The Lattice B45
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 13:10:49 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
DET
BASIS (in Magma)
GRAM (in Magma)
MINIMAL_NORM
PROPERTIES
COMMENTS
LAST_LINE
-
NAME
B45
-
DIMENSION
45
-
DET
1
-
BASIS (in Magma)
B45:=LatticeWithBasis(45, \[ IntegerRing() | 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 1, -3, 5, 0, 3, 1, -1, 0, -1, 0, -1, 2, 0,
-1, -1, 1, -1, -1, 1, 0, -1, 0, -1, 1, -1, -1, -1, -1, -1, 0, -2, 0, 1, 0, 0, -1, 0, 0, -1, -1, 1, 1, 0, 0, 0, 0, 1, -1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2,
0, 0, 2, -1, 1, 0, -1, 0, -3, -1, 1, 0, -1, -1, 0, 1, -1, 0, -1, -1, 0, 0, 0, 0, -2, 0, 1, 0, 0, 0, -1, 0, 0, 0, 1, 1, 0,
-1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, -1, 2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1,
0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0,
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, -3, -2, 0, 1, 0, -1, 1, -2, 0, -4, 0, 3, 1, -1, -1, 0, 0, 0, -1, 0, 0, 0, -1, 1, 2, -2, 0, 1, 1, 0, 0,
-2, 0, 0, 0, 1, 2, -1, -1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0,
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, -1, 0, 1, -1, 2, 1, 3, 0, -2, -1, 1, 1, 0, 1, 0, 1,
0, 0, 0, 0, 0, -1, 2, 0, -1, -1, 0, 0, 2, 0, 1, 0, -1, -2, 1, 1, 0, 0, 0, -1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0,
1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0,
-1, -1, -2, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0,
1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, -3, 4, 0, 2, 1, -5, 2, 1, -1, -4, 3, 0, 5, -3, 1, -2, -1, -1, -1, 2, 0,
-3, 0, 0, -2, -2, -1, -1, 0, -1, 0, 1, -1, -1, -2, 0, -1, 2, 0, 2, 1, 0, 0, 0, 0, 0, -1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 0, -1, 0, -2, -1, 1, 1, 2, 2,
0, 0, -1, 2, 1, 3, -1, 0, 0, 0, -2, 1, 2, -2, 1, 0, -1, -1, -2, 0, 1, 0, 1, 0, 0, -2, 0, -1, 0, -1, 2, -2, 1, 0, 0, 0, 0,
0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, -2, 0, 0, 1, -1, 1, 0, -1, 1, -2, 0, 1, 0, -1, 0, 0, 1, -1, 0, -1, -1, 0, -1, 1, 0, -1, 0, 1, 1, 0, 0, -1, 0, 0, 1, 1,
1, 0, -1, 0, 0, 1, 1, 0, 1, 3, 0, 0, -1, 0, 1, -1, 1, 0, 2, 0, -1, 1, 1, 1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, -1, 0, 0,
0, 1, 0, 0, 0, -1, -1, 0, 1, 0, 0, -1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 1, 1, 1, 0, -1, 0, 0, -3, -1, 0, 1, -2, 0, -2, 0,
