LatticeTester::NormaPalpha Class Reference

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

#include <latticetester/NormaPalpha.h>

Inheritance diagram for LatticeTester::NormaPalpha:

Public Member Functions
	NormaPalpha (const std::int64_t m, int64_t alpha, int64_t s, NormType norm=L2NORM)
	Constructor for the bounds \(B_{\alpha}(s)\) obtained for lattices, in all dimensions \(\le s\), where \(\alpha= {}\)alpha.
double	calcBound (int64_t alpha, int64_t s)
	Computes and returns the bound \(B_{\alpha}(s)\) given in [12]  (p.
void	computeBounds (int64_t alpha)
	Initializes the bounds for the Palpha normalization.
void	computeBounds ()
	Initializes the bounds for the Palpha normalization.
int64_t	getAlpha () const
	Returns the value of \(\alpha\).
virtual void	computeBounds (double logDensity)
	Preventing the compiler from raising the warning: '' latticetester/NormaPalpha.h:59:9: warning: 'LatticeTester::NormaPalpha::init' hides overloaded virtual function [-Woverloaded-virtual] void init (int64_t alpha); ''.
void	computeBounds (double logm, int64_t k)
	Preventing the compiler from raising the warning: '' latticetester/NormaPalpha.h:59:9: warning: 'LatticeTester::NormaPalpha::init' hides overloaded virtual function [-Woverloaded-virtual] void init (int64_t alpha); ''.
Public Member Functions inherited from LatticeTester::Normalizer
	Normalizer (double logm, int64_t k, int64_t maxDim, std::string name="", NormType norm=L2NORM)
	This is the preferred constructor in all the subclasses.
	Normalizer (double logDensity, int64_t maxDim, std::string name="", NormType norm=L2NORM)
	***** DEPRECATED *****
	Normalizer (int64_t maxDim, std::string name, NormType norm=L2NORM)
	This constructor creates a Normalizer object without computing any bounds.
virtual	~Normalizer ()
	Destructor.
void	computeBounds (double logm, int64_t k)
	This method is called by the constructors of subclasses, in which the constants gamma_t are known.
std::string	toString () const
	Returns a string that describes this object.
NormType	getNorm () const
	Returns the norm associated with this object.
int64_t	getMaxDim () const
	Returns the maximal dimension for this object.
virtual double *	getBounds () const
	Returns the array that contains the bounds on the lengths of the shortest nonzero vectors.
virtual double	getBound (int64_t j) const
	Returns the bound on the length of the shortest nonzero vector in j dimensions.
virtual double	getGamma (int64_t j) const
	Returns the value of a lattice constant \(\gamma\) in dimension \(j\).

Additional Inherited Members
Static Public Attributes inherited from LatticeTester::Normalizer
static const int64_t	MAX_DIM = 48
	The maximum dimension of the lattices for which this class can give an upper bound.
Protected Attributes inherited from LatticeTester::Normalizer
std::string	m_name
	Name of the normalizer.
NormType	m_norm
	Norm type associated with this object.
int64_t	m_maxDim
	Maximum dimension.
double *	m_bounds
	Contains the bounds on the length of the shortest nonzero vector in the lattice in each dimension up to maxDim.

Detailed Description

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

Constructor & Destructor Documentation

NormaPalpha()

LatticeTester::NormaPalpha::NormaPalpha	(	const std::int64_t	m,
		int64_t	alpha,
		int64_t	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()

double LatticeTester::NormaPalpha::calcBound	(	int64_t	alpha,
		int64_t	s )

Computes and returns the bound \(B_{\alpha}(s)\) given in [12]  (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 \(-1\).

computeBounds() [1/4]

void LatticeTester::NormaPalpha::computeBounds

(

)

Initializes the bounds for the Palpha normalization.

computeBounds() [2/4]

void LatticeTester::Normalizer::computeBounds ( double logDensity )

virtual

Preventing the compiler from raising the warning: '' latticetester/NormaPalpha.h:59:9: warning: 'LatticeTester::NormaPalpha::init' hides overloaded virtual function [-Woverloaded-virtual] void init (int64_t alpha); ''.

Reimplemented from LatticeTester::Normalizer.

computeBounds() [3/4]

void LatticeTester::Normalizer::computeBounds	(	double	logm,
		int64_t	k )

Preventing the compiler from raising the warning: '' latticetester/NormaPalpha.h:59:9: warning: 'LatticeTester::NormaPalpha::init' hides overloaded virtual function [-Woverloaded-virtual] void init (int64_t alpha); ''.

computeBounds() [4/4]

void LatticeTester::NormaPalpha::computeBounds

(

int64_t

alpha

)

Initializes the bounds for the Palpha normalization.

getAlpha()

int64_t LatticeTester::NormaPalpha::getAlpha ( ) const

inline

Returns the value of \(\alpha\).

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

• include/latticetester/NormaPalpha.h