docs/doc/dox/background.dox File Reference

Functions
of cite mCON99a *for f $t le24 and can be approximated with f	$O (1/t)\f$ error and *approximately 4 decimal digits of precision
in cite rLEC99c gives the ratio *	f (\gamma_t^{\mathrm{L}}/\gamma_t^{\mathrm{R}})^
*We can view the lattice as a way of packing the space by spheres of *radius f ell_t with one sphere centered at each lattice point In *the dual this gives f we obtain *f an upper bound on f ell_t f can be obtained in terms *of an upper bound on f delta_t and vice versa Let f delta_t *f be the *largest possible value of f delta_t f for a	lattice (i.e., the densest *packing by non-overlapping spheres arranged in a lattice). The quantity *\f$\gamma_t
*We can view the lattice as a way of packing the space by spheres of *radius f ell_t with one sphere centered at each lattice point In *the dual this gives f we obtain *f an upper bound on f ell_t f can be obtained in terms *of an upper bound on f delta_t and vice versa Let f delta_t *f be the *largest possible value of f delta_t f for a cite mGRU87a It gives the *upper bound f ell_t	le (\ell_t^ *(n))^2
*We can view the lattice as a way of packing the space by spheres of *radius f ell_t with one sphere centered at each lattice point In *the dual this gives f we obtain *f an upper bound on f ell_t f can be obtained in terms *of an upper bound on f delta_t and vice versa Let f delta_t *f be the *largest possible value of f delta_t f for a cite mGRU87a It gives the *upper bound f ell_t or good approximations of is useful because it *allows us to normalize f ell_t f to a value between and by taking *f ell_t ell_t *m k f This is convenient for comparing values for *different values of f $t f and f $m k f Good values are close to and *bad values are close to **The Hermite constants are known exactly only for f $t le8 in which *case the densest lattice packings are attained by the *laminated *lattices *cite mCON99a Conway and Sloane cite and provide lower *and upper bounds on f delta_t *f for other values of f $t f The *largest value of f ell_t which we denote *by f in terms of f delta *f The laminated which give the lower bound *f ell_t where the constants f lambda_t f are given in *cite	mCON88a (Table 6.1, page 158) for \f $t\le48\f$
*We can view the lattice as a way of packing the space by spheres of *radius f ell_t with one sphere centered at each lattice point In *the dual this gives f we obtain *f an upper bound on f ell_t f can be obtained in terms *of an upper bound on f delta_t and vice versa Let f delta_t *f be the *largest possible value of f delta_t f for a cite mGRU87a It gives the *upper bound f ell_t or good approximations of is useful because it *allows us to normalize f ell_t f to a value between and by taking *f ell_t ell_t *m k f This is convenient for comparing values for *different values of f $t f and f $m k f Good values are close to and *bad values are close to **The Hermite constants are known exactly only for f $t le8 in which *case the densest lattice packings are attained by the *laminated *lattices *cite mCON99a Conway and Sloane cite	mCON99a (Table 1.2) *give the values of \f$\delta_t^ *\f$ for \f $t\le8\f$