0, 0, -1, 2, 0, -3, 2, -1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 2, -1, -1, 4, -1, -1, -2, 1, -2, -1, 0, -2, 5,
0, 0, 1, 2, -2, 0, -1, 1, 1, 1, 1, 0, 0, 1, 0, 4, 0, -2, 0, 0, 1, 2, 0, 0, 0, -1, -1, 0, 2, 0, 0, -2, -1, 0, -2, -7, 0, 0,
1, 0, -4, 1, 0, 3, -3, 2, 2, -1, -2, 2, 0, 4, -1, -1, -1, 0, -2, 0, 3, -1, -3, 0, 1, 0, -2, -1, -2, 0, 0, 0, 1, 0, -1, -2,
1, -1, 2, 0, 1, 3, 1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 2, 0, -1, -1, 0, 2, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, -1, -1, 0,
-1, 1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, 1, 0, 0, -1, 1, 1, -1, 3, 1, 0, 1, -4, -1, 0, 0, -1, 0, 0, 0, -1, 0, -1, -1, 1, 0,
-1, 1, -3, 0, 1, -1, 0, -1, -1, 0, -1, 0, 1, 1, 0, -1, 0, 0, 1, 1, 0, 1, 5, 1, 0, 0, 0, 3, 0, 2, 1, 1, -1, -3, -1, 1, 1,
0, -1, 0, 1, 0, -1, 1, 0, -2, 0, 1, 0, 0, -1, 1, 0, 2, 0, 0, 0, -1, -1, 1, 1, 0, 0, 0, -1, -1, -1, -3, 1, 0, -1, 1, -2, 1,
1, 2, 0, 2, -1, 0, 0, 4, 0, 0, 0, -1, 0, 0, -1, 0, 1, 0, -1, 0, 1, 1, 0, -1, 1, -1, 1, 0, -1, -2, 0, 0, 1, -1, 2, -2, -1,
1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, -2, -2, 0, 2, -1, 0, -2, 1, 0, 0, 0, 1, 0, -2, -1, 0, 0, 1, 0, 1, 0, 1, 0,
1, 0, -1, 0, 1, 1, 0, 0, 0, 0, -1, 0, 4, -1, 0, -1, 0, 1, 0, 1, -1, 3, 0, -1, 0, 1, 0, 0, -2, 1, 1, 1, 0, 1, 0, -1, 1, 2,
0, -2, -1, 0, 1, 1, 0, -1, 0, 0, 0, 0, 1, 0, 0, -2, -1, 0, -1, 1, 1, 0, -1, 1, 0, 0, 1, -1, 3, -1, -3, 0, 2, 1, 0, -2, 1,
0, 1, 0, 1, 1, -2, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, -1, -1, 1, 1, 0, 0, 0, -1, -1, -1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,
0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -1, 0,
0, -1, 1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1,
1, 0, 0, 2, 0, 1, 1, 2, -2, -4, -1, 2, 2, 0, -1, 0, 0, 0, 0, 1, 2, -2, -2, -1, 0, 1, -1, 1, -1, 1, 0, 1, 0, -1, -1, 1, 0,
0, 0, 1, -1, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 1, 0, 1, -1, 0, 1, 1, 4, -5, 1, -1, 0, -1, 4, 0, 2, -2, -1, -2, -1, -1, 1, 1, -1, -4,
0, 3, 0, 0, -1, 0, -1, 1, 1, 0, -1, 1, -2, 1, -1, 5, -1, -1, 3, 5, 0, 0, 0, 0, 5, 0, 0, -1, 1, -2, -2, -1, 1, 0, 0, -3, 2,
1, 1, 0, 2, -1, -3, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, -1, -1, 0, 1, 2, -1, 1, -2, 0, -1, -2 ], MatrixRing(IntegerRing(), 45) !
\[ IntegerRing() | 4, 0, 0, 1, 0, -2, 1, -1, -2, -1, 1, -1, 1, 0, 0, 0, 1, 1, -1, 0, -1, 1, 0, 0, -2, -1, 0, 1, 2, -1, -2,
