 m_bounds bellow will be pre-computed.
double	m_beta
	Beta factor used to give more or less importance to some of the dimensions.
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.

References LatticeTester::Normalizer< RedDbl >::MAX_DIM, and LatticeTester::Normalizer< RedDbl >::Normalizer().

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 [31]  (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().

Referenced by init().

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

• latticetester/include/latticetester/NormaPalpha.h