# MAPLE. June 5. 2006. # # Take dual of coset graph of a code of length 21 derived from # the extended binary Golay code. (see (A16) on p365 in [BCN].) # This is Q-antipodal with r=4. So delete one class to get a new scheme. with(linalg): d := 6; v := [1 , 112 , 210 , 672 , 280 , 240 , 21]; verts := 1536; Q := matrix([ [1 , 21 , 210 , 840 , 420 , 42 , 2], [1 , 9 , 30 , 0 , -30 , -9 , -1], [1 , 5 , 2 , -24 , 4 , 10 , 2], [1 , 1 , -10 , 0 , 10 , -1 , -1], [1 , -3 , -6 , 24 , -12 , -6 , 2], [1 , -7 , 14 , 0 , -14 , 7 , -1], [1 , -11 , 50 , -120 , 100 , -22 , 2] ]); # This tridiagonal matrix, L_1-star, allows us to fill out the cols of Q # if we are only given the cosines (normalized column 1 of Q). L := matrix([ [0 , 21 , 0 , 0 , 0 , 0 , 0], [1 , 0 , 20 , 0 , 0 , 0 , 0], [0 , 2 , 3 , 16 , 0 , 0 , 0], [0 , 0 , 4 , 9 , 8 , 0 , 0], [0 , 0 , 0 , 16 , 3 , 2 , 0], [0 , 0 , 0 , 0 , 20 , 0 , 1], [0 , 0 , 0 , 0 , 0 , 21 , 0] ]); [0 112 0 0 0 0 0] [ ] [1 20 45 36 10 0 0] [ ] [0 24 0 80 0 8 0] [ ] L_1 = [0 6 25 40 30 10 1] [ ] [0 4 0 72 0 36 0] [ ] [0 0 7 28 42 28 7] [ ] [0 0 0 32 0 80 0] [0 0 210 0 0 0 0] [ ] [0 45 0 150 0 15 0] [ ] [1 0 110 0 96 0 3] [ ] L_2 = [0 25 0 140 0 45 0] [ ] [0 0 72 0 126 0 12] [ ] [0 7 0 126 0 77 0] [ ] [0 0 30 0 160 0 20] [0 0 0 672 0 0 0] [ ] [0 36 150 240 180 60 6] [ ] [0 80 0 448 0 144 0] [ ] L_3 = [1 40 140 216 180 80 15] [ ] [0 72 0 432 0 168 0] [ ] [0 28 126 224 196 84 14] [ ] [0 32 0 480 0 160 0] [0 0 0 0 280 0 0] [ ] [0 10 0 180 0 90 0] [ ] [0 0 96 0 168 0 16] [ ] L_4 = [0 30 0 180 0 70 0] [ ] [1 0 126 0 144 0 9] [ ] [0 42 0 196 0 42 0] [ ] [0 0 160 0 120 0 0] [0 0 0 0 0 240 0] [ ] [0 0 15 60 90 60 15] [ ] [0 8 0 144 0 88 0] [ ] L_5 = [0 10 45 80 70 30 5] [ ] [0 36 0 168 0 36 0] [ ] [1 28 77 84 42 8 0] [ ] [0 80 0 160 0 0 0] [0 0 0 0 0 0 21] [ ] [0 0 0 6 0 15 0] [ ] [0 0 3 0 16 0 2] [ ] L_6 = [0 1 0 15 0 5 0] [ ] [0 0 12 0 9 0 0] [ ] [0 7 0 14 0 0 0] [ ] [1 0 20 0 0 0 0] [1 112 210 672 280 240 21] [ ] [1 48 50 32 -40 -80 -11] [ ] [1 16 2 -32 -8 16 5] [ ] P = [1 0 -6 0 8 0 -3] [ ] [1 -8 2 16 -8 -8 5] [ ] [1 -24 50 -16 -40 40 -11] [ ] [1 -56 210 -336 280 -120 21] [1 21 210 840 420 42 2] [ ] [1 9 30 0 -30 -9 -1] [ ] [1 5 2 -24 4 10 2] [ ] Q = [1 1 -10 0 10 -1 -1] [ ] [1 -3 -6 24 -12 -6 2] [ ] [1 -7 14 0 -14 7 -1] [ ] [1 -11 50 -120 100 -22 2] [0 21 0 0 0 0 0] [ ] [1 0 20 0 0 0 0] [ ] [0 2 3 16 0 0 0] [ ] Ls1 = [0 0 4 9 8 0 0] [ ] [0 0 0 16 3 2 0] [ ] [0 0 0 0 20 0 1] [ ] [0 0 0 0 0 21 0] [0 0 210 0 0 0 0] [ ] [0 20 30 160 0 0 0] [ ] Ls2 = [0 4 24 126 48 8 0] [ ] [0 0 32 96 78 3 1] [ ] [0 0 0 160 30 20 0] [ ] [0 0 0 0 210 0 0] [0 0 0 840 0 0 0] [ ] [0 0 160 360 320 0 0] [ ] [0 16 96 504 192 32 0] [ ] Ls3 = [1 9 126 432 252 18 2] [ ] [0 16 96 504 192 32 0] [ ] [0 0 160 360 320 0 0] [ ] [0 0 0 840 0 0 0] [0 0 0 0 420 0 0] [ ] [0 0 0 320 60 40 0] [ ] [0 0 64 192 156 6 2] [ ] Ls4 = [0 8 48 252 96 16 0] [ ] [1 3 78 192 142 3 1] [ ] [0 20 30 320 30 20 0] [ ] [0 0 210 0 210 0 0] [0 0 0 0 0 42 0] [ ] [0 0 0 0 40 0 2] [ ] [0 0 0 32 6 4 0] [ ] Ls5 = [0 0 8 18 16 0 0] [ ] [0 2 3 32 3 2 0] [ ] [1 0 20 0 20 0 1] [ ] [0 21 0 0 0 21 0] [0 0 0 0 0 0 2] [ ] [0 0 0 0 0 2 0] [ ] [0 0 0 0 2 0 0] [ ] Ls6 = [0 0 0 2 0 0 0] [ ] [0 0 1 0 1 0 0] [ ] [0 1 0 0 0 1 0] [ ] [1 0 0 0 0 0 1]