#  MAPLE. June 1, ,2006.
# 
# Example 28 in HCD table of spherical designs by Munemasa.
#
# These are the shortest vectors in the E6 lattice.
d := 4;

v :=   [1 ,  20 ,  30 ,   20 ,   1];
verts := 72;

Q := matrix([
[1  ,  6 ,  20 ,   30 ,  15],
[1  ,  3 ,   2 ,   -3 ,  -3],
[1  ,  0 ,  -4 ,    0 ,   3],
[1  , -3 ,   2 ,    3 ,  -3],
[1  , -6 ,  20 ,  -30 ,  15]]);
 

#  This tridiagonal matrix is L1-star.
L := matrix(5,5,[
0,   6,  0,   0,  0,
1,   0,  5,   0,  0,
0, 3/2,  0, 9/2,  0,
0,   0,  3,   0,  3,
0,   0,  0,   6,  0]);


         [0    20    0     0    0]
         [                       ]
         [1     9    9     1    0]
         [                       ]
  L_1 =  [0     6    8     6    0]
         [                       ]
         [0     1    9     9    1]
         [                       ]
         [0     0    0    20    0]
# Note: This graph appears as E_{6,2} on p307 of the book [BCN].

          [0    0    30    0    0]
          [                      ]
          [0    9    12    9    0]
          [                      ]
  L_2 =   [1    8    12    8    1]
          [                      ]
          [0    9    12    9    0]
          [                      ]
          [0    0    30    0    0]

         [0     0    0    20    0]
         [                       ]
         [0     1    9     9    1]
         [                       ]
  L_3 =  [0     6    8     6    0]
         [                       ]
         [1     9    9     1    0]
         [                       ]
         [0    20    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    20    30     20     1]
     [                          ]
     [1    10     0    -10    -1]
     [                          ]
P  = [1     2    -6      2     1]
     [                          ]
     [1    -2     0      2    -1]
     [                          ]
     [1    -4     6     -4     1]



  [1     6    20     30    15]
  [                          ]
  [1     3     2     -3    -3]
  [                          ]
Q=[1     0    -4      0     3]
  [                          ]
  [1    -3     2      3    -3]
  [                          ]
  [1    -6    20    -30    15]



        [0     6     0     0     0]
        [                         ]
        [1     0     5     0     0]
        [                         ]
Ls1  =  [0    3/2    0    9/2    0]
        [                         ]
        [0     0     3     0     3]
        [                         ]
        [0     0     0     6     0]

         [0    0    20     0    0]
         [                       ]
         [0    5     0    15    0]
         [                       ]
Ls2  =   [1    0    10     0    9]
         [                       ]
         [0    3     0    17    0]
         [                       ]
         [0    0    12     0    8]

       [0     0      0     30      0]
       [                            ]
       [0     0     15     0      15]
       [                            ]
Ls3 =  [0    9/2     0    51/2     0]
       [                            ]
       [1     0     17     0      12]
       [                            ]
       [0     6      0     24      0]

         [0    0    0     0    15]
         [                       ]
         [0    0    0    15     0]
         [                       ]
Ls4  =   [0    0    9     0     6]
         [                       ]
         [0    3    0    12     0]
         [                       ]
         [1    0    8     0     6]