0, 1, 2, 1, 1, 1, 1, -2, 2, 1, -1, -2, 1, 2, 0, 4, 0, -1, 0, -1, 0, 0, 0, -2, 0, 0, 0, 0, -1, -1, -2, 2, 2, 0, -1, 2, 1,
1, -1, 0, 0, -1, 0, -1, 0, -2, 1, 0, 0, 0, 0, -1, 1, 1, -1, 1, 0, 0, -1, 0, 0, 5, 0, 1, -1, -1, 0, -1, -2, 0, 0, -1, 0, 0,
0, -2, -1, 1, 0, 0, 0, -2, 1, -2, 1, 0, -1, -1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, -1, 0, 0, -1, 0, 1, -1, 0, 5, 0, 0, 0, 0,
1, 1, 2, 0, 2, 0, 0, 1, 1, 0, -1, 2, 1, 1, 1, 0, 0, -1, -2, 1, 1, -1, 0, 2, 0, -1, 0, 0, -1, 1, 0, 0, -1, -1, 1, 0, 0, 0,
0, 1, 0, 5, 2, -1, 2, -1, -1, -1, 1, -1, 2, 0, 0, 0, -1, -1, 1, 1, -1, -1, 2, -1, 0, 0, -1, 0, 1, 0, 0, -1, 0, 1, 1, 1, 0,
-2, 0, 1, 0, -1, 0, 0, -2, -1, -1, 0, 2, 5, 0, 2, 2, 2, 0, 0, -1, 2, -2, 2, 1, 0, -1, 2, 2, -2, 1, 2, 3, 0, -1, -1, 1, 2,
2, 0, -1, 0, -2, 0, 0, -2, -1, -2, 1, 2, 0, 1, 1, 1, 0, -1, 0, -1, 0, 5, -2, 0, 1, -1, 0, 1, -1, 1, 0, 2, 1, -1, -2, 0, 1,
2, -1, 1, 2, 0, 1, 0, -1, 1, -1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, -1, 2, 1, -1, 0, 0, 0, 2, 2, -2, 5, -1, 0, 0, 2, -1, 2,
-1, 1, 0, 1, 1, 2, 0, -1, 0, 3, 0, -1, -2, 0, 0, 3, -1, 0, 1, 1, 0, 1, 1, -1, -1, 0, 0, 0, 0, -2, 0, -2, 0, -1, 1, -1, 2,
0, -1, 5, 1, 1, -1, 1, 0, -2, 0, 0, 0, -1, 1, 0, -1, 2, -1, 2, -1, -1, -1, 1, -1, 3, 1, -1, -2, -2, -1, -2, -2, 2, -3, -1,
0, 1, 1, -2, -1, -2, -2, 1, -1, 2, 1, 0, 1, 5, 1, 0, 1, -1, 0, 2, 2, -1, 0, 1, 1, -1, 0, 0, 3, -1, 0, 2, 0, 2, 1, 1, 0,
-2, -3, -2, 0, 0, -1, -2, 2, 1, 1, 0, 2, 1, 0, 0, 2, -1, 0, -1, 0, 1, 1, 5, 0, 1, 1, -2, 1, 0, 0, 1, 2, 0, 1, 0, -1, -1,
-3, -2, -1, 2, 0, -2, 2, 1, 0, -1, -1, 1, 0, -1, 0, -1, 0, 1, -1, 1, -1, 0, 0, 0, 1, 0, 0, 2, -1, 0, 0, 5, -1, 1, 2, -1,
0, 0, 1, 0, -1, 0, 1, -1, -1, 1, 0, 1, -2, 1, -1, 1, 0, -1, 2, -1, 1, 2, 0, 0, -1, -1, 1, -1, -1, 1, 0, -1, 2, -1, -1, 1,
-1, 1, 1, 1, -1, 5, -1, 1, 0, 2, 1, -1, 0, -1, 0, 1, -1, -1, -2, -1, 1, 0, -1, 0, 2, -1, -1, 0, 0, 0, 1, 0, 1, 0, -3, 2,
0, -1, 0, 0, 0, 0, 2, 2, -1, 2, 0, -1, 1, 1, -1, 5, -1, 1, 1, 1, 0, 1, 0, -1, 0, 1, 0, -1, -1, -2, 1, 1, -1, 0, -1, 1, 0,
1, 2, -1, -1, -1, 0, 1, -1, 1, 0, 0, -1, 0, 0, 0, -2, 1, -1, -2, 0, -2, 2, 1, -1, 5, -1, 1, -1, 0, -2, 0, 0, 0, -2, -1, 2,
2, 2, -3, -1, 0, 1, -1, -1, 2, 0, 1, 3, 1, 2, 0, -2, 2, 0, -1, 0, -1, 0, 1, 0, 2, 0, 1, 0, 2, 1, -1, 0, 1, -1, 6, 2, 2, 0,
1, 1, 1, 0, 2, 3, 0, -1, 1, 1, 2, 1, -1, 2, 0, -2, -1, 0, -1, -1, 0, 2, 2, -1, 1, 3, 1, -2, -2, 1, 0, 1, 2, 0, 0, 2, 0, 0,
