The Lattice dim56min6
An entry from the Catalogue of Lattices, which is a joint project of
Gabriele Nebe, RWTH Aachen University
(nebe@math.rwth-aachen.de)
and
Neil J. A. Sloane
(njasloane@gmail.com)
Last modified Fri Jul 18 13:15:04 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
DET
GRAM (in Magma)
MINIMAL_NORM
PROPERTIES
REFERENCES
NOTES
LAST_LINE
-
NAME
dim56min6
-
DIMENSION
56
-
DET
1
-
GRAM (in Magma)
M56:=KMatrixSpace(Rationals(),56, 56) ! [ 6, 2, 1, 1, 1, 0, 0, 1, 1, 1,
0, 2, 1, 2, 1, 1, 0, 3, 0, -3, 3, -1, -1, -2, -1, 0, -2, -3, 0, -2, 0, -1, -1,
-3, -3, -3, 3, -3, -3, 3, 2, 2, 0, 2, 0, 2, 2, 3, 1, 0, 1, 3, -1, 0, -2, 1, 2,
6, 1, 1, 0, 0, 0, 2, 0, 1, 3, -1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 0, -1, 1, 1, 0,
0, -1, 0, 0, 1, 0, -1, 0, -1, 0, -1, 1, 0, 0, 1, 3, 0, -1, 0, 0, 1, 1, 0, 0, 0,
1, 1, 1, 1, 1, 1, 6, 1, 1, 2, 1, 2, 0, 1, 1, 0, 0, 0, 1, 1, 2, 0, 1, 1, -1, 0,
1, 0, 0, 0, 0, 2, -1, 0, 1, 1, 1, 0, 2, 1, 0, 1, 2, -1, 1, 2, 0, 1, 1, 0, 1, -1,
1, 1, 0, 0, 2, 1, 0, 3, 1, 1, 1, 6, 1, 3, 1, -1, 0, 1, 2, 0, 2, -1, -1, 0, 0, 0,
1, 1, -1, 0, 0, 1, 1, 0, 2, 1, -1, 0, 1, -1, 1, 2, 0, 1, -2, 0, 1, 0, 2, 1, 1,
-1, 1, 0, -1, 0, 1, 0, 2, 1, -1, 0, 0, 1, 1, 0, 1, 1, 6, 1, 1, 1, 0, -1, 1, 0,
-1, 0, 3, 1, 1, 1, 0, 0, -1, -1, 1, 1, 0, -1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0,
1, 1, -1, 1, 0, 0, 0, 0, 1, 1, 1, -1, 0, 1, -1, 0, 0, 1, 2, 0, 0, 2, 3, 1, 6, 1,
0, 1, 2, 2, -1, 1, -1, 1, -1, 0, -1, 0, 2, 0, -1, 2, 3, 2, 0, 2, 2, -1, 0, 0,
-1, 0, 2, 1, 1, -2, 1, 1, 0, 0, -1, 1, -1, -1, -1, 0, -2, -1, -1, 0, 1, 0, 0, 1,
2, 0, 0, 1, 1, 1, 1, 6, 3, 1, 2, 1, -1, 0, 1, 1, 0, 0, -2, 0, 2, 0, 1, 1, 1, 1,
1, 3, 0, -1, 1, -1, -1, 0, 2, 1, 1, -2, 1, 2, -2, 2, 0, 0, -1, 0, -1, 1, -3, 0,
-2, 0, -1, -1, 0, 2, -1, 1, 2, 2, -1, 1, 0, 3, 6, 0, 3, 1, -2, 0, 1, 2, 0, 1,
-1, -1, 1, 1, 1, 0, -1, 0, 1, 1, -1, -1, 0, -1, -1, 0, 0, 1, -1, 0, 1, 1, -1, 0,
