The Lattice dim56min6

An entry from the Catalogue of Lattices, which is a joint project of
Gabriele Nebe, University of Ulm (nebe@mathematik.uni-ulm.de) and
Neil J. A. Sloane, AT&T Shannon Labs (njas@research.att.com)

Last modified Sun Aug 5 08:38:38 EDT 2001

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
  
  

INDEX FILE | ABBREVIATIONS

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