# MAPLE.   June 7, 2006.
# 
# Example 20 in HCD table of spherical designs by Munemasa.
#
#  Shortest vectors in Martinet lattice in 16-D.
d := 4;

v := [1, 135, 240, 135, 1];
verts := 512;



Q := matrix([
[1 ,   16   ,  135 ,   240  ,  120],
[1 ,  16/3  ,    7 ,  -16/3 ,   -8],
[1 ,    0   ,   -9 ,    0   ,    8],
[1 ,  -16/3 ,    7 ,  16/3  ,   -8],
[1 ,   -16  ,  135 ,  -240  ,  120]
]);

#   strengths of spherical designs are, 5, 2, 3, 2


#  This tridiagonal  matrix, L_1-star,  allows us to fill out the cols of Q
L := matrix(d+1,d+1,[
[0 ,   16  ,   0 ,    0   ,  0],
[1 ,   0   ,  15 ,    0   ,  0],
[0 ,  16/9 ,   0 ,  128/9 ,  0],
[0 ,   0   ,   8 ,    0   ,  8],
[0 ,   0   ,   0 ,   16   ,  0]
]);



      [0    135     0      0    0]
      [                          ]
      [1     56    64     14    0]
      [                          ]
L_1 = [0     36    63     36    0]
      [                          ]
      [0     14    64     56    1]
      [                          ]
      [0      0     0    135    0]

      [0     0    240     0    0]
      [                         ]
      [0    64    112    64    0]
      [                         ]
L_2 = [1    63    112    63    1]
      [                         ]
      [0    64    112    64    0]
      [                         ]
      [0     0    240     0    0]

      [0      0     0    135    0]
      [                          ]
      [0     14    64     56    1]
      [                          ]
L_3 = [0     36    63     36    0]
      [                          ]
      [1     56    64     14    0]
      [                          ]
      [0    135     0      0    0]

      [0    0    0    0    1]
      [                     ]
      [0    0    0    1    0]
      [                     ]
L_4 = [0    0    1    0    0]
      [                     ]
      [0    1    0    0    0]
      [                     ]
      [1    0    0    0    0]



      [1    135    240    135     1]
      [                            ]
      [1     45      0    -45    -1]
      [                            ]
P =   [1      7    -16      7     1]
      [                            ]
      [1     -3      0      3    -1]
      [                            ]
      [1     -9     16     -9     1]

      [1     16      135     240     120]
      [                                 ]
      [1    16/3       7    -16/3     -8]
      [                                 ]
Q =   [1      0       -9      0        8]
      [                                 ]
      [1    -16/3      7    16/3      -8]
      [                                 ]
      [1     -16     135    -240     120]



      [0     16      0      0      0]
      [                             ]
      [1     0      15      0      0]
      [                             ]
Ls1 = [0    16/9     0    128/9    0]
      [                             ]
      [0     0       8      0      8]
      [                             ]
      [0     0       0     16      0]

      [0     0    135      0     0]
      [                           ]
      [0    15      0    120     0]
      [                           ]
Ls2 = [1     0     70      0    64]
      [                           ]
      [0     8      0    127     0]
      [                           ]
      [0     0     72      0    63]

      [0      0        0     240        0]
      [                                  ]
      [0      0      120      0       120]
      [                                  ]
Ls3 = [0    128/9      0    2032/9      0]
      [                                  ]
      [1      0      127      0       112]
      [                                  ]
      [0     16        0     224        0]

      [0    0     0      0    120]
      [                          ]
      [0    0     0    120      0]
      [                          ]
Ls4 = [0    0    64      0     56]
      [                          ]
      [0    8     0    112      0]
      [                          ]
      [1    0    63      0     56]