
Lattice packings. Assume the packing is a lattice packing defined by the generator matrix U. We can take r=1 so that and leading to independent of . The density of the lattice packing is . For example, for d=2 and with the generator matrix
the density is
which is known to be the highest density in two dimensions.
Packings with radius give the lattices . Take and the standard basis . We have
With and J=[0,1/2], we get
and the density is
corresponding to a center density .
These packings are lattice packings called .
In dimensions d=3,4,5, these packings are the packings with the
highest known density like for d=3, the density is
and
in four dimensions
.
Packings with radius .
With , and , we get
and a center density .
This density is larger than the density of for d;SPMgt;12. These packings were
the densest we found numerically for d=2,3,4,5 in the class of quasiperiodic
packings with .
Packings with radius r=2.
We have the problem to find integers p and
such that in the group , no sum gives zero
if . In other words, all
sums are different from zero modulo p.
We can build solutions by defining recursively a sequence with
the linear difference equation
, and define the vector
. In the
case or ,
or , these were the densest 2packings
we found in our class. For d=5, the 2packing
was denser than the 2packing determined by
.
This construction of families of packings with increasing dimension can be generalized for any r. The problem is to find the smallest p=p(d,r), such that there exist d numbers such that for any with , the equation in has no solution. The center density of the corresponding packing is then . We rephrase the result in the following proposition.
Remark.
For prime p, a packing in the proposition
defines a code in the vector space . In coding theory, one considers
however rather the packing problem with the
Hamming metric instead of the Euclidean metric.
We found some good packings in the special case, when r is an integer and .
Remarks.
1) Because of the periodicity of the packing, we have for even r
that pr/2 is a multiple of r and that for odd r,
p(r+1)/2 is a multiple of r.
Example: for r=89, , this gives
a packing with density 0.73386212.
2) The construction can be modified by taking for example
with suitable p',
which gives for some r denser packings than .
Fig. 5. Packing densities for special packings in the case d=2,d=3. For each r, we plot the density obtained by taking , where p=p(r) is the integer described in corollary 4.2. The densities seem to accumulate at the maximal known density in two dimensions and to in three dimensions. 
Some good packings were obtained by taking a good solution
and choosing so that
the packing has maximal density.
This is motivated (evenso there is no direct relation) by the laminated
lattice construction (see [3]), which also constructs ddimensional
packings by building up layers of (d1)dimensional packings which
are known to be dense.
We never found a denser packing while using two intervals. The search is also algorithmically more expensive, since we have to order the set .
Table 1. Some examples of packings in three dimension. The packing with
radius is the Kepler packing. The packing with
radius 120 gives a slightly denser packing as the packing reported
in [17] using Penrose tilings which have densities accumulating
by 0.7341.
Table 2. Some examples of packings in four dimension. The first packing is
the packing which is believed to be the densest.
Table 3. Some examples of packings in five dimension. The first packing is the packing which is believed to be the densest.