0, 0, 1, -1, 0, 0, -2, -1, -2, -1, -1, 0, -1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 6,
0, 0, 0, 0, 0, 1, -1, 0, 0, -1, 1, 1, 0, 0, 1, 1, -1, 0, 0, 1, 0, 0, 0, -1, 1,
1, 0, -1, 0, -1, -1, -1, -1, 2, 1, 1, 0, -1, -1, -2, 0, 0, 0, 1, 1, 2, 0, 1, 1,
1, 1, -1, 2, 2, 3, 0, 6, 0, -1, 1, 1, 0, -1, -1, -2, -1, 1, 1, 0, 0, 0, 0, 2, 2,
0, -1, 1, -2, -2, 0, 0, 0, -1, -1, 1, -1, 0, 0, 1, 1, 0, -1, 0, -1, -2, -1, -1,
-1, 0, -1, -2, -1, -1, 0, 3, 1, 2, 1, 2, 1, 1, 0, 0, 6, -2, 2, 0, 1, 1, 2, 1, 2,
2, 0, 1, 1, 1, 2, 1, 2, 1, -1, -1, 1, -1, -1, 1, 1, 1, -2, 1, 2, 0, 0, 0, 2, -2,
-1, 1, 0, -1, 0, -1, -1, 0, 0, 1, 1, 1, 2, -1, 0, 0, 0, -1, -1, -2, 0, -1, -2,
6, 0, 0, -1, 0, 1, 2, 1, -2, 0, 0, 0, 0, 1, -1, -2, -1, 1, 0, 1, 1, 0, -1, -1,
0, 2, -2, -2, 2, 2, 2, -1, 1, 3, 1, 1, 3, 2, 2, 1, 1, -2, 1, -1, 0, 1, 1, 0, 2,
-1, 1, 0, 0, 0, 1, 2, 0, 6, -1, -1, -1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 2, 1, 0,
1, 0, -1, 0, 1, 0, 0, -1, -1, -1, 1, 0, 1, -1, 0, 0, 1, 0, 0, 2, 0, 0, -1, -2,
-1, 0, 2, 2, 1, 0, -1, 0, -1, 1, 1, 0, 1, 0, 0, -1, 6, 2, 1, 0, 1, 1, -1, 1, 0,
-2, -1, -2, 0, 0, -2, 0, -1, -2, 0, -2, -2, -2, -2, 2, 0, -1, 0, 1, 1, 1, 0, -2,
1, 3, 0, -1, 0, -1, 2, -1, 0, 0, -2, 1, 1, 1, -1, 3, 1, 1, 2, 1, 0, 1, -1, -1,
2, 6, 1, 2, 2, 1, 0, 0, -2, 0, 0, -1, 0, 1, -1, 0, -1, 0, 0, -2, 0, 1, -1, 1, 1,
0, 0, -1, -1, 1, 0, -1, 0, 1, -1, -2, -1, 0, 1, 0, 0, 2, 0, 1, 1, 1, 0, 1, -1,
0, 0, -1, -1, 1, 0, -1, 1, 1, 6, 1, 1, 1, 0, 0, 0, 0, -1, -1, -1, 0, -1, 1, -2,
-1, 0, -1, 0, -1, 1, 1, 1, 1, 1, 3, 1, 0, 1, 1, 2, 1, 1, 2, 0, 1, 0, 2, 1, -1,
0, 0, 2, 2, 0, 1, 0, 0, 1, 0, -1, 2, 1, 1, 0, 2, 1, 6, 3, 3, 1, -1, 0, -1, 0, 1,
1, 0, 1, 0, 1, 2, 2, -1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, -1,
0, 0, 3, 2, 1, 3, 2, 0, 0, 1, -1, -2, -1, 0, -2, 1, 2, 0, 1, 2, 1, 3, 6, 3, -2,
2, -1, -2, -2, 0, 0, -2, -2, 0, -2, 3, 1, -2, -3, -2, -3, 3, -3, -2, 3, 0, 0, 1,
0, 0, 1, 0, 3, 0, -1, 0, 3, -1, 3, 0, -1, 0, 2, 1, 1, 0, 0, 0, -1, -1, -1, 2, 1,
1, 1, 1, 1, 3, 3, 6, 1, -1, 0, -1, 0, 1, 0, 1, 1, 0, 0, 2, 2, -1, 0, 1, 0, 0, 0,
1, 0, 1, 0, 1, -2, 1, -1, 0, 1, 1, 0, 0, 0, -1, 2, 2, -1, -3, 1, 1, 1, 0, 2, 2,
1, 1, 1, 2, -2, 0, -1, 0, 0, 1, -2, 1, 6, 0, 3, 2, 3, 2, 1, 3, 3, 0, 2, 0, 1, 1,
3, 3, 3, -3, 3, 3, -3, 1, -2, 2, 0, 0, -2, -2, -3, -2, -1, -2, -3, 3, 2, 3, 0,
3, 1, -1, -1, -1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, -1, 2, -1, 0, 6, 1, 0, -1, 0,
1, -1, -3, 0, -2, 0, -1, -1, -3, -3, -3, 3, -3, -3, 3, 1, -2, 0, 2, -2, 0, 0, 1,
-2, -3, -2, 2, 1, 2, -1, -1, -1, 1, 0, 0, -1, -1, 1, 1, 0, 0, 1, 0, 1, 0, -2, 0,
0, -1, 0, 3, 1, 6, 0, 0, 1, 1, 1, 0, -1, 1, 0, 0, 1, 0, 1, 2, -1, 0, 1, -1, 1,
-1, 1, 2, 0, 1, 0, 0, -1, 0, -1, -2, 1, 2, 1, 0, -1, 0, 1, 0, 1, 2, 1, 0, 0, 0,
1, 0, 0, -2, 0, 0, -1, -2, -1, 2, 0, 0, 6, 3, 2, 0, 1, 2, 0, 1, -1, 0, 2, 2, 2,
3, -1, 1, 2, -1, 1, 0, -1, 0, 0, -1, 1, -1, 1, 0, -1, -2, 2, -1, 1, 3, -2, -1,
0, 1, 1, 3, 1, -1, 1, 0, 1, 0, 0, -1, 0, -1, 0, -2, 0, 3, -1, 0, 3, 6, 3, -1, 1,
3, 1, 2, 0, 1, 1, 3, 1, 3, -2, 2, 2, -2, 0, -1, 1, 0, -1, -2, 1, -1, 0, 0, -2,
-2, 0, 0, 2, 1, -1, 1, 0, 1, 0, 2, 1, 0, 1, 0, 2, 1, 1, -2, -1, -1, 1, 0, 1, 2,
0, 1, 2, 3, 6, 0, 0, 2, 0, 1, 1, 1, 1, 2, 1, 1, -2, 0, 2, 0, 0, -1, 1, -2, 0,
-1, -1, -1, 1, -1, 0, -1, -1, 1, 2, 1, 0, 1, 0, 0, -1, 0, 1, 1, -1, 2, 1, -1, 2,
0, 0, -1, 1, 0, 0, 1, 1, 1, 0, -1, 0, 6, 1, 1, -1, 2, 0, -1, 1, -1, 0, -1, -1,
0, -1, 1, 0, 1, 0, -1, -1, 0, -1, -1, 0, 0, -1, 0, 0, 0, -2, 0, -2, 0, 0, 2, 0,
2, 3, 1, 0, 2, 2, -2, 2, 0, 1, 0, 0, -2, 1, 3, -1, 1, 1, 1, 0, 1, 6, 1, -1, 1,
-1, -2, -1, 3, 2, 2, -3, 2, 1, -2, 0, -1, 1, -1, 0, 0, -1, -3, -1, -2, -1, -2,
-1, -1, 2, -2, -3, 0, 2, 1, 0, 2, 0, -1, 0, 0, 1, -1, 1, -2, -1, -1, 1, -2, 1,
3, -3, 0, 2, 3, 2, 1, 1, 6, 1, 3, 1, 2, 2, 3, 3, 3, -3, 3, 3, -3, 0, 1, 0, -2,
