scc is an implementation of Voronoi's reduction theory
by
Frank Vallentin
and
Achill Schürmann.
It allows to cruise through the secondary cones of Delone triangulation
(also known as Ltype domains).
See the paper
"Computational approaches to lattice packing and covering problems"
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.
Its new version,
scc 2.0
by
Alexey Garber,
can also compute the list of all Delone decompositions, by cruising through all faces of fulldimensional secondary cones. The test whether or not two Delone decompositions are equivalent is now done by testing arithmetic equivalence of central reduced quadratic forms, using the program
ISOM by Bernd Souvignier.
As a consequence the program is much faster now and allows to
obtain a complete list of 110244 inequivalent Delone decompositions
in dimension 5.
See the paper
"The complete classification of fivedimensional DirichletVoronoi polyhedra of translational lattices"
for further information.
USAGE:
scc [options] < file
Options:
h show this help
d D work in dimension D (required)
a find all Delone decompositions
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
v write central form information for every cone (only if a option is chosen)
EXAMPLES:

scc d4
*** sc ***
*** c ***
Time: 19:37:47
Looking at #0
* 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)
Time: 19:37:47
Looking at #1
* 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)
Time: 19:37:47
Looking at #2
* 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)


scc d4 a
*** sc ***
*** c ***
Time: 19:28:39
Looking at #0
* 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)
Time: 19:28:39
Looking at #1
* 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)
Time: 19:28:39
Looking at #2
* 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 of Delone triangulations completed!
There are 3 nonequivalent Delone triangulations in dimension 4.
Starting classification of arbitrary Delone decompositions
There are 3 forms of codimension 0
Time: 19:28:39
There are 4 forms of codimension 1
Time: 19:28:39
There are 7 forms of codimension 2
Time: 19:28:39
There are 11 forms of codimension 3
Time: 19:28:39
There are 10 forms of codimension 4
Time: 19:28:39
There are 8 forms of codimension 5
Time: 19:28:39
There are 5 forms of codimension 6
Time: 19:28:39
There are 2 forms of codimension 7
Time: 19:28:39
There are 1 forms of codimension 8
Time: 19:28:39
There are 1 forms of codimension 9
Time: 19:28:39
Number of differrent decompositions: 52
*******

scc d6 r4 w
*** sc ***
*** c ***
Time: 19:38:56
Looking at #0 (normalized covering density >= 0.493668)
* There are 21 neighbours.
* Write output for COOP
* Proceed with number 1.
Time: 19:38:56
Looking at #1 (normalized covering density >= 0.499392)
* There are 21 neighbours.
* Write output for COOP
* Proceed with number 4.
Time: 19:38:56
Looking at #2 (normalized covering density >= 0.502569)
* There are 21 neighbours.
* Write output for COOP
* Proceed with number 9.
Time: 19:38:57
Looking at #3 (normalized covering density >= 0.508167)
* There are 21 neighbours.
* Write output for COOP
* Proceed with number 19.

scc d6 g i <sc6_3.coop
*** sc ***
*** c ***
Time: 19:51:13
Looking at #0 (normalized covering density >= 0.508167)
* There are 21 neighbours.
Time: 19:51:13
Looking at #1 (normalized covering density >= 0.502569)
* There are 21 neighbours.
Time: 19:51:14
Looking at #2 (normalized covering density >= 0.499392)
* There are 21 neighbours.
Time: 19:51:14
Looking at #3 (normalized covering density >= 0.493668)
* There are 21 neighbours.
INTERPRETATION:

Classifies all secondary cones in dimension 4.

Classifies all secondary cones in dimension 4.
After the classification of full dimensional secondary cones
(Delone triangulations)
their faces are classified as well
(giving all Delone decompositions).

Takes a random walk of length 4 from Voronoi's first type and writes
all found secondary cones into coopformat 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 GMP first. On a linux system this can usually be done by running
aptget install libgmp3dev.
Then you have to install
NTL (A Library for doing Number
Theory) of Victor Shoup; you should allow GMP support in NTL, so you should configure it with
NTL_GMP_LIP=on.
Then run
make of
scc.
You may have to edit the variable
CXXFlags in the
Makefile so that it finds NTL.
Also you might need to remove the
m64 flag from the
Makefile if you use non 64bit architecture OS.