# MAPLE. May 30, 2006. # # Example 17 in HCD table of spherical designs by Munemasa. # # This is the Q-bipartite double cover of the dual polar graph # in Example 3. d := 7; v := [1 , 42, 336, 512, 512, 336, 42, 1]; verts := 1782; Q := matrix([ [1 , 22 , 252 , 616 , 616 , 252 , 22 , 1], [1 , 11 , 54 , 44 , -44 , -54 , -11 , -1], [1 , 11/2 , 9/2 , -11 , -11 , 9/2 , 11/2 , 1], [1 , 11/4 , -63/8 , -77/8 , 77/8 , 63/8 , -11/4 , -1], [1 , -11/4 , -63/8 , 77/8 , 77/8 , -63/8 , -11/4 , 1], [1 , -11/2 , 9/2 , 11 , -11 , -9/2 , 11/2 , -1], [1 , -11 , 54 , -44 , -44 , 54 , -11 , 1], [1 , -22 , 252 , -616 , 616 , -252 , 22 , -1] ]); # This tridiagonal matrix, L_1-star, allows us to fill out the cols of Q L := matrix(d+1,d+1,[ [0 , 22 , 0 , 0 , 0 , 0 , 0 , 0], [1 , 0 , 21 , 0 , 0 , 0 , 0 , 0], [0 , 11/6 , 0 , 121/6 , 0 , 0 , 0 , 0], [0 , 0 , 33/4 , 0 , 55/4 , 0 , 0 , 0], [0 , 0 , 0 , 55/4 , 0 , 33/4 , 0 , 0], [0 , 0 , 0 , 0 , 121/6 , 0 , 11/6 , 0], [0 , 0 , 0 , 0 , 0 , 21 , 0 , 1], [0 , 0 , 0 , 0 , 0 , 0 , 22 , 0] ]); [0 42 0 0 0 0 0 0] [ ] [1 0 40 0 0 0 1 0] [ ] [0 5 0 32 0 5 0 0] [ ] [0 0 21 0 21 0 0 0] L_1 = [ ] [0 0 0 21 0 21 0 0] [ ] [0 0 5 0 32 0 5 0] [ ] [0 1 0 0 0 40 0 1] [ ] [0 0 0 0 0 0 42 0] [0 0 336 0 0 0 0 0] [ ] [0 40 0 256 0 40 0 0] [ ] [1 0 170 0 160 0 5 0] [ ] [0 21 0 210 0 105 0 0] L_2 = [ ] [0 0 105 0 210 0 21 0] [ ] [0 5 0 160 0 170 0 1] [ ] [0 0 40 0 256 0 40 0] [ ] [0 0 0 0 0 336 0 0] [0 0 0 512 0 0 0 0] [ ] [0 0 256 0 256 0 0 0] [ ] [0 32 0 320 0 160 0 0] [ ] [1 0 210 0 280 0 21 0] L_3 = [ ] [0 21 0 280 0 210 0 1] [ ] [0 0 160 0 320 0 32 0] [ ] [0 0 0 256 0 256 0 0] [ ] [0 0 0 0 512 0 0 0] [0 0 0 0 512 0 0 0] [ ] [0 0 0 256 0 256 0 0] [ ] [0 0 160 0 320 0 32 0] [ ] [0 21 0 280 0 210 0 1] L_4 = [ ] [1 0 210 0 280 0 21 0] [ ] [0 32 0 320 0 160 0 0] [ ] [0 0 256 0 256 0 0 0] [ ] [0 0 0 512 0 0 0 0] [0 0 0 0 0 336 0 0] [ ] [0 0 40 0 256 0 40 0] [ ] [0 5 0 160 0 170 0 1] [ ] [0 0 105 0 210 0 21 0] L_5 = [ ] [0 21 0 210 0 105 0 0] [ ] [1 0 170 0 160 0 5 0] [ ] [0 40 0 256 0 40 0 0] [ ] [0 0 336 0 0 0 0 0] [0 0 0 0 0 0 42 0] [ ] [0 1 0 0 0 40 0 1] [ ] [0 0 5 0 32 0 5 0] [ ] [0 0 0 21 0 21 0 0] L_6 = [ ] [0 0 21 0 21 0 0 0] [ ] [0 5 0 32 0 5 0 0] [ ] [1 0 40 0 0 0 1 0] [ ] [0 42 0 0 0 0 0 0] [0 0 0 0 0 0 0 1] [ ] [0 0 0 