The Lattice BW16 (The Barnes-Wall Lattice)

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 Mon Oct 3 09:47:02 EDT 2005

INDEX FILE | ABBREVIATIONS

Contents of this file

NAME
DIMENSION
GRAM
BASIS
DET
DIVISORS
MINIMAL_NORM
KISSING_NUMBER
GROUP_ORDER
GROUP_NAME
GROUP_GENERATORS
PROPERTIES
REFERENCES
NOTES
MAGMA
LAST_LINE

• NAME
  BW16 (The Barnes-Wall Lattice)
  
• DIMENSION
  16
  
• GRAM
  16 16
  4 2 2 1 2 1 -2 -2 -1 -1 0 0 -1 2 2 2
  2 4 1 2 1 2 -1 -1 -2 1 1 -1 1 0 0 1
  2 1 4 2 2 1 0 0 0 0 -1 -1 -1 0 0 2
  1 2 2 4 1 2 0 0 0 0 1 0 0 -1 0 1
  2 1 2 1 4 2 -2 -2 -1 -1 1 1 -2 2 0 2
  1 2 1 2 2 4 -1 -1 -2 1 2 1 -1 1 0 0
  -2 -1 0 0 -2 -1 4 2 2 0 0 0 0 -2 -2 -2
  -2 -1 0 0 -2 -1 2 4 2 0 -2 -2 2 -2 -1 -1
  -1 -2 0 0 -1 -2 2 2 4 -2 -1 0 0 -1 0 -1
  -1 1 0 0 -1 1 0 0 -2 4 0 -1 1 -1 0 -1
  0 1 -1 1 1 2 0 -2 -1 0 4 2 -1 1 0 -1
  0 -1 -1 0 1 1 0 -2 0 -1 2 4 -2 1 0 -1
  -1 1 -1 0 -2 -1 0 2 0 1 -1 -2 4 -2 0 0
  2 0 0 -1 2 1 -2 -2 -1 -1 1 1 -2 4 2 0
  2 0 0 0 0 0 -2 -1 0 0 0 0 0 2 4 0
  2 1 2 1 2 0 -2 -1 -1 -1 -1 -1 0 0 0 4
  
• BASIS
  16 16
  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
  0 2 0 0 0 0 0 2 0 0 0 2 0 2 0 0
  0 0 2 0 0 0 0 2 0 0 0 2 0 0 2 0
  0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 2
  0 0 0 0 2 0 0 2 0 0 0 0 0 2 2 0
  0 0 0 0 0 2 0 2 0 0 0 0 0 2 0 2
  0 0 0 0 0 0 2 2 0 0 0 0 0 0 2 2
  0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0
  0 0 0 0 0 0 0 0 2 0 0 2 0 2 2 0
  0 0 0 0 0 0 0 0 0 2 0 2 0 2 0 2
  0 0 0 0 0 0 0 0 0 0 2 2 0 0 2 2
  0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0
  0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2
  0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4
  
• DET
  256
  
• DIVISORS
  1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
  
• MINIMAL_NORM
  4
  
• KISSING_NUMBER
  4320
  
• GROUP_ORDER
  89181388800
  
• GROUP_NAME
  2+^(1+8).O+(8,2)
  
• GROUP_GENERATORS
  2
  16
  -1 1 0 0 -1 0 1 0 1 1 0 1 0 2 0 2
  -1 1 0 0 -1 0 0 1 1 1 0 1 0 2 0 2
  -1 1 1 0 -1 0 0 0 1 1 0 1 0 1 0 1
  -1 1 1 0 -1 -1 -1 1 1 1 1 1 0 1 0 1
  0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 1
  0 -1 -1 1 1 0 1 0 0 1 0 0 1 1 0 1
  0 -1 1 0 -1 1 -1 -1 0 -1 0 -1 0 -1 0 -1
  1 -1 -1 1 1 0 0 -1 0 0 -1 -1 0 -1 0 -1
  0 0 1 0 -1 0 -1 -1 0 -1 0 0 0 -1 0 -1
  -1 1 0 0 0 0 -1 1 0 0 0 0 -1 0 0 0
  0 -1 1 0 -1 0 -1 0 0 0 1 0 1 1 -1 0
  -1 0 1 -1 0 0 0 0 -1 -1 1 0 0 -1 1 0
  0 1 0 0 0 -1 -1 1 0 0 0 0 -1 0 0 0
  1 -1 -1 0 0 1 1 -1 1 1 0 0 1 1 -1 1
  0 1 0 0 -1 -1 -1 1 1 1 1 1 -1 1 -1 1
  0 1 0 0 0 -1 1 1 0 1 0 1 0 1 0 1
  16
  -1 1 0 0 -1 0 0 1 1 1 0 1 0 2 0 2
  -1 1 0 0 -1 0 1 0 1 1 0 1 0 2 0 2
  0 0 0 1 0 -1 0 1 0 1 0 1 0 1 0 1
  0 0 1 0 -1 -1 0 1 0 1 1 1 0 1 0 1
  0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1
  0 -1 0 0 0 1 1 -1 1 1 0 0 1 1 0 1
  0 0 1 0 -1 0 -1 0 -1 -1 0 0 -1 -1 0 -1
  1 -1 -1 1 1 -1 0 0 -1 0 0 -1 -1 -1 0 -1
  0 0 1 0 -1 -1 -2 1 -1 -1 1 0 -1 -1 0 -1
  0 -1 -1 1 1 1 2 -2 0 0 -1 -1 1 0 0 0
  -1 1 2 -2 -2 1 -1 0 1 0 1 1 0 0 0 1
  -1 0 2 -1 -1 1 -1 -1 0 -1 0 0 1 0 0 -1
  -1 1 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0
  1 -1 -1 0 0 1 0 0 1 1 0 0 1 1 -1 1
  0 0 0 0 -1 0 -1 1 1 1 1 0 0 1 -1 1
  0 1 0 0 0 -1 0 1 1 1 0 1 0 1 0 1
  
• PROPERTIES
  INTEGRAL=1
  EVEN=1
  MODULAR=2
  
• REFERENCES
  SPLAG p. 129
  
• NOTES
  Bais should be divided by 2
  
• MAGMA
  BW16 := LatticeWithGram(16, [
  4,
  2,4,
  2,1,4,
  1,2,2,4,
  2,1,2,1,4,
  1,2,1,2,2,4,
  -2,-1,0,0,-2,-1,4,
  -2,-1,0,0,-2,-1,2,4,
  -1,-2,0,0,-1,-2,2,2,4,
  -1,1,0,0,-1,1,0,0,-2,4,
  0,1,-1,1,1,2,0,-2,-1,0,4,
  0,-1,-1,0,1,1,0,-2,0,-1,2,4,
  -1,1,-1,0,-2,-1,0,2,0,1,-1,-2,4,
  2,0,0,-1,2,1,-2,-2,-1,-1,1,1,-2,4,
  2,0,0,0,0,0,-2,-1,0,0,0,0,0,2,4,
  2,1,2,1,2,0,-2,-1,-1,-1,-1,-1,0,0,0,4
  ]);
  BW16`Minimum := 4;
  
• LAST_LINE
  

INDEX FILE | ABBREVIATIONS