Variables
of the *lower bound over the upper bound on f ell_t for *f le t le48 f This ratio tends to decrease with f $t but not *monotonously **Computing the shortest vector in terms of the Euclidean norm is *	convenient
*We can view the lattice as a way of packing the space by spheres of *radius f ell_t with one sphere centered at each lattice point In *the dual this gives f we obtain *f	delta_t
of the *lower bound over the upper bound on f ell_t for *f le t le48 f This ratio tends to decrease with f $t but not *monotonously **Computing the shortest vector in terms of the Euclidean norm is e for computational but one can also use another *norm instead For	example
*We can view the lattice as a way of packing the space by spheres of *radius f ell_t	f
of the *lower bound over the upper bound on f ell_t for *f le t le48 f This ratio tends to decrease with f $t but not *monotonously **Computing the shortest vector in terms of the Euclidean norm is e	g
*We can view the lattice as a way of packing the space by spheres of *radius f ell_t with one sphere centered at each lattice point In *the dual this gives f we obtain *f an upper bound on f ell_t f can be obtained in terms *of an upper bound on f delta_t and vice versa Let f delta_t *f be the *largest possible value of f delta_t f for a cite mGRU87a It gives the *upper bound f ell_t or good approximations of is useful because it *allows us to normalize f ell_t f to a value between and by taking *f ell_t ell_t *m k f This is convenient for comparing values for *different values of f $t f and f $m k f Good values are close to and *bad values are close to **The Hermite constants are known exactly only for f $t le8 in which *case the densest lattice packings are attained by the *laminated *lattices *cite mCON99a Conway and Sloane cite and provide lower *and upper bounds on f delta_t *f for other values of f $t f The *largest value of f ell_t which we denote *by f	gamma_t
*We can view the lattice as a way of packing the space by spheres of *radius f ell_t with one sphere centered at each lattice point In *the dual	lattice
*We can view the lattice as a way of packing the space by spheres of *radius f ell_t with one sphere centered at each lattice point In *the dual this gives f we obtain *f an upper bound on f ell_t f can be obtained in terms *of an upper bound on f delta_t and vice versa Let f delta_t *f be the *largest possible value of f delta_t f for a cite mGRU87a It gives the *upper bound f ell_t or good approximations of is useful because it *allows us to normalize f ell_t f to a value between and by taking *f ell_t ell_t *m k f This is convenient for comparing values for *different values of f $t f and f $m k f Good values are close to and *bad values are close to **The Hermite constants are known exactly only for f $t le8 in which *case the densest lattice packings are attained by the *laminated *lattices *cite mCON99a Conway and Sloane cite and provide lower *and upper bounds on f delta_t *f for other values of f $t f The *largest value of f ell_t which we denote *by f in terms of f delta *f The laminated *	lattices
of the *lower bound over the upper bound on f ell_t for *f le t le48 f This ratio tends to decrease with f $t but not *monotonously **Computing the shortest vector in terms of the Euclidean norm is e for computational but one can also use another *norm instead For one can take the *f	mathcal {L}_p\f$-norm
*We can view the lattice as a way of packing the space by spheres of *radius f ell_t with one sphere centered at each lattice point In *the dual this gives f	n
of cite mCON99a *for f $t le24 and can be approximated with f for f $t ge25 by *f *[*R(t)=\frac{t}{2} \log_2\left(\frac{t}{4\pi e}\right)+\frac{3}{2} \log_2(t) - \log_2 \left(\frac{e}{\sqrt{\pi}}\right)+\frac{5.25}{t+2.5}. *\f] Table &	nbsp
*We can view the lattice as a way of packing the space by spheres of *radius f ell_t with one sphere centered at each lattice point In *the dual this gives f we obtain *f an upper bound on f ell_t f can be obtained in terms *of an upper bound on f delta_t and vice versa Let f delta_t *f be the *largest possible value of f delta_t f for a cite mGRU87a It gives the *upper bound f ell_t or good approximations of is useful because it *allows us to normalize f ell_t f to a value between and by taking *f ell_t ell_t *m k f This is convenient for comparing values for *different values of f $t f and f $m k f Good values are close to and *bad values are close to **The Hermite constants are known exactly only for f $t le8 in which *case the densest lattice packings are attained by the *laminated *lattices *cite mCON99a Conway and Sloane cite and provide lower *and upper bounds on f delta_t *f for other values of f $t f The *largest value of f ell_t which we denote *by f	page
of the *lower bound over the upper bound on f ell_t for *f le t le48 f This ratio tends to decrease with f $t but not *monotonously **Computing the shortest vector in terms of the Euclidean norm is e for computational	reasons
*We can view the lattice as a way of packing the space by spheres of *radius f ell_t with one sphere centered at each lattice point In *the dual this gives f we obtain *f an upper bound on f ell_t f can be obtained in terms *of an upper bound on f delta_t and vice versa Let f delta_t *f be the *largest possible value of f delta_t f for a cite mGRU87a It gives the *upper bound f ell_t or good approximations of	them
*We can view the lattice as a way of packing the space by spheres of *radius f ell_t with one sphere centered at each lattice point In *the dual this gives f we obtain *f an upper bound on f ell_t f can be obtained in terms *of an upper bound on f delta_t and vice versa Let f delta_t *f be the *largest possible value of f delta_t f for a cite mGRU87a It gives the *upper bound f ell_t or good approximations of is useful because it *allows us to normalize f ell_t f to a value between and by taking *f ell_t ell_t *m k f This is convenient for comparing values for *different values of f $t f and f $m k f Good values are close to and *bad values are close to **The Hermite constants are known exactly only for f $t le8 in which *case the densest lattice packings are attained by the *laminated *lattices *cite mCON99a Conway and Sloane cite and provide lower *and upper bounds on f delta_t *f for other values of f $t f The *largest value of f ell_t which we denote *by f in terms of f delta *f The laminated which give the lower bound *f ell_t where the constants f lambda_t f are given in *cite are the *best constructions in dimensions	to

