Lattice Builder Manual Software Package for Constructing Rank-1 Lattices

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LatCommon::NormaRogers Class Reference

This class implements the Rogers bounds on the density of sphere packing. More...

`#include <NormaRogers.h>`

Inherits LatCommon::Normalizer.

## Public Member Functions
| NormaRogers (const MScal &m, int k, int t, double beta=1)
| Constructor for the Rogers bounds. More...
| ~NormaRogers ()
| Destructor.
double | getGamma (int j) const throw (std::out_of_range)
| Returns the value of the Rogers lattice constant \(\gamma_j\) in dimension \(j\).
Public Member Functions inherited from LatCommon::Normalizer
| Normalizer (const MScal &m, int k, int t, std::string Name, NormType norm=L2NORM, double beta=1)
virtual | ~Normalizer ()
| Destructor.
void | init (const MScal &m, int k, 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 | setNorm (NormType norm)
| Sets the norm associated with this object to `norm`.
int | getDim () const
| Returns the maximal dimension for this object.
double & | getCst (int j)
| Returns the bound on the length of the shortest nonzero vector in dimension \(j\). More...

## Additional Inherited Members
Static Public Attributes inherited from LatCommon::Normalizer
static const int | MAX_DIM = 48
| Constructor for the bounds. More...
Protected Attributes inherited from LatCommon::Normalizer
std::string | m_name
| Name of the normalizer.
NormType | m_norm
| Norm associated with this object.
MScal | m_m
| Number of points of the lattice per unit volume.
int | m_rank
| Rank of the lattice.
int | m_maxDim
| Only elements 1 to `m_maxDim` (inclusive) of arrays are defined.
double | m_beta
| Beta factor.
double * | m_cst
| Contains the bounds on the length of the shortest nonzero vector in the lattice in each dimension.

## Detailed Description

This class implements the Rogers bounds on the density of sphere packing.

The length of vectors is computed using the \({\cal L}_2\) norm. The bounding lengths, for a lattice of rank \(k\) containing \(m\) points per unit volume in dimension \(t\), are given by \(\ell_t^* = \gamma_t m^{k/t}\) for \(t \ge k\), where the \(\gamma_t\) are the Rogers lattice constants.

## Constructor & Destructor Documentation

LatCommon::NormaRogers::NormaRogers | ( | const MScal & | m,
|  | int | k,
|  | int | t,
|  | double | beta = `1`
| ) |  |

Constructor for the Rogers bounds.

The lattices are those of rank \(k\), with \(m\) points per unit volume, in all dimensions \(\le t\). The bias factor `beta` \(= \beta\) gives more weight to some of the dimensions. There is no restriction on the dimension \(t\) which can be larger than 48.

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

- latcommon/include/latcommon/NormaRogers.h