2, 1, 1, 2, 6, 1, -2, -1, 0, 0, 2, -1, 2, -1, 0, 1, 2, 0, 0, 1, -1, 0, 0, -1, 2, 1, -1, -1, 3, -2, -2, 0, 1, 1, 2, -1, 0,
-1, 0, 1, 1, 0, -1, 0, 0, 1, 1, -1, 2, 1, 6, 1, 0, -2, 2, 3, 2, 1, 0, -1, 1, 1, 0, 0, -2, 2, 1, 0, 0, 0, -1, 0, 2, 0, 1,
-1, 2, 0, -1, 2, 1, -1, -1, -1, -1, 1, -1, 0, 1, 1, -1, 0, 0, 0, -2, 1, 6, 1, 0, 1, 0, 1, 0, 0, 1, 0, -1, 1, -1, -1, 2, 0,
-1, -1, 1, -1, 1, 1, -1, 2, 2, -2, 1, 0, 0, 0, 2, 1, 2, -2, 2, 1, 1, 2, 0, 0, 1, -2, 1, -1, 0, 1, 6, 0, -2, 0, 2, 0, -2,
-1, 0, 1, 2, 0, 1, 1, -1, -1, 0, -1, -1, -1, 0, 0, 1, 2, 0, 2, -1, -1, 0, 1, 1, 2, 0, 0, 0, 1, 0, -1, -1, 0, 0, 1, 0, -2,
0, 0, 6, 1, -1, 0, 2, 2, 0, -1, 0, 0, 0, 1, -1, 1, -1, 1, 0, -1, -1, 0, 0, 2, 1, 0, 2, 1, 2, 0, 1, -1, -2, 1, -1, -1, -1,
1, 0, 0, -1, 0, 1, 0, 2, 1, -2, 1, 6, 1, 0, 0, 1, -1, 0, 1, -2, -1, -1, 2, 0, 0, -1, 0, 1, 0, 2, -1, 1, -1, 0, 0, 0, 1,
-2, 1, -1, 1, 2, 0, 2, 0, 0, 1, 1, 0, 0, 0, 2, 3, 0, 0, -1, 1, 6, 0, 2, 0, 0, 1, 2, -2, 2, 0, 1, 0, 0, -1, 0, 0, 2, 0, -1,
-1, 0, 1, -1, 0, 1, 1, 0, 2, 2, -1, 3, -1, 0, -1, -1, -1, 1, -2, 2, -1, 2, 1, 2, 0, 0, 0, 6, 1, -1, -1, 0, 1, 2, 1, -2, 2,
1, -2, 1, 0, -1, -1, 1, 1, 2, -1, 0, 1, -2, -1, -2, 0, -1, 3, 1, 0, 2, 3, -1, -1, -1, 0, -1, 3, 2, 1, 0, 0, 2, 0, 2, 1, 6,
1, 1, 1, 1, 1, 3, -1, 1, -1, -3, -2, -1, -2, 1, -3, 2, 3, -1, 2, 2, -1, 0, 1, -1, 0, 0, 2, -1, -1, -1, -3, 1, -2, -1, 2,
0, -1, 0, 0, -2, 2, 1, 0, -1, 1, 6, 1, 1, -3, 0, 1, -2, -1, 1, 1, 1, -1, 0, 1, 1, -1, 2, 0, 2, 1, 0, 0, 0, -2, 0, -1, 0,
-2, -1, 0, -2, 0, -1, -1, 2, -1, 0, -1, 1, -1, 0, -1, 0, -1, 1, 1, 6, 1, -1, -1, 1, 0, 0, -1, 1, -2, 1, 0, 1, 0, 2, 1, 0,
1, 1, 1, -1, -1, 1, -1, -1, 1, 0, -1, 2, -1, 1, 1, -2, 2, 1, 1, 1, 0, 0, -1, 0, 1, 0, 1, 1, 1, 6, -1, 0, 1, 0, 2, -1, 0,
0, -1, 1, 0, 1, 1, -1, 0, 1, 1, 2, 0, -1, 1, 0, 1, 0, 0, 1, 0, 2, -2, 0, 1, -3, 1, 2, 1, -1, 1, 0, 1, 2, 1, 1, -3, -1, -1,
6, -1, 0, 0, 2, 1, -1, 0, 0, -1, -1, -1, 1, 0, -3, 0, 1, -1, -1, 0, -1, 1, 2, -1, 3, -1, 2, 0, 1, -1, 1, -1, 2, 0, 0, 1,
2, 0, -2, -2, 2, 1, 0, -1, 0, -1, 6, -1, -1, 1, 0, -2, 0, 0, -1, -2, -1, 1, 2, 0, 0, 3, -2, 0, 0, 0, 0, 2, 1, -1, 3, 1,
-2, -1, 0, -1, 0, 1, 0, 0, -1, 0, 0, -1, 2, 1, 3, 1, 1, 1, 0, -1, 6, -1, 0, -2, -2, -1, -2, -1, 3, -2, 1, 1, 0, 2, -1, 0,
-2, 0, 2, 0, 0, -1, 0, 1, 1, 2, 1, 2, 0, 1, -1, 1, -2, -1, 1, 1, -1, 0, -2, -1, -2, 0, 0, 0, -1, -1, 6, -1, -1, 1, -1, 1,
