# MAPLE.   May 31, 2006.
# 
# Example 26 in HCD table of spherical designs by Munemasa.
#
# These are 126 of the shortest vectors in the E8 lattice.
#
d := 4;

v := [1 ,  32 , 60 , 32 , 1 ];
verts := 126;


Q := matrix([
[1  ,  7   ,   27  ,   56  ,   35  ],
[1  , 7/2  ,  27/8 ,  -7/2 ,  -35/8],
[1  ,  0   ,  -9/2 ,   0   ,   7/2 ],
[1  , -7/2 ,  27/8 ,  7/2  ,  -35/8],
[1  ,  -7  ,   27  ,  -56  ,   35  ]
 ]);



#  This tridiagonal  matrix, L_1-star,  allows us to fill out the cols of Q
L := matrix(d+1,d+1,[
[0  ,  7   ,   0   ,   0   ,   0  ],
[1  ,  0   ,   6   ,   0   ,   0  ],
[0  , 14/9 ,   0   ,  49/9 ,   0  ],
[0  ,  0   ,  21/8 ,   0   ,  35/8],
[0  ,  0   ,   0   ,   7   ,   0  ]
]);


     [0    32     0     0    0]
     [                        ]
     [1    15    15     1    0]
     [                        ]
L_1 =[0     8    16     8    0]
     [                        ]
     [0     1    15    15    1]
     [                        ]
     [0     0     0    32    0]
# Note: This graph appears as E_{7,1} on p313 of the book [BCN].
 
     [0     0    60     0    0]
     [                        ]
     [0    15    30    15    0]
     [                        ]
L_2 =[1    16    26    16    1]
     [                        ]
     [0    15    30    15    0]
     [                        ]
     [0     0    60     0    0]
 
     [0     0     0    32    0]
     [                        ]
     [0     1    15    15    1]
     [                        ]
L_3 =[0     8    16     8    0]
     [                        ]
     [1    15    15     1    0]
     [                        ]
     [0    32     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    32     60     32     1]
     [                           ]
     [1    16      0    -16    -1]
     [                           ]
P =  [1     4    -10      4     1]
     [                           ]
     [1    -2      0      2    -1]
     [                           ]
     [1    -4      6     -4     1]
 
      [1     7       27      56      35  ]
      [                                  ]
      [1    7/2     27/8    -7/2    -35/8]
      [                                  ]
 Q := [1     0      -9/2     0       7/2 ]
      [                                  ]
      [1    -7/2    27/8    7/2     -35/8]
      [                                  ]
      [1     -7      27     -56      35  ]
 

          [0     7       0       0       0  ]
          [                                 ]
          [1     0       6       0       0  ]
          [                                 ]
Ls1 =     [0    14/9     0      49/9     0  ]
          [                                 ]
          [0     0      21/8     0      35/8]
          [                                 ]
          [0     0       0       7       0  ]
 
          [0     0      27       0       0 ]
          [                                ]
          [0     6       0      21       0 ]
          [                                ]
          [             171             245]
          [1     0      ---      0      ---]
Ls2 =     [             16              16 ]
          [                                ]
          [0    21/8     0     195/8     0 ]
          [                                ]
          [             189             243]
          [0     0      ---      0      ---]
          [             16              16 ]
 
        [0     0        0       56        0  ]
        [                                    ]
        [0     0       21        0       35  ]
        [                                    ]
Ls3 =   [0    49/9      0      455/9      0  ]
        [                                    ]
        [1     0      195/8      0      245/8]
        [                                    ]
        [0     7        0       49        0  ]
 
          [0     0       0       0      35 ]
          [                                ]
          [0     0       0      35       0 ]
          [                                ]
          [             245             315]
          [0     0      ---      0      ---]
Ls4 =     [             16              16 ]
          [                                ]
          [0    35/8     0     245/8     0 ]
          [                                ]
          [             243             301]
          [1     0      ---      0      ---]
          [             16              16 ]