Corrected version of November 14th, 2006 (in PART II, the structure {1,2,3,4,2^2,6} was not previously quoted). ************************************************ * STATISTICS ON PERFECT 8-DIMENSIONAL LATTICES * ************************************************ We display in these tables (I) the numbers of known (*) perfect 8-dimensional lattices with given invariant s, with given minimal norm N and with given order g of the automorphism group (indeed, the kissing number is 2s). (II) the list of the possible structures for L/L' for L perfect and L' generated by 8 minimal vectors of L. (*) The collection of lattices displayed under the names of Laihem, Baril, Napias and Batut is indeed complete (Dutour-Sch\"urmann-Vallentin, 2005; Math. arXiv, NT/0609388). ============================================================================ ============================================================================ PART I ------ 1. LAIHEM LATTICES (1175 lattices) We recall that LH1172 to LH1175 are E_8, A_8^2, D_8 and A_8 respectively. Half kissing numbers: ==================== s = 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 54 56 58 71 120 nb = 330 231 234 120 98 54 43 12 16 8 11 1 4 1 2 1 1 4 1 1 1 1 Minimal norms: ============= N = 2 4 6 8 10 12 16 nb = 3 84 728 328 26 5 1 Order of the automorphism groups: ================================ g = 2 4 6 8 12 16 24 32 48 56 64 72 96 128 144 160 192 288 nb = 545 357 1 126 4 43 12 23 11 2 4 1 5 2 4 1 11 1 g = 384 432 512 576 648 672 768 864 960 1536 3456 5184 62208 80640 nb = 3 1 1 2 1 1 1 1 1 2 1 1 1 1 g = 161280 725760 10321920 696729600 nb = 1 1 1 1 ============================================================================ 2. BARIL LATTICES (53 lattices) Half kissing numbers: ==================== s= 36 37 38 39 40 41 42 51 nb =11 12 10 11 3 3 1 2 Minimal norms: ============= N = 6 8 10 12 nb = 2 45 1 5 Order of the automorphism groups: ================================ g = 2 4 8 12 16 32 48 96 144 288 n =23 15 5 1 2 3 1 1 1 1 ============================================================================ 3. NAPIAS LATTICES (9542 lattices) Half kissing numbers: ==================== s = 36 37 38 39 40 41 42 43 44 45 46 47 nb = 5815 1779 1251 347 232 64 23 14 8 5 1 3 Minimal norms: ============= N = 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 nb = 1 619 3425 3195 1501 509 194 54 29 5 4 2 1 1 2 Order of the automorphism groups: ================================ g = 2 4 8 12 16 20 24 28 32 40 48 56 64 96 160 nb = 6330 2461 542 35 67 1 64 1 20 2 8 2 3 5 1 ============================================================================ 4. BATUT LATTICES (146 lattices) This list was established July, 1999; the numbers given here may be subject to modification. Half kissing numbers: ==================== s = 36 37 38 39 40 42 nb = 98 11 3 23 9 2 Minimal norms: ============= N = 6 8 10 12 14 16 18 20 22 nb = 6 65 45 16 9 2 1 1 1 Order of the automorphism groups: ================================ g = 2 4 8 12 16 24 32 48 72 96 128 192 480 576 nb = 31 57 28 4 2 9 6 3 1 1 1 1 1 1 ============================================================================ 5. ALL LATTICES TOGETHER (1175+9542+53+146 = 10916 known perfect lattices) Half kissing numbers: ==================== s= 36 37 38 39 40 41 42 43 44 45 46 47 48 49 nb= 6254 2033 1498 501 342 121 69 26 24 13 12 4 4 1 s = 50 51 52 54 56 58 71 120 nb = 2 3 1 4 1 1 1 1 Minimal norms: ============= N = 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 nb = 3 85 1355 3863 3267 1527 518 197 55 30 6 4 2 1 1 2 Order of the