1, -1, 0, 0, -3, 3, -1, 0, 1, 1, 0, 0, -1, -1, -1, 1, -1, 0, 1, 0, -1, -1, -1, 2, -1, 2, 2, 1, -1, 2, 1, 2, 1, -1, 0, 2,
2, 1, 0, -1, 6, 0, -1, -1, -1, 0, 0, 1, 0, 1, -1, 0, 2, 2, 0, 1, -1, 0, 0, 1, 1, -2, -2, 0, -1, -1, 1, -1, 0, 0, 1, 0, -1,
1, 0, 0, 1, -1, 1, -1, -1, 1, 0, -2, -1, 0, 6, 1, 3, 2, -1, -1, 2, 0, 0, -2, 0, 2, 1, 0, 1, 0, 1, -2, 0, 0, -2, -3, -1, 2,
0, 0, 2, -2, 0, 0, -1, -1, -1, 0, 0, -2, -3, 1, 1, 0, -1, -2, -2, 1, -1, 1, 6, 1, 0, 2, 0, 2, -1, -3, 0, -1, -2, 1, 0, 1,
0, 1, 0, 0, 1, -1, -2, -1, -1, 0, 1, 0, -1, -1, 0, -1, 0, 1, -1, -1, 1, -2, 1, -2, 0, 0, 0, -1, -1, -1, 3, 1, 6, -1, 0, 0,
2, -2, -1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, 1, -2, 0, 1, 1, 0, 2, 1, 0, 2, 0, 1, -1, 0, 0, 0, 0, -1, -1, 1, -1, 0, 0, -2,
1, -1, 2, 0, -1, 6, 0, -1, 1, 2, 0, -1, -1, 1, 1, -1, 1, 1, 0, -2, 1, -1, -2, 0, 0, 2, 1, -1, 3, -1, 1, -1, -1, -1, -1, 1,
0, -1, -2, 0, 0, 1, -1, -1, -1, 1, 0, -1, 2, 0, 0, 6, 0, 2, -1, -3, 1, -1, -1, -2, 1, 1, 0, -2, -1, 0, -1, 2, -1, -1, 0,
0, -1, 1, -1, -1, 0, 1, -1, -1, 0, 2, -1, 1, 1, 1, 0, -1, -2, 3, -1, 0, -1, 0, 0, -1, 0, 6, -1, -2, 0, 1, 0, -3, 2, 1, 1,
0, 0, -2, 0, 0, -3, -2, 0, 0, 1, -1, 2, 0, -1, 2, 1, 0, 0, 2, 0, 1, -3, 1, 0, 1, -1, -1, -2, 0, 1, 2, 2, 2, 1, 2, -1, 7,
-1, -1, 2, 0, 0, 1, -1, -1, -1, 1, 1, 1, 0, -1, 2, -1, -1, 0, 0, 0, 2, 3, 0, -1, 0, 0, -1, -1, 1, 2, -1, 2, 1, 1, 1, 1, 0,
0, 0, -1, -2, 2, -1, -2, -1, 7, 1, -3, 1, 3, -1, 1, 0, -1, 0, 2, 0, 0, 0, 1, 0, -1, -3, 1, -2, 2, -2, 1, 2, 1, 2, 1, -1,
2, 3, 2, 1, -1, 0, 2, 1, -3, 1, 0, -3, -1, 0, -3, 0, -1, 1, 7, -1, 3, 3, -2, 0, 0, 1, -1, 0, -1, 0, 1, 1, 1, 1, 2, -1, 2,
-1, -2, -1, 2, 2, 1, -1, 0, -1, -1, 0, 0, 0, -3, 0, 0, 3, -1, -2, 0, 0, -1, 1, 1, 2, -3, -1, 7, -1, -1, 1, 0, -1, 0, 0, 1,
2, -2, 1, 0, -1, -1, 0, 1, 0, 1, 0, 2, -2, 0, 0, 0, 1, 0, 2, 2, 1, 1, 0, 0, 2, -1, 0, 0, -1, 0, -1, -1, 0, 0, 1, 3, -1, 7,
2, 2, -1, 0, 0, 0, 1, 1, 0, -2, 2, 1, -1, -1, 0, -1, 3, 1, 0, 1, 2, 2, 0, -1, 1, 2, 1, 1, 1, 1, 3, -1, 0, 2, 2, -2, -1, 1,
-1, -3, 0, 3, 3, -1, 2, 8 ]);
-
GRAM (in Magma)
G45:=MatrixAlgebra(IntegerRing(), 45) ! \[ 4, -1, -1, 1, 0, 1, 0, 0, 0, 0, 0, 1, -1, 0, -1, 0, -1, -1, -1, -1, 0, 0, -1, 1, 0,
0, -1, -1, 0, 1, 1, 1, -1, -1, -1, 1, -1, 1, 0, 1, 0, -1, 0, -1, -1, -1, 4, 1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 0, -1, 0, 1, 0,
