User's Guide for scc (secondary cone cruiser)
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 the paper
"Computational approaches to lattice packing and covering problems"
(math.MG/0403272) by Achill Schürmann and Frank Vallentin
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.
The version 2.0 can also compute the list of all Delone decompositions, so it is also cruises through faces of full-dimensional secondary cones. The test whether two quadratic form are arithmetically equivalent is using ISOM software by Bernd Souvignier, so it is much faster now. These features were added by Alexey Garber.
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. If you have trouble to install our software, please feel free to contact us.
Also you might need to remove -m64 flag from the Makefile if you use non 64-bit architecture OS.
The version 2.0 can also compute the list of all Delone decompositions, so it is also cruises through faces of full-dimensional secondary cones. The test whether two quadratic form are arithmetically equivalent is using ISOM software by Bernd Souvignier, so it is much faster now. These features were added by Alexey Garber.
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) --
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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 non-equivalent 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 *******
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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.
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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 that classifies all Delone decompositions 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 GMP first. Usually you can do it by running apt-get install libgmp3-dev.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. If you have trouble to install our software, please feel free to contact us.
Also you might need to remove -m64 flag from the Makefile if you use non 64-bit architecture OS.