automorphism groups: ================================ g = 2 4 6 8 12 16 20 24 28 32 40 48 56 64 72 96 128 144 160 nb = 6929 2890 1 701 44 114 1 85 1 52 2 23 4 7 2 12 3 5 2 g = 192 288 384 432 480 512 576 648 672 768 864 960 1536 3456 5184 nb = 12 2 3 1 1 1 3 1 1 1 1 1 2 1 1 g = 62208 80640 161280 725760 10321920 696729600 nb = 1 1 1 1 1 1 ============================================================================ COMMENTS AVERAGE VALUES: ============== Half kissing number : 36.91 Minimal norm : 9.41 Number of automorphisms : 64872.02 Remark: 6 lattices with more than 50 000 automorphisms suffice to make ====== the average value very large. Kissing numbers point to 36 - 37 and minimal norms to 8 - 10. LARGE VALUES: ============ * The nine largest values for the kissing number occur for Laihem lattices: LH1172 = E_8, LH1173= A_8^2, LH1, LH1174 = D_8, ... * A remark of the same kind applies to automorphism groups: the largest correspond to LH1172 = E_8, LH1174 = D_8, LH1175= A_8, LH7, LH1173= A_8^2, LH2, ... ============================================================================ ============================================================================ PART II ------- We consider the set of possible quotients L/L' where L is an 8-dimensional perfect lattice and L' is a sublattice of L generated by 8 independent minimal vectors. A group of oder 4 is written 4 or 2^2 according to whether it is cyclic or non-cyclic, ... We refer to [M3] for the basic facts on the theory (initiated by Watson) of indices. In particular, it is proved there that all existing quotients L/L' can be realized by perfect lattices, and that, up to dimension 8, the root lattices A_n (n=1,2,3), D_4,D_5,D_6,E_6,E_7, and E_8 exhaust all possibilities. The main experimental result for dimension 8 is that among the known perfect lattices in this dimension, all existing systems of quotients consist up to 7 exceptions of the sequences 1,2,3,4,2^2 , 1,2,3,4,2^2,5 , 1,2,3,4,2^2,6 or 1,2,3,4,2^2,5,6 . Here are the exceptional lattices, which all belong to Laihem's list, consisting of the four extensions of perfect, 7-dimensional root lattices and of three lattices having a low density: LH1172 = E_8 : 1, 2, 3, 4, 2^2, 5, 6, 4.2, 2^3, 3^2, 2^4 ; LH1173 = A_8^2 : 1, 2, 3, 4, 2^2, 2^3 (compare E_7) ; LH1174 = D_8 : 1, 2, 2^2, 2^3 ; LH1154, LH1164 : 1, 2, 3 ; LH1171 : 1, 2, 2^2 ; LH1175 = A_8 : 1 . ================================================ We give below the number of lattices in each category corresponding to a maximal index equal to 4, 5, or 6. 1. LAIHEM LATTICES (1168 lattices; 7 other possibilities, see above) maximal index 4 : 708 ; maximal index 5 : 444 ; maximal index 6 : 16 2. BARIL LATTICES (53 lattices) maximal index 4 : 32 ; maximal index 5 : 20 ; maximal index 6 : 1 3. NAPIAS LATTICES (9542 lattices) maximal index 4 : 7227 ; maximal index 5 : 2306 ; maximal index 6 : 9 4. BATUT LATTICES (146 lattices) maximal index 4 : 105 ; maximal index 5 : 39 ; maximal index 6 : 2 5. ALL LATTICES TOGETHER (1168+9542+53+146 = 10909 ; 7 extra lattices) maximal index 4 : 8072 ; maximal index 5 : 2809 ; maximal index 6 : 28 [M3] Sur l'indice d'un sous-r\'eseau (with an appendix by C. Batut), preprint, 1997/98 (see this WEB page), and in R\'e\-seaux euclidiens, designs sph\'eriques et formes modulaires, J. Martinet, \'ed., Monographie de l'Enseignememt Math\'ematique nu. 37, Geneva, 2001, pp. 163 - 211. ============================================================================ ============================================================================