0, 2, 1, 0, 0, 1, -2, -1, 0, -1, 1, 0, 0, 1, -1, 0, 1, 1, -1, 0, 0, 1, -2, 1, 0, 0, 0, 1, -1, 1, 4, -2, 0, -1, 1, 1, -1,
0, 1, 1, 1, 0, -1, 0, 0, 0, 2, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 2, 1, 0, -2, 1, 1, -1, 0, -1, 1, -1, 1, 1, 1, -1,
-2, 4, -1, 0, 0, -1, 1, -1, -1, -1, -1, 0, 0, 0, -1, 0, -2, 0, 0, 0, -1, 0, 0, -1, 0, -1, 1, 1, 1, 1, -1, 0, 0, 2, -1, 0,
1, 1, -1, 0, 1, -1, -1, 0, 0, 0, -1, 4, 1, 1, -1, 0, -1, 1, 0, 0, 0, 0, 0, 1, -1, 0, -1, -1, 0, 1, 0, -1, 1, 0, 0, 0, 0,
0, 0, 0, 1, -2, 0, 0, -2, -1, -1, 1, 0, 0, 1, -2, 1, 0, -1, 0, 1, 4, 0, -1, 0, -1, 0, 1, 0, -1, 0, 0, 1, -1, 0, -2, -2, 1,
1, 0, -1, -1, -1, -1, 0, -2, 0, 2, -1, -1, -1, 1, -1, -2, 1, -2, 1, -2, 2, -2, -2, 0, 0, 1, 0, 1, 0, 4, -1, -1, -2, 1, 0,
-2, 0, -1, 1, -1, -2, 1, 1, -1, 0, -1, 1, -2, 0, -1, 0, 1, 1, 0, 1, 1, 2, -2, 2, 1, -1, -1, 2, -2, 2, 2, 1, -2, 0, 0, 1,
-1, -1, -1, -1, 5, -1, 1, -1, 1, 1, 0, -1, 0, 0, 1, 0, 0, 1, 1, -1, 2, 0, 0, 0, 0, 1, 0, 0, -1, 2, -1, 0, -2, 0, 3, 0, 0,
0, 0, -2, 2, 0, 0, 0, -1, 1, 0, 0, -1, -1, 5, 0, -1, -2, -1, 0, 0, 0, -1, -1, -2, 0, -1, 1, 1, -2, -1, -1, 2, -1, -1, 2,
0, 0, -2, -1, 1, 1, -2, 0, 2, 0, 2, -1, 1, 0, -1, 0, 0, 0, -1, -1, -1, -2, 1, 0, 5, 1, 1, 1, -1, 1, -1, 0, 1, 0, 1, 2, -1,
-1, 0, 2, 2, -1, 1, -1, -1, -1, -1, -1, -1, 2, -2, 0, 1, 0, 0, 0, 0, -1, 0, 2, 0, 1, 1, -1, 1, 0, 1, -1, -1, 1, 5, 1, 1,
-1, 0, 0, -1, 0, 1, 2, 1, -1, -1, 1, 1, 1, -2, 1, -1, -1, 0, 1, 0, 1, 0, -2, 0, 0, -1, 0, -1, 2, -1, 0, 0, 1, 1, 1, -1, 0,
1, 0, 1, -2, 1, 1, 5, 2, -2, 0, -1, -1, 0, 2, 0, 0, -2, 0, 1, 0, 0, -2, 2, 0, -2, -1, 0, 1, 1, 0, -1, 0, 0, -2, -2, -2,
-2, -1, 0, 1, -1, 0, 1, -1, 0, 0, -2, 1, -1, 1, 1, 2, 5, 0, 0, -1, 0, 1, 1, 0, 1, -2, 2, 1, 2, -1, 1, 2, -1, -2, 0, 0, 1,
1, 1, -3, -1, 1, -2, -2, -1, -2, -3, 0, 3, 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, -1, -2, 0, 5, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2,
0, 1, -1, -1, 1, 1, 0, 1, -1, -1, -1, 1, 1, 0, 2, 0, 1, -2, 1, 0, -1, 0, -1, 0, 0, 0, -1, -1, 0, 1, 0, 0, 0, 0, 5, 1, 1,
1, 0, 0, 0, -1, 1, -1, 2, 1, -1, 1, -2, -1, -2, -1, -2, 1, 0, -1, 1, -1, 0, -1, -1, -1, -1, -1, 2, 0, 1, 0, 0, 0, 0, 1, 0,
0, -1, 0, -1, -1, 1, 1, 5, 0, -1, 0, 0, -1, 1, 0, 0, 1, 0, -1, 1, 0, 1, 0, 1, 0, 1, -2, 0, 1, 0, 0, 1, -1, 2, 0, -1, 1,
-1, 0, 0, -1, 1, 1, -1, 0, -1, 0, -1, -1, 0, 1, 1, 0, 5, 2, 1, -2, 1, 1, 1, -1, 1, 1, 0, -2, 1, 0, -1, -2, 0, -2, 1, -1,
