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 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.
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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.
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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.