Function Documentation

$O()

of cite mCON99a* for f $t le24 and can be approximated with f $O

(

1/

t

)

f()

in cite rLEC99c gives the ratio* f

(

\gamma_t^{\mathrm{L}}/\gamma_t^{\mathrm{R}}

)

lattice()

* We can view the lattice as a way of packing the space by spheres of* radius f ell_t with one sphere centered at each lattice point In* the dual this gives f we obtain* f an upper bound on f ell_t f can be obtained in terms* of an upper bound on f delta_t and vice versa Let f delta_t* f be the* largest possible value of f delta_t f for a lattice	(	i.	e.,
		the densest *packing by non-overlapping spheres arranged in a	lattice
	)

le()

* We can view the lattice as a way of packing the space by spheres of* radius f ell_t with one sphere centered at each lattice point In* the dual this gives f we obtain* f an upper bound on f ell_t f can be obtained in terms* of an upper bound on f delta_t and vice versa Let f delta_t* f be the* largest possible value of f delta_t f for a cite mGRU87a It gives the* upper bound f ell_t le

(

\ell_t^ *

n

)

mCON88a()

* We can view the lattice as a way of packing the space by spheres of* radius f ell_t with one sphere centered at each lattice point In* the dual this gives f we obtain* f an upper bound on f ell_t f can be obtained in terms* of an upper bound on f delta_t and vice versa Let f delta_t* f be the* largest possible value of f delta_t f for a cite mGRU87a It gives the* upper bound f ell_t or good approximations of is useful because it* allows us to normalize f ell_t f to a value between and by taking* f ell_t ell_t* m k f This is convenient for comparing values for* different values of f $t f and f $m k f Good values are close to and* bad values are close to* * The Hermite constants are known exactly only for f $t le8 in which* case the densest lattice packings are attained by the* laminated* lattices* cite mCON99a Conway and Sloane cite and provide lower* and upper bounds on f delta_t* f for other values of f $t f The* largest value of f ell_t which we denote* by f in terms of f delta* f The laminated which give the lower bound* f ell_t where the constants f lambda_t f are given in* cite mCON88a	(	Table 6.	1,
		page	158
	)

mCON99a()

* We can view the lattice as a way of packing the space by spheres of* radius f ell_t with one sphere centered at each lattice point In* the dual this gives f we obtain* f an upper bound on f ell_t f can be obtained in terms* of an upper bound on f delta_t and vice versa Let f delta_t* f be the* largest possible value of f delta_t f for a cite mGRU87a It gives the* upper bound f ell_t or good approximations of is useful because it* allows us to normalize f ell_t f to a value between and by taking* f ell_t ell_t* m k f This is convenient for comparing values for* different values of f $t f and f $m k f Good values are close to and* bad values are close to* * The Hermite constants are known exactly only for f $t le8 in which* case the densest lattice packings are attained by the* laminated* lattices* cite mCON99a Conway and Sloane cite mCON99a

(

Table 1.