2, -2, 1, -2, 2, 0, 0, -1, 0, -1, 0, 0, 0, -1, -1, -2, 1, -1, 1, 0, 0, 1, 0, 1, -1, 2, 5, 0, 0, 2, 0, -1, 0, 2, 1, 0, -1,
1, -1, -1, -1, 0, -2, 2, -2, 1, 0, 0, -1, 0, 0, -2, 0, 1, -1, 2, 2, -2, 0, 0, 1, 0, -2, 0, 1, 2, 1, 0, 0, 0, 1, 0, 5, 1,
1, -1, 1, 0, 0, 0, -1, 1, 0, -1, 0, -1, 1, 1, 1, -1, 2, 0, -1, -1, 0, 0, 0, 0, 2, -1, 1, 1, 0, -1, -2, 1, 0, 0, 1, 2, 0,
0, 0, 0, 0, -2, 0, 1, 5, 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, -1, 0, 1, -1, 1, -2, 1, 0, 1, 1, 0, 0, 1, 0, -1,
-2, -1, 1, -1, 2, 1, 0, 1, 0, 0, -1, 1, 2, 1, 1, 5, -1, -1, 1, 2, 1, -1, -1, 0, 1, 1, -2, -1, -1, 2, -3, 1, 2, 0, 1, 0, 1,
-2, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 1, -1, -1, -2, -2, 0, -1, 1, 1, 0, -1, -1, -1, 5, -1, 0, -1, 0, 0, -2, 1, 0, 0, 1, 1,
-2, -1, 1, 0, 0, 1, 0, 2, 1, 2, -1, -2, -1, 1, 0, -1, 1, 1, -1, -1, 1, -1, -1, 0, 2, 0, 1, 0, 1, -1, 1, -1, -1, -1, 5, -2,
-1, -2, 2, 1, -1, 0, 1, -1, -1, 2, 1, 0, -1, -1, 0, -3, 1, -3, -1, -1, 2, 1, -2, 1, 0, 0, 0, 1, 2, -2, 0, 1, 1, 1, 0, -1,
0, -1, 0, 0, 0, 1, 0, -2, 5, 1, 0, -1, 0, 0, -1, 0, 2, 1, 0, -2, -1, 0, 2, -2, 2, -2, 1, -1, 0, -1, 0, -1, 0, 0, -1, -1,
-2, 0, -1, 2, 1, 0, 2, 2, 2, 1, 1, 2, 0, 0, 2, -1, -1, 1, 5, 1, -1, 1, -2, -1, -1, 0, 0, -1, 0, -3, 1, 1, -1, 1, -1, 1,
-3, -1, 3, 0, 0, 0, -1, 1, -1, 0, 0, -1, 2, 1, 0, -1, 0, 1, 0, 1, 1, 0, 1, 1, 0, -2, 0, 1, 5, -1, 0, 1, 0, -2, -1, 0, -1,
0, -1, 2, -1, -1, 0, 0, 1, 1, 0, 0, -1, -1, 0, 0, 0, -1, -1, 0, 2, -1, -2, -2, 1, 1, -1, -1, 0, 0, -1, 0, -1, 0, 2, -1,
-1, -1, 6, 0, 0, 2, 0, -1, 0, 0, 1, 1, -1, 0, 0, 0, 2, -1, 0, 1, 0, -1, 1, 0, -1, 0, -1, 0, 0, -1, 1, 1, 2, 2, -1, 1, 1,
-2, -1, 1, 1, -1, -2, 1, 0, 1, 0, 0, 6, -2, -1, 0, -1, 1, 3, -1, 0, -1, 0, -3, 0, -1, 0, -1, 0, 3, 0, 0, 0, 1, 0, 0, 1, 1,
-1, -1, -1, 0, -1, -1, -2, 0, 1, 1, 0, 0, 0, 1, -1, 0, -2, 1, 0, -2, 6, 0, -1, 0, 2, 0, 1, 2, 2, -1, 0, -1, -1, 1, 2, 0,
-1, 1, 0, 0, 1, 0, -2, 1, 0, 2, -1, -1, -2, -2, 1, -1, 1, 0, -1, -1, 0, 1, 0, 0, -1, -1, 0, 2, -1, 0, 6, 1, -2, -1, 0, 0,
1, 0, 1, 1, 2, 1, 1, 0, 1, -1, 1, 1, -1, 1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1, -2, 0, -1, -1, 0, 0, 1, 0, 1, 0, -1, -2, 0, 0,
-1, 1, 6, 0, -1, 0, 0, 0, -2, 2, 1, 1, 2, 0, -1, 0, -1, 1, -1, 0, 1, 0, 2, 1, -1, 0, -1, 1, 0, 0, 0, -1, 1, -2, -1, -1, 0,
-2, 1, -1, 2, 0, -1, -1, -1, 0, -2, 0, 6, 0, 0, -2, 1, -1, 0, 0, 1, -2, 0, 1, -1, -2, -1, 0, 2, -1, 0, -1, 1, 2, -2, -1,
