scc is an implementation of Voronoi's reduction theory.
It allows to cruise through the secondary cones of Delone triangulation
(also known as L-type domains). See our paper
"Computational approaches to lattice packing and covering problems"
(
math.MG/0403272)
for a detailed description and applications to the
construction of lattices with extremal properties.
scc can in particular produce input files
for
coop
and
pacoop.
USAGE:
scc [options] < file
Options:
-h show this help
-d D work in dimension D (required)
-r N do a random walk of length N
-g greedy approach
-i read triangulation from file
-w write coop file for every secondary cone
EXAMPLES:
-
scc -d4
*** sc ***
*** c ***
Looking at #0 (normalized covering density >= 0.357771)
* Remaining: 0
* There are 10 neighbours.
* (0->1) (1=1) (2=1) (3=1) (4=1) (5=1) (6=1) (7=1) (8=1) (9=1)
Looking at #1 (normalized covering density >= 0.362039)
* Remaining: 0
* There are 10 neighbours.
* (0=1) (1=1) (2=1) (3=1) (4=1) (5=0) (6->2) (7=2) (8=2) (9=1)
Looking at #2 (normalized covering density >= 0.3584)
* Remaining: 0
* There are 10 neighbours.
* (0=1) (1=1) (2=1) (3=1) (4=1) (5=1) (6=1) (7=1) (8=2) (9=1)
--
Classification completed!
There are 3 non-equivalent Delone triangulations in dimension 4.
-
scc -d6 -r4 -w
*** sc ***
*** c ***
Looking at #0 (normalized covering density >= 0.493668)
* There are 21 neighbours.
* Write output for COOP
* Proceed with number 0.
Looking at #1 (normalized covering density >= 0.499392)
* There are 21 neighbours.
* Write output for COOP
* Proceed with number 11.
Looking at #2 (normalized covering density >= 0.503561)
* There are 21 neighbours.
* Write output for COOP
* Proceed with number 9.
Looking at #3 (normalized covering density >= 0.505423)
* There are 21 neighbours.
* Write output for COOP
* Proceed with number 14.
-
scc -d6 -g -i <sc6_3.coop
*** sc ***
*** c ***
Looking at #0 (normalized covering density >= 0.505423)
* There are 21 neighbours.
Looking at #1 (normalized covering density >= 0.503561)
* There are 21 neighbours.
Looking at #2 (normalized covering density >= 0.499392)
* There are 21 neighbours.
Looking at #3 (normalized covering density >= 0.493668)
* There are 21 neighbours.
INTERPRETATION:
-
Classifies all secondary cones in dimension 4.
-
Takes a random walk of length 4 from Voronoi's first type and writes
all found secondary cones into coop-format with file names
sc6_0.coop ... sc6_3.coop
-
Reads sc6_3.coop and takes a greedy approach to find a
local minimum of the lower bound.
INSTALLATION:
To install
scc you have to install
NTL (A Library for doing Number
Theory) of Victor Shoup before. Then run
make.
You may have to
edit the variable
CXXFlags in the
Makefile so that it finds NTL.
If you have trouble to install our software,
please feel free to contact us.