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Software Package For Testing The Uniformity Of Integral Lattices In The Real Space
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LatticeTester::NormaBestLat Class Reference

This Normalizer class implements approximate upper bounds on the length of the shortest nonzero vector in a lattice. More...

+ Inheritance diagram for LatticeTester::NormaBestLat:

Public Member Functions

 NormaBestLat (double logm, int64_t k, int64_t maxDim, NormType norm=L2NORM)
 This constructor assumes that the rescaled primal lattice has scaling factor \(m\) and order \(k\), so its density is \(m^{k-t}\) for \(t\geq k\), and cannot exceed 1 for projections in \(s < k\) dimensions.
 
 NormaBestLat (double logDensity, int64_t maxDim, NormType norm=L2NORM)
 Constructs a NormaBestLat for up to maxDim dimensions, by assuming that the log density is logDensity in all dimensions and the lattice was not rescaled.
 
double getGamma (int64_t j) const
 Returns the value of the bound on the Hermite's constant \(\gamma_j\) in dimension \(j\).
 
- Public Member Functions inherited from LatticeTester::Normalizer
 Normalizer (double logm, int64_t k, int64_t maxDim, NormType norm=L2NORM)
 This is the preferred constructor in all the subclasses.
 
 Normalizer (double logDensity, int64_t maxDim, NormType norm=L2NORM)
 This old (legacy) constructor assumes that the lattice is not rescaled and that the (log)density is the same in all dimensions, which is practically never true when rescaling.
 
 Normalizer (int64_t maxDim, 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 computes the bounds that this normalizer will return, by assuming that the primal lattice was rescaled by the factor \(m\) and has order \(k\), so its density is \(m^{k-t}\) for \(t\geq k\), and cannot exceed \(m^s\) for projections in \(s < k\) dimensions.
 
virtual void computeBounds (double logDensity)
 This method computes bounds by assuming that the log density is logDensity for all dimensions up to the maximal dimension maxDim, which is rarely true.
 
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.
 

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 compute a 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 Normalizer class implements approximate upper bounds on the length of the shortest nonzero vector in a lattice.

In the dimensions in which the Hermite constants \(\gamma_s\) are known, the exact values are used. For the other dimensions, they are approximated by lower bounds that correspond to the largest known center density values \(\delta_s\) for lattices sphere packings, by using the relationship

\[ \gamma_s = 4 \delta_s^{2/s}. \]

These values are taken from Table I.1(a) in the preface of [2]. See also Table 1 in [mCOH17a]. These approximations are stored in a table accessible via the getGamma method. The values given here are for the L2NORM. For the L1NORM, the bounds must be multiplied by s in s dimensions.

Constructor & Destructor Documentation

◆ NormaBestLat() [1/2]

LatticeTester::NormaBestLat::NormaBestLat ( double logm,
int64_t k,
int64_t maxDim,
NormType norm = L2NORM )

This constructor assumes that the rescaled primal lattice has scaling factor \(m\) and order \(k\), so its density is \(m^{k-t}\) for \(t\geq k\), and cannot exceed 1 for projections in \(s < k\) dimensions.

◆ NormaBestLat() [2/2]

LatticeTester::NormaBestLat::NormaBestLat ( double logDensity,
int64_t maxDim,
NormType norm = L2NORM )

Constructs a NormaBestLat for up to maxDim dimensions, by assuming that the log density is logDensity in all dimensions and the lattice was not rescaled.

Restriction: maxDim \( \le 48\).

Member Function Documentation

◆ getGamma()

double LatticeTester::NormaBestLat::getGamma ( int64_t j) const
inlinevirtual

Returns the value of the bound on the Hermite's constant \(\gamma_j\) in dimension \(j\).

Reimplemented from LatticeTester::Normalizer.


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