2

)

Variable Documentation

convenient

of the* lower bound over the upper bound on f ell_t for* f le t le48 f This ratio tends to decrease with f $t but not* monotonously* * Computing the shortest vector in terms of the Euclidean norm is* convenient

delta_t

* We can view the lattice as a way of packing the space by spheres of* radius f ell_t with one sphere centered at each lattice point In* the dual this gives f we obtain* f delta_t

Initial value:

= (\ell_t/2)^t/n\f$ spheres per unit volume. This number
 * \f$\delta_t\f$ is called the *center density* of the lattice. For a given
 * value of \f$n\f$

example

of the* lower bound over the upper bound on f ell_t for* f le t le48 f This ratio tends to decrease with f $t but not* monotonously* * Computing the shortest vector in terms of the Euclidean norm is e for computational but one can also use another* norm instead For example

f

f for Korobov lattices of dimension</center > f

g

of the* lower bound over the upper bound on f ell_t for* f le t le48 f This ratio tends to decrease with f $t but not* monotonously* * Computing the shortest vector in terms of the Euclidean norm is e g

gamma_t

*We can view the lattice as a way of packing the space by spheres of *radius f ell_t with one sphere centered at each lattice point In *the dual this gives f we obtain *f an upper bound on f ell_t f can be obtained in terms *of an upper bound on f delta_t and vice versa Let f delta_t *f be the *largest possible value of f delta_t f for a cite mGRU87a It gives the *upper bound f ell_t or good approximations of is useful because it *allows us to normalize f ell_t f to a value between and by taking *f ell_t ell_t *m k f This is convenient for comparing values for *different values of f $t f and f $m k f Good values are close to and *bad values are close to **The Hermite constants are known exactly only for f $t le8 in which *case the densest lattice packings are attained by the *laminated *lattices *cite mCON99a Conway and Sloane cite and provide lower *and upper bounds on f delta_t *f for other values of f $t f The *largest value of f ell_t which we denote *by f in terms of f delta *f The laminated which give the lower bound *f ell_t where the constants f lambda_t f are given in *cite are the *best constructions in dimensions except for dimensions to *One has f gamma_t

Initial value:

{\mathrm{B}}\f$. Such values are given in Table 1.2 of
 * \cite mCON99a

lattice

* We can view the lattice as a way of packing the space by spheres of* radius f ell_t with one sphere centered at each lattice point In* the dual lattice

lattices

* We can view the lattice as a way of packing the space by spheres of* radius f ell_t with one sphere centered at each lattice point In* the dual this gives f we obtain* f an upper bound on f ell_t f can be obtained in terms* of an upper bound on f delta_t and vice versa Let f delta_t* f be the* largest possible value of f delta_t f for a cite mGRU87a It gives the* upper bound f ell_t or good approximations of is useful because it* allows us to normalize f ell_t f to a value between and by taking* f ell_t ell_t* m k f This is convenient for comparing values for* different values of f $t f and f $m k f Good values are close to and* bad values are close to* * The Hermite constants are known exactly only for f $t le8 in which* case the densest lattice packings are attained by the* laminated* lattices* cite mCON99a Conway and Sloane cite and provide lower* and upper bounds on f delta_t* f for other values of f $t f The* largest value of f ell_t which we denote* by f in terms of f delta* f The laminated* lattices

mathcal

of the* lower bound over the upper bound on f ell_t for* f le t le48 f This ratio tends to decrease with f $t but not* monotonously* * Computing the shortest vector in terms of the Euclidean norm is e for computational but one can also use another* norm instead For one can take the* f mathcal {L}_p\f$-norm

n

*We can view the lattice as a way of packing the space by spheres of *radius f ell_t with one sphere centered at each lattice point In *the dual this gives f we obtain *f an upper bound on f ell_t f can be obtained in terms *of an upper bound on f delta_t and vice versa Let f delta_t *f be the *largest possible value of f delta_t f for a cite mGRU87a It gives the *upper bound f ell_t or good approximations of is useful because it *allows us to normalize f ell_t f to a value between and by taking *f ell_t ell_t *m k f This is convenient for comparing values for *different values of f $t f and f $m k f Good values are close to and *bad values are close to **The Hermite constants are known exactly only for f $t le8 in which *case the densest lattice packings are attained by the *laminated *lattices *cite mCON99a Conway and Sloane cite and provide lower *and upper bounds on f delta_t *f for other values of f $t f The *largest value of f ell_t which we denote *by f in terms of f delta *f The laminated which give the lower bound *f ell_t n

Initial value:

= m^{-k}\f$ spheres per unit volume.
 * If we rescale so that the radius of each sphere is 1

nbsp

of cite mCON99a* for f $t le24 and can be approximated with f for f $t ge25 by* f* [ * R(t) = \frac{t}{2} \log_2\left(\frac{t}{4\pi e}\right) + \frac{3}{2} \log_2 (t) - \log_2 \left(\frac{e}{\sqrt{\pi}}\right) + \frac{5.25}{t + 2.5}. * \f] Table& nbsp

page

* We can view the lattice as a way of packing the space by spheres of* radius f ell_t with one sphere centered at each lattice point In* the dual this gives f we obtain* f an upper bound on f ell_t f can be obtained in terms* of an upper bound on f delta_t and vice versa Let f delta_t* f be the* largest possible value of f delta_t f for a cite mGRU87a It gives the* upper bound f ell_t or good approximations of is useful because it* allows us to normalize f ell_t f to a value between and by taking* f ell_t ell_t* m k f This is convenient for comparing values for* different values of f $t f and f $m k f Good values are close to and* bad values are close to* * The Hermite constants are known exactly only for f $t le8 in which* case the densest lattice packings are attained by the* laminated* lattices* cite mCON99a Conway and Sloane cite and provide lower* and upper bounds on f delta_t* f for other values of f $t f The* largest value of f ell_t which we denote* by f page

reasons

of the* lower bound over the upper bound on f ell_t for* f le t le48 f This ratio tends to decrease with f $t but not* monotonously* * Computing the shortest vector in terms of the Euclidean norm is e for computational reasons

them

* We can view the lattice as a way of packing the space by spheres of* radius f ell_t with one sphere centered at each lattice point In* the dual this gives f we obtain* f an upper bound on f ell_t f can be obtained in terms* of an upper bound on f delta_t and vice versa Let f delta_t* f be the* largest possible value of f delta_t f for a cite mGRU87a It gives the* upper bound f ell_t or good approximations of them

to

* We can view the lattice as a way of packing the space by spheres of* radius f ell_t with one sphere centered at each lattice point In* the dual this gives f we obtain* f an upper bound on f ell_t f can be obtained in terms* of an upper bound on f delta_t and vice versa Let f delta_t* f be the* largest possible value of f delta_t f for a cite mGRU87a It gives the* upper bound f ell_t or good approximations of is useful because it* allows us to normalize f ell_t f to a value between and by taking* f ell_t ell_t* m k f This is convenient for comparing values for* different values of f $t f and f $m k f Good values are close to and* bad values are close to* * The Hermite constants are known exactly only for f $t le8 in which* case the densest lattice packings are attained by the* laminated* lattices* cite mCON99a Conway and Sloane cite and provide lower* and upper bounds on f delta_t* f for other values of f $t f The* largest value of f ell_t which we denote* by f in terms of f delta* f The laminated which give the lower bound* f ell_t where the constants f lambda_t f are given in* cite are the* best constructions in dimensions to