0 0 0 1 0] [ ] [0 0 0 0 0 1 0 0] [ ] [0 0 0 0 1 0 0 0] L_7 = [ ] [0 0 0 1 0 0 0 0] [ ] [0 0 1 0 0 0 0 0] [ ] [0 1 0 0 0 0 0 0] [ ] [1 0 0 0 0 0 0 0] [1 42 336 512 512 336 42 1] [ ] [1 21 84 64 -64 -84 -21 -1] [ ] [1 9 6 -16 -16 6 9 1] [ ] [1 3 -6 -8 8 6 -3 -1] P = [ ] [1 -3 -6 8 8 -6 -3 1] [ ] [1 -9 6 16 -16 -6 9 -1] [ ] [1 -21 84 -64 -64 84 -21 1] [ ] [1 -42 336 -512 512 -336 42 -1] [1 22 252 616 616 252 22 1] [ ] [1 11 54 44 -44 -54 -11 -1] [ ] [1 11/2 9/2 -11 -11 9/2 11/2 1] [ ] [1 11/4 -63/8 -77/8 77/8 63/8 -11/4 -1] Q = [ ] [1 -11/4 -63/8 77/8 77/8 -63/8 -11/4 1] [ ] [1 -11/2 9/2 11 -11 -9/2 11/2 -1] [ ] [1 -11 54 -44 -44 54 -11 1] [ ] [1 -22 252 -616 616 -252 22 -1] [0 22 0 0 0 0 0 0] [ ] [1 0 21 0 0 0 0 0] [ ] [0 11/6 0 121/6 0 0 0 0] [ ] [0 0 33/4 0 55/4 0 0 0] Ls1 = [ ] [0 0 0 55/4 0 33/4 0 0] [ ] [0 0 0 0 121/6 0 11/6 0] [ ] [0 0 0 0 0 21 0 1] [ ] [0 0 0 0 0 0 22 0] [0 0 252 0 0 0 0 0] [ ] [0 21 0 231 0 0 0 0] [ ] [1 0 399/4 0 605/4 0 0 0] [ ] [0 33/4 0 1455/8 0 495/8 0 0] Ls2= [ ] [0 0 495/8 0 1455/8 0 33/4 0] [ ] [0 0 0 605/4 0 399/4 0 1] [ ] [0 0 0 0 231 0 21 0] [ ] [0 0 0 0 0 252 0 0] [0 0 0 616 0 0 0 0] [ ] [0 0 231 0 385 0 0 0] [ ] [ 5335 ] [0 121/6 0 ---- 0 605/4 0 0] [ 12 ] [ ] [1 0 1455/8 0 3355/8 0 55/4 0] Ls3=[ ] [0 55/4 0 3355/8 0 1455/8 0 1] [ ] [ 5335 ] [0 0 605/4 0 ---- 0 121/6 0] [ 12 ] [ ] [0 0 0 385 0 231 0 0] [ ] [0 0 0 0 616 0 0 0] [0 0 0 0 616 0 0 0] [ ] [0 0 0 385 0 231 0 0] [ ] [ 5335 ] [0 0 605/4 0 ---- 0 121/6 0] [ 12 ] [ ] [0 55/4 0 3355/8 0 1455/8 0 1] Ls4=[ ] [1 0 1455/8 0 3355/8 0 55/4 0] [ ] [ 5335 ] [0 121/6 0 ---- 0 605/4 0 0] [ 12 ] [ ] [0 0 231 0 385 0 0 0] [ ] [0 0 0 616 0 0 0 0] [0 0 0 0 0 252 0 0] [ ] [0 0 0 0 231 0 21 0] [ ] [0 0 0 605/4 0 399/4 0 1] [ ] [0 0 495/8 0 1455/8 0 33/4 0] Ls5= [ ] [0 33/4 0 1455/8 0 495/8 0 0] [ ] [1 0 399/4 0 605/4 0 0 0] [ ] [0 21 0 231 0 0 0 0] [ ] [0 0 252 0 0 0 0 0] [0 0 0 0 0 0 22 0] [ ] [0 0 0 0 0 21 0 1] [ ] [0 0 0 0 121/6 0 11/6 0] [ ] [0 0 0 55/4 0 33/4 0 0] Ls6 = [ ] [0 0 33/4 0 55/4 0 0 0] [ ] [0 11/6 0 121/6 0 0 0 0] [ ] [1 0 21 0 0 0 0 0] [ ] [0 22 0 0 0 0 0 0] [0 0 0 0 0 0 0 1] [ ] [0 0 0 0 0 0 1 0] [ ] [0 0 0 0 0 1 0 0] [ ] [0 0 0 0 1 0 0 0] Ls7 = [ ] [0 0 0 1 0 0 0 0] [ ] [0 0 1 0 0 0 0 0] [ ] [0 1 0 0 0 0 0 0] [ ] [1 0 0 0 0 0 0 0]