0, -2, -1, -2, 1, 2, -1, -3, 2, 0, 1, 2, 0, -1, -1, -1, 1, -1, -1, -1, 1, -1,
-1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0, -1, -1, 1, 6, 0, 1, 1, -1, 1, -1,
1, 1, 1, -1, -1, 1, 0, -1, 1, 1, 1, 0, 2, 0, 1, 0, -2, 1, 1, 0, 0, -2, 0, 0, 0,
0, 0, 1, 0, 0, 1, -1, 0, 1, -1, -1, -2, 1, -2, 0, 2, -2, 1, 1, 2, 1, 2, 1, 3, 0,
6, 0, 2, 2, 1, 2, 2, -3, 1, 1, -3, -1, 1, 1, 0, -1, -1, 0, -1, 0, 1, -1, -3, 0,
0, 1, 1, 0, 0, 1, 1, 1, 0, -1, -1, 0, -2, 1, 1, 0, -2, 0, -1, 2, 3, 2, 0, 0, 0,
-1, 0, 1, 0, -1, 1, 1, 0, 6, 2, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, 1, 0, 0, 0, -1,
2, -1, -1, 0, 0, -1, 3, 0, 0, -1, 1, 1, -1, 0, -1, -1, -1, 0, -2, -1, 1, -1, 0,
0, 0, 2, 1, 2, 1, -1, 0, 0, 1, 1, -1, -2, 2, 1, 2, 2, 6, 2, 0, 1, 0, 0, 0, 2,
-2, 0, 0, 1, 0, -1, -3, 0, 0, 0, 0, 0, -1, 1, 2, 2, 1, -1, 0, 1, 1, 0, 0, 0, 0,
-1, 0, -1, 0, 0, -2, -2, -1, -1, -2, -1, 1, -1, 1, 2, 1, 1, 1, -1, 2, -1, 2, 0,
2, 6, 1, 2, 1, -1, 0, 2, -1, 0, 0, -1, 0, 0, -3, -1, -1, 1, 1, 1, -1, 1, -2, 0,
3, -3, -1, 0, 2, 1, 2, 2, 0, 1, 0, 1, -1, 1, -2, 0, 0, 0, -3, 0, 3, -3, 0, 2, 3,
2, -1, 3, 3, 1, 1, -1, 0, 1, 6, 3, 3, -3, 3, 3, -3, 1, -1, 0, -2, 2, -2, -2, -3,
1, 0, 1, -3, 0, -1, 3, 1, -3, 0, 2, 0, 1, 1, 1, 1, 1, 0, 1, -1, 0, -2, 1, -1, 1,
-2, 1, 3, -3, 1, 2, 1, 1, 0, 2, 3, -1, 2, 0, 1, 2, 3, 6, 3, -3, 3, 3, -3, -1,
-1, 0, -1, 2, -2, -2, -3, -1, 1, 0, -3, 2, -1, 3, 2, -3, -1, 1, 1, 1, 1, 1, -1,
0, -1, 1, 0, 0, -2, -1, 1, 0, -3, 0, 3, -3, 2, 3, 3, 1, -1, 2, 3, 1, 2, 0, 0, 1,
3, 3, 6, -3, 3, 3, -3, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 0, -4, 2, 0, 1, 2, 3, 0, 0,
-2, 0, -2, -2, 0, -1, -1, -2, 2, -1, 2, 1, 1, 0, 3, 0, -3, 3, -1, -1, -2, -2,
-1, -3, -3, 1, -3, 0, 0, -1, -3, -3, -3, 6, -3, -3, 3, 1, 0, -2, 2, 0, 0, 2, 3,
1, 0, -1, 2, 0, 1, -1, 0, -3, -1, 1, 0, 1, 1, 1, 1, 0, 1, 1, -2, -1, 0, 1, 1, 0,
-3, 0, 3, -3, 0, 1, 2, 0, 0, 2, 3, 1, 1, -1, 0, 0, 3, 3, 3, -3, 6, 3, -3, 1, 1,
0, -1, 1, 0, -1, -3, -1, 1, -1, -3, 2, -2, 1, 0, -3, 1, 2, 1, 1, 1, 2, 1, -1,
-1, 2, -2, -1, -1, 0, 1, 1, -2, 1, 3, -3, 1, 2, 2, 2, -1, 1, 3, -1, 1, 0, 2, 2,
3, 3, 3, -3, 3, 6, -3, 1, 0, 0, -2, 0, -2, 0, -3, 1, 0, 1, -2, 2, 0, 3, 2, 3, 0,
-1, 0, -1, 0, -2, -1, -1, 0, 0, 2, 1, 0, 0, 1, 0, 3, 0, -3, 3, -1, -1, -2, 0, 1,
-2, -3, -1, -3, 0, -2, -1, -3, -3, -3, 3, -3, -3, 6, 1, 0, -2, 0, 0, 1, 0, 2, 1,
0, 1, 4, -1, 0, -3, 0, 2, 0, 1, 2, 1, 0, 2, 0, -1, 0, 0, 2, 0, 1, -1, 3, 0, 0,
1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, -1, -1, 0, 0, 1, -1, 1, 1, 1, 1, 1, 8, 2, -3,
0, 2, 0, 1, 1, 2, 1, 1, 0, 2, 1, -1, 1, 2, 1, 2, 1, 0, -1, 0, 0, -1, 1, 0, 2, 1,
1, -1, 1, 1, 0, 0, -2, -2, -1, 0, -1, -1, 1, -1, 1, 0, 1, 0, 0, 0, -1, -1, 1, 0,
1, 0, 0, 2, 8, 0, 0, 2, 3, 2, 2, 4, 3, 1, -1, -1, -2, -3, 2, 0, 3, 0, 1, 0, 1,
0, 0, 2, 1, 2, -1, -1, 1, 1, 0, 1, 1, 1, 2, 0, 1, -1, 1, 1, 0, 1, 0, -1, 1, 1,
1, -1, 0, 0, 0, -2, 0, 0, -2, -3, 0, 8, 0, -1, 1, -1, 0, -2, -1, 0, 0, -1, 2, 1,
-2, 2, 0, 1, -1, 0, -1, -1, 1, 1, 0, -2, 1, 0, 0, 0, 1, 0, 0, -2, 0, 2, 2, 0, 0,
-2, -1, -1, -2, 1, 0, 0, 0, 0, -2, -1, 1, 2, -1, -2, 0, 0, 0, 0, 8, 1, 2, 2, 3,
0, 0, -2, -1, 2, 1, -1, 1, 0, -1, 1, 1, 0, -1, 0, -1, 1, -1, -1, 3, 0, -2, -1,
1, 1, 0, 1, 0, -2, 0, 0, -1, 0, -1, 0, 0, 1, -1, 0, -1, 0, 2, 2, 2, 0, 1, 0, 0,
2, 2, -1, 1, 8, 1, -2, 1, 3, 3, 2, -1, 1, 0, 0, 0, 2, 0, 0, 0, 1, -1, -1, 0, 0,
0, 1, 1, 1, 1, 0, 2, 0, 1, -1, -2, 0, 1, -1, -2, -1, 0, 0, -2, 1, -1, 0, -3, -3,
-2, -2, 1, 0, 0, -2, 1, 0, 3, 1, 2, 1, 8, 2, 3, 0, 1, 1, 0, -1, 0, -3, 0, 2, 0,
1, -1, 1, 0, 1, 0, -1, -1, 0, 1, 0, 3, 1, 1, 0, 0, 0, -2, 0, 0, 1, 1, -1, -1,
-1, -1, 0, 0, -1, 0, -1, -2, -2, 1, 2, -1, 0, 0, 1, 2, -1, 2, -2, 2, 8, 2, 2, 1,
-1, 0, -1, 0, 0, 2, 3, 1, -1, 0, 1, -2, -3, -2, -1, -2, -1, 3, 0, 0, -1, 1, 0,
3, 1, -3, 1, 0, -1, -1, -1, -1, -3, -2, 2, -1, 2, 0, -1, -3, -3, 0, 3, -3, -3,
2, 1, 2, 0, 3, 1, 3, 2, 8, 2, 2, 1, 0, -1, 1, -3, 1, 1, 1, 1, 1, -1, -1, 0, -1,
-2, -1, 0, 2, 2, -1, -2, 2, 1, 0, 1, -2, -2, -1, 1, 0, 1, 0, -1, 1, 0, 0, -1, 0,
1, 1, -1, 1, 1, -1, 1, 1, 2, 4, -2, 0, 3, 0, 2, 2, 8, 2, 1, -1, -1, -1, -1, 2,
0, 0, 1, 0, 0, -1, -2, -2, 0, -1, -1, 2, 0, 0, -1, 0, 0, -1, 0, -1, -3, 0, 0, 0,
-1, 0, -2, 2, 1, 1, -1, 0, 1, 0, 1, 2, 0, 1, 0, 0, 1, 3, -1, 0, 3, 1, 1, 2, 2,
8, 2, 0, 2, -1, -2, 2, 1, 0, 0, 2, 1, 0, 0, -1, 0, -1, -1, 1, 0, -1, 0, 1, -1,
0, 0, -2, -2, -1, -1, -2, 0, -1, -1, -1, 0, -1, 0, 0, 1, 1, 0, 0, -1, -1, 1, 1,
1, 1, 0, -2, 2, 1, -1, 1, 1, 2, 8, 2, -1, -1, -1, 2, 3, 0, 0, 1, -1, 1, -1, -1,
0, 0, 0, 1, -1, 2, 1, 0, 0, 3, 0, -3, 2, -2, -2, -2, -1, 0, -2, -3, -2, -3, 0,
-1, -1, -3, -3, -4, 2, -3, -2, 4, 0, -1, 0, -1, -1, 0, 0, 0, -1, 0, 2, 8, -1, 1,
-2, -2, -1, 1, 2, -1, 0, 0, -1, 0, 1, -1, 0, -2, -2, -1, 0, 2, 0, -1, -1, 3, 1,
1, 2, 0, -1, 0, -1, 2, 1, 0, -1, 1, 1, 0, 2, 2, 0, 2, 2, -1, 2, -1, -1, 2, 1,
-1, -1, -1, -1, 2, -1, -1, 8, 2, 0, 2, 0, 1, 1, 0, 0, 0, 0, -1, 1, -2, 1, 1, -1,
0, 0, 1, 3, 3, 2, 2, 2, 2, -1, 0, 1, 0, -1, 0, 1, 0, 3, 2, -2, -1, -1, 0, 1, -2,
0, 0, 1, -2, 2, 1, 0, 0, 0, 1, -1, -1, -1, 1, 2, 8, 1, -1, -2, 1, 0, 0, 1, 1, 2,
1, 2, -1, 1, -1, 0, 0, 2, -1, 2, 0, 2, 3, -1, 1, 1, 2, 2, -2, 2, 1, 0, 1, 0, 2,
0, 3, 3, 1, -1, 1, 3, -3, -1, -3, 1, -1, 0, -3, 0, -3, -1, -2, -1, -2, 0, 1, 8,
0, 1, 1, 3, 1, 2, 2, -1, 1, 0, -1, 1, 0, 2, -2, 0, 0, 1, -1, -1, 0, -1, 0, 3, 1,
1, 0, -2, 2, 0, 1, 0, 1, 3, 1, 2, 2, 0, 0, 2, 0, 1, 2, -2, 1, 0, 0, 2, 1, 2, 2,
2, -2, 2, -1, 0, 10 ];
-
MINIMAL_NORM
6
-
PROPERTIES
UNIMODULAR = 1
EXTREMAL = 1
-
REFERENCES
Philippe Gaborit:
Construction of new extremal unimodular lattices (Preprint)
-
NOTES
Construction desccribed by Ozeki but starting from
a [56,28,15] ternary self-dual code rather than by subtracting from XQ59 or PS60.
-
LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe