# MAPLE. May 30, 2006. # # 196,560 shortest vectors in the Leech lattice. # Example 1 in HCD table of spherical designs by Munemasa. # # # These are the shortest vectors in the Leech lattice. d := 6; v := [1 , 4600 , 47104 , 93150 , 47104 , 4600 , 1]; verts := 196560; Q := matrix([ [1 , 24 , 299 , 2576 , 17250 , 95680 , 80730], [1 , 12 , 65 , 196 , 285 , -208 , -351 ], [1 , 6 , 13/2 , -77/2 ,-1875/16 , 65/2 ,1755/16], [1 , 0 , -13 , 0 , 90 , 0 , -78 ], [1 , -6 , 13/2 , 77/2 , -1875/16, -65/2 ,1755/16], [1 , -12 , 65 , -196 , 285 , 208 , -351 ], [1 , -24 , 299 , -2576 , 17250 , -95680 , 80730]]); # This tridiagonal matrix, L_1-star, allows us to obtain all # parameters from just the cosines. L := matrix(d+1,d+1,[ [0 , 24 , 0 , 0 , 0 , 0 , 0 ], [1 , 0 , 23 , 0 , 0 , 0 , 0 ], [0 ,24/13, 0 ,288/13 , 0 , 0 , 0 ], [0 , 0 , 18/7, 0 , 150/7 , 0 , 0 ], [0 , 0 , 0 , 16/5 , 0 , 104/5 , 0 ], [0 , 0 , 0 , 0 , 15/4 , 0 , 81/4], [0 , 0 , 0 , 0 , 0 , 24 , 0 ] ]); [0 4600 0 0 0 0 0] [ ] [1 891 2816 891 0 1 0] [ ] [0 275 2025 2025 275 0 0] [ ] L_1 = [0 44 1024 2464 1024 44 0] [ ] [0 0 275 2025 2025 275 0] [ ] [0 1 0 891 2816 891 1] [ ] [0 0 0 0 0 4600 0] [0 0 47104 0 0 0 0] [ ] [0 2816 20736 20736 2816 0 0] [ ] [1 2025 15400 22275 7128 275 0] [ ] L_2 = [0 1024 11264 22528 11264 1024 0] [ ] [0 275 7128 22275 15400 2025 1] [ ] [0 0 2816 20736 20736 2816 0] [ ] [0 0 0 0 47104 0 0] [0 0 0 93150 0 0 0] [ ] [0 891 20736 49896 20736 891 0] [ ] [0 2025 22275 44550 22275 2025 0] [ ] L_3 = [1 2464 22528 43164 22528 2464 1] [ ] [0 2025 22275 44550 22275 2025 0] [ ] [0 891 20736 49896 20736 891 0] [ ] [0 0 0 93150 0 0 0] [0 0 0 0 47104 0 0] [ ] [0 0 2816 20736 20736 2816 0] [ ] [0 275 7128 22275 15400 2025 1] [ ] L_4 = [0 1024 11264 22528 11264 1024 0] [ ] [1 2025 15400 22275 7128 275 0] [ ] [0 2816 20736 20736 2816 0 0] [ ] [0 0 47104 0 0 0 0] [0 0 0 0 0 4600 0] [ ] [0 1 0 891 2816 891 1] [ ] [0 0 275 2025 2025 275 0] [ ] L_5 = [0 44 1024 2464 1024 44 0] [ ] [0 275 2025 2025 275 0 0] [ ] [1 891 2816 891 0 1 0] [ ] [0 4600 0 0 0 0 0] [0 0 0 0 0 0 1] [ ] [0 0 0 0 0 1 0] [ ] [0 0 0 0 1 0 0] [ ] L_6 = [0 0 0 1 0 0 0] [ ] [0 0 1 0 0 0 0] [ ] [0 1 0 0 0 0 0] [ ] [1 0 0 0 0 0 0] [1 4600 47104 93150 47104 4600 1] [ ] [1 2300 11776 0 -11776 -2300 -1] [ ] [1 1000 1024 -4050 1024 1000 1] [ ] P = [1 350 -704 0 704 -350 -1] [ ] [1 76 -320 486 -320 76 1] [ ] [1 -10 16 0 -16 10 -1] [ ] [1 -20 64 -90 64 -20 1] [1 24 299 2576 17250 95680 80730] [ ] [1 12 65 196 285 -208 -351 ] [ ] [ -1875 1755 ] [1 6 13/2 -77/2 ----- 65/2 ---- ] [ 16 16 ] [ ] Q = [1 0 -13 0 90 0 -78 ] [ ] [ -1875 1755 ] [1 -6 13/2 77/2 ----- -65/2 ---- ] [ 16 16 ] [ ] [1 -12 65 -196 285 208 -351 ] [ ] [1 -24 299 -2576 17250 -95680 80730] [0 24 0 0 0 0 0 ] [ ] [1 0 23 0 0 0 0 ] [ ] [ 24 288 ] [0 -- 0 --- 0 0 0 ] [ 13 13 ] [ ] Ls1 = [0 0 18/7 0 150/7 0 0 ] [ ] [0 0 0 16/5 0 104/5 0 ] [ ] [0 0 0 0 15/4 0 81/4] [ ] [0 0 0 0 0 24 0 ] [0 0 299 0 0 0 0 ] [ ] [0 23 0 276 0 0 0 ] [ ] [1 0 286/7 0 1800/7 0 0 ] [ ] Ls2 = [0 18/7 0 55 0 1690/7 0 ] [ ] [ 156 1859 4563 ] [0 0 --- 0 ---- 0 ---- ] [ 35 28 20 ] [ ] [0 0 0 13/2 0 585/2 0 ] [ ] [0 0 0 0 195/4 0 1001/4] [0 0 0 2576 0 0 0 ] [ ] [0 0 276 0 2300 0 0 ] [ ] [ 288 6160 ] [0 --- 0 ---- 0 2080 0 ] [ 13 13 ] [ ] Ls3 = [1 0 55 0 2475/4 0 7605/4] [ ] [0 16/5 0 462/5 0 12402/5 0 ] [ ] [ 7155 33957 ] [0 0 13/2 0 ---- 0 ----- ] [ 16 16 ] [ ] [0 0 0 182/3 0 7546/3 0 ] [0 0 0 0 17250 0 0 ] [ ] [0 0 0 2300 0 14950 0 ] [ ] [ 53625 ] [0 0 1800/7 0 ----- 0 26325/2] [ 14 ] [ ] [ 465075 ] Ls4 = [0 150/7 0 2475/4 0 ------ 0 ] [ 28 ] [ ] [ 1859 686745 451737 ] [1 0 ---- 0 ------ 0 ------ ] [ 28 224 32 ] [ ] [ 7155 268785 ] [0 15/4 0 ---- 0 ------ 0 ] [ 16 16 ] [ ] [ 96525 453915 ] [0 0 195/4 0 ----- 0 ------ ] [ 32 32 ] [0 0 0 0 0 95680 0 ] [ ] [0 0 0 0 14950 0 80730 ] [ ] [0 0 0 2080 0 93600 0 ] [ ] [ 465075 ] Ls5 = [0 0 1690/7 0 ------ 0 315315/4] [ 28 ] [ ] [0 104/5 0 12402/5 0 465894/5 0 ] [ ] [ 268785 1257399 ] [1 0 585/2 0 ------ 0 ------- ] [ 16 16 ] [ ] [0 24 0 7546/3 0 279422/3 0 ] [0 0 0 0 0 0 80730 ] [ ] [0 0 0 0 0 80730 0 ] [ ] [0 0 0 0 26325/2 0 135135/2] [ ] Ls6 = [0 0 0 7605/4 0 315315/4 0 ] [ ] [ 4563 451737 10621611] [0 0 ---- 0 ------ 0 --------] [ 20 32 160 ] [ ] [ 33957 1257399 ] [0 81/4 0 ----- 0 ------- 0 ] [ 16 16 ] [ ] [ 453915 2121405 ] [1 0 1001/4 0 ------ 0 ------- ] [ 32 32 ]