0, 1, 1, 1, -2, 0, 0, 0, 1, 0, -1, 1, -1, 1, 0, 0, 0, 1, 2, -1, -1, 0, 6, 1, -1, 0, 2, 1, -2, 0, -1, 2, -1, 2, 0, -1, 1,
1, 0, 1, -1, 2, -1, -1, -1, 1, 1, 1, -1, 1, 1, -2, -2, 1, 1, -1, -2, 2, 0, -1, -1, 0, 3, 0, 0, 0, 0, 1, 6, -1, 1, 0, 0,
-2, 0, -3, 0, -1, 1, 2, -1, 1, 0, 0, -2, -1, -2, 0, 1, 2, 0, 0, 1, -1, 0, -2, 1, 2, 1, 1, 2, -1, 1, -2, 0, 0, 1, -1, 1, 0,
0, -2, -1, -1, 6, -1, 0, 0, 2, -2, 1, -1, 0, 0, 2, 1, -1, -2, 2, 0, 1, 2, -2, 1, -2, -2, -1, -3, -1, -1, 0, -1, -2, -1,
-1, -3, 1, 0, -1, -3, -1, 1, 0, 2, 1, 0, 1, 0, 1, -1, 6, 0, -2, 0, 1, 0, 0, 4, -1, -2, -1, 0, 1, -1, 0, -1, 1, 0, -2, 0,
0, 0, -1, 1, 1, 1, 2, 1, 2, 0, 1, 0, -1, 0, 1, 2, -1, -1, 2, 0, -2, -1, 2, 0, 0, 0, 6, -1, -1, 0, -1, 3, 0, 0, 1, 1, 0, 1,
0, -2, -2, -1, 3, 0, 1, 0, 0, 1, 1, -1, 0, -2, 0, 0, 1, 2, 0, -1, 2, 1, -1, 0, 0, -1, 1, 2, 0, 1, 0, 0, -2, -1, 6, 0, 2,
0, 0, -3, 1, 1, 0, 1, -1, 1, -1, 1, -1, 0, 2, 0, -1, -2, -2, 0, 0, 0, 1, 0, -1, -1, 0, 1, 0, -2, -1, -1, 0, -3, 0, 1, 1,
0, -2, -2, 2, 0, -1, 0, 6, 0, 3, 0, 1, 1, -2, 1, -2, 0, 1, -1, -2, 2, 0, 0, 0, 0, -2, -2, 2, -1, 1, -2, -1, -1, 1, 1, 0,
-3, 2, 1, 0, 0, 0, -1, 2, 1, 1, 0, 0, -2, 1, 0, 2, 0, 6, -1, 3, 0, 2, -1, 0, 1, -1, -1, 1, 1, -2, 0, 2, 0, -1, -2, -1, 0,
-1, -1, 2, 0, 0, -2, 0, 2, 1, -2, -1, 0, 2, -1, -1, 1, 2, -2, -1, -3, 1, 0, -1, 0, 3, -1, 7, 0, 1, 0, -2, -1, 0, 1, 0, 0,
-2, 2, 0, -1, 0, 2, -2, -2, 1, -1, 2, 0, 0, 0, 1, 1, 1, -3, 1, 1, 1, -1, 0, 1, 1, 0, 0, 2, 0, -1, 0, 3, 0, 0, 3, 0, 7, 0,
1, -1, 0, 0, -1, 1, 0, 2, 2, -2, 1, -1, -1, -1, -3, -2, -1, 0, 0, -2, 0, 0, -2, 2, -1, -1, -3, 1, 0, -1, 2, 0, -1, 1, -1,
-1, 0, 4, 0, -3, 1, 0, 1, 0, 7, -2, -2, -1, 0, 1, -1, 1, -2, 1, 2, 0, 0, 0, 0, 0, 1, -1, -1, -1, 0, 0, 1, 0, -1, -1, 0,
-1, 0, 1, 0, 0, 1, 0, -1, 2, 1, 0, -1, 0, 1, 1, 2, 0, 1, -2, 7, -2, -1, 1, 1, -1, -2, -2, -2, 0, -1, 2, 0, 1, 3, 0, 2, 1,
0, 1, 2, 1, 2, -2, 2, -1, 3, 0, 0, 3, -1, -1, -1, -2, 0, 2, 2, -2, 1, 1, -2, -1, -2, -1, -2, -2, 8 ];
-
MINIMAL_NORM
4
-
PROPERTIES
UNIMODULAR = 1
EXTREMAL = 1
-
COMMENTS
Date: Wed, 27 Jun 2001 18:00:18 +0200 (MET DST)
From: gaborit@unilim.fr (Philippe Gaborit)
Here is an extremal 45-dimensional unimodular lattice.
The construction uses self-dual codes over the ring Z/25Z.
This will be a part of a forthcoming paper "Construction of extremal unimodular lattices with
self-dual codes".
-
LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe