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 13:21:19 CEST 2014

INDEX FILE | ABBREVIATIONS

Contents of this file

NAME
DIMENSION
DET
MINIMAL_NORM
KISSING_NUMBER
THETA_SERIES
PROPERTIES
NOTES
REFERENCES
GRAM
GROUP_ORDER
GROUP_GENERATORS
LAST_LINE

• NAME
  Odd Barnes-Wall
  

• DIMENSION
  16
  

• DET
  256
  

• MINIMAL_NORM
  3
  

• KISSING_NUMBER
  256
  

• THETA_SERIES
  1,0,0,256,2016,9216,32768,96768,246240,563200,1179648,...
  (see sequence A014711), or in other words
  (theta_3(z)theta_3(2z))^6((theta_3(z)theta_3(2z))^2-4(theta_4(z)theta_2(2z))^2).
  

• PROPERTIES
  INTEGRAL=1
  MODULAR=2
  

• NOTES
  Odd 16-dim Barnes-Wall lattice
  

• REFERENCES
  E. M. Rains and N. J. A. Sloane,
  The Shadow Theory of Modular and Unimodular Lattices,
  J. Number Theory, 73 (1998), pp. 359-389;
  http://neilsloane.com/doc/shad.pdf or shad.ps
  

• GRAM
  16 16
  3 1 -1 -1 -1 0 1 1 -1 0 0 0 0 -1 -1 0
  1 3 0 -1 -1 2 -1 0 0 -1 0 -1 1 -1 0 0
  -1 0 3 0 1 -1 0 -1 -1 0 -1 1 1 1 1 -1
  -1 -1 0 3 0 0 -1 0 0 1 1 0 1 0 -1 0
  -1 -1 1 0 3 -1 0 -1 -1 1 -1 0 0 0 0 -1
  0 2 -1 0 -1 4 -2 -1 1 -1 1 -2 1 -1 0 0
  1 -1 0 -1 0 -2 3 1 -1 0 -1 1 -1 0 0 1
  1 0 -1 0 -1 -1 1 3 0 -1 -1 1 -1 -1 0 1
  -1 0 -1 0 -1 1 -1 0 3 0 1 0 -1 1 1 0
  0 -1 0 1 1 -1 0 -1 0 3 1 0 0 1 -1 -1
  0 0 -1 1 -1 1 -1 -1 1 1 3 -1 0 0 -1 0
  0 -1 1 0 0 -2 1 1 0 0 -1 3 -1 1 1 -1
  0 1 1 1 0 1 -1 -1 -1 0 0 -1 3 0 -1 0
  -1 -1 1 0 0 -1 0 -1 1 1 0 1 0 3 1 -1
  -1 0 1 -1 0 0 0 0 1 -1 -1 1 -1 1 3 -1
  0 0 -1 0 -1 0 1 1 0 -1 0 -1 0 -1 -1 3
  

• GROUP_ORDER
  3096576 = 2^14 * 3^3 * 7
  

• GROUP_GENERATORS
  2
  16 16
  0 -1 0 -1 1 1 0 1 0 0 1 1 1 0 0 0
  0 -1 1 -1 0 1 0 1 0 1 0 0 0 0 0 0
  1 0 1 1 0 0 0 0 0 -1 0 -1 -1 1 0 0
  -1 1 -1 0 0 0 0 0 -1 0 0 0 0 1 0 0
  0 1 0 1 0 -1 0 -1 0 -1 0 0 -1 0 0 0
  0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0
  1 0 0 1 1 0 0 0 1 -1 0 0 0 0 0 0
  -1 0 0 0 0 1 1 1 0 0 0 0 0 0 -1 -1
  0 0 0 0 -1 0 0 -1 0 1 -1 0 0 -1 0 0
  0 0 -1 0 0 0 -1 -1 -1 0 0 1 0 0 1 1
  0 0 -1 -1 0 0 -1 0 -1 1 0 1 1 0 1 1
  0 0 0 1 0 0 1 0 0 -1 0 -1 0 0 0 -1
  0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
  1 0 0 1 0 0 0 -1 0 0 0 0 0 0 1 1
  1 0 1 1 0 0 0 0 0 0 0 -1 -1 0 0 0
  0 0 0 0 0 0 0 0 1 0 -1 0 0 0 -1 0
  16 16
  0 0 0 0 0 -1 0 -1 1 0 0 0 0 -1 0 0
  0 0 1 0 0 0 0 0 1 0 0 0 0 -1 0 0
  0 -1 0 -1 0 0 -1 1 0 0 0 0 1 0 0 0
  -1 1 -1 0 0 0 0 0 -1 0 0 0 0 1 0 0
  0 -1 0 -1 -1 0 -1 0 -1 0 0 0 0 0 0 0
  0 1 1 1 0 0 1 0 1 -1 0 -1 -1 0 -1 -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 1 0 0 0 0 1 1
  0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0
  -1 0 -1 -1 0 0 0 0 -1 0 1 1 1 0 0 0
  -1 1 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1
  0 0 -1 0 0 -1 0 0 0 0 0 0 1 0 1 0
  0 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 1 1 0 0 0
  0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
  1 0 1 1 1 1 0 0 0 0 0 0 -1 1 0 1
  

• LAST_LINE
  

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  Haftungsausschluss/Disclaimer
  
  
  Gabriele Nebe