Keywords: unimodular lattices, tables, minimal norm, quadratic forms
Part of the Catalogue of Lattices,
which is a joint project of
Gabriele Nebe, RWTH Aachen university
(nebe@math.rwth-aachen.de)
and
Neil J. A. Sloane,
(njasloane@gmail.com).
See also our home pages:
Gabriele Nebe
and
Neil Sloane.
A unimodular lattice is an integral lattice which is its own dual. In other words, det L = 1 and u.v is an integer for all u, v in L.
If a lattice is unimodular its entry should indicate this by saying:
%DETERMINANT
1
%PROPERTIES
INTEGRAL=1
The table give the highest possible minimal norm (mu) of an n-dimensional unimodular lattice and the names of lattices meeting the bound (also, whenever possible, links to files containing these lattices).
NOTE: THIS TABLE IS UNDER CONSTRUCTION.
NOTE: THIS TABLE IS UNDER CONSTRUCTION.
NOTE: THIS TABLE IS UNDER CONSTRUCTION.
NOTE: THIS TABLE IS UNDER CONSTRUCTION.
Footnote (a): For these dimensions I have written down in one of my notebooks that minimal norm 4 exists. But I cannot recall the construction -- perhaps some reader of this page can help? - NJAS
| Dim n | mu | Lattice(s) | Remarks |
|---|---|---|---|
| 1 | 1 | Z | The 1-dim integer lattice |
| 2 | 1 | Z^2 | simple square lattice |
| 3 | 1 | Z^3 | simple cubic lattice |
| 4 | 1 | Z^4 | 4-dim simple cubic lattice |
| 5 | 1 | Z^5 | |
| 6 | 1 | Z^6 | |
| 7 | 1 | Z^7 | |
| 8 | 2 | E8 | The root lattice E8 |
| 9 | 1 | E8+Z | |
| 10 | 1 | E8+Z^2 | |
| 11 | 1 | E8+Z^3 | |
| 12 | 2 | D12+ | The root lattice D12 glued up |
| 13 | 1 | E8+Z^5 | |
| 14 | 2 | E7^2+ | |
| 15 | 2 | A15+ | |
| 16 | 2 | N | - |
| 17 | 2 | N | - |
| 18 | 2 | N | - |
| 19 | 2 | N | - |
| 20 | 2 | N | - |
| 21 | 2 | N | - |
| 22 | 2 | N | - |
| 23 | 3 | O23 | The shorter Leech lattice |
| 24 | 4 | LAMBDA24 | Leech lattice (see also the 23 Niemeier lattices) |
| 25 | 2 | 25MIN2, 25MIN2a | Many lattices: see Borcherds' complete list |
| 26 | 3 | Borcherds' S_26 | Unique lattice |
| 27 | 3 | Borcherds' T_27 | 3 lattices. |
| 28 | 3 | 28MIN3 | 38 lattices |
| 29 | 3 | dim29odd | |
| 30 | 3 | N | - |
| 31 | 3 | dim31odd | |
| 32 | 4 | Koch-Venkov partial list | |
| 33 | 3 | 33MIN3 | There are probably 1020 lattices with min. norm 3. Reference: postscript, pdf. |
| 34 | 3 | 34MIN3 | 4 is impossible |
| 35 | 3 | 35MIN3 | 4 is impossible |
| 36 | 4 | Sp4(4)D8.4, dim36min4b | Found by G. Nebe and by Philippe Gaborit |
| 37 | 3-4 | N | - |
| 38 | 4 | dim38min4 | Found by Philippe Gaborit |
| 39 | 4 | GH39 | Found by T. A. Gulliver and M. Harada, Nov. 1998 |
| 40 | 4 | (U5(2) x 2^(1+4)_-.Alt_5).2 | One of several even examples known. An odd example. |
| 41 | 4 | Har41 | Harada from code mod 4 |
| 42 | 4 | P42.1, dim42min4 | |
| 43 | 4 | R43 | Found by Gaborit and Otmani |
| 44 | 4 | HKO44 | Found by M. Harada and M. Ozeki, Apr. 1998 |
| 45 | 4 | B45 | Found by Philippe Gaborit |
| 46 | 4 | H46 | Found by M. Harada, Jun 19, 2001 |
| 47 | 4 | H47 | Found by M. Harada, Jun 19, 2001 |
| 48 | 6 | P_48p, P_48q, P_48n, and P_48m | At least 4 lattices |
| 49 | 4-5 | N | - |
| 50 | 4-5 | N | - |
| 51 | 4-5 | N | - |
| 52 | 5 | dim52min5 | Found by Philippe Gaborit |
| 53 | 4-5 | N | - |
| 54 | 5 | dim54min5 | Found by Philippe Gaborit |
| 55 | -5 | N | - |
| 56 | 6 | L_56,2(M), L_56,2(tilde(M)), dim56min6 | |
| 57 | M | N | - |
| 58 | M | N | - |
| 59 | M | N | - |
| 60 | 6 | P60q, HKO60 | Found by Gaborit and by Harada-Kitazume-Ozeki |
| 61 | M | N | - |
| 62 | M | N | - |
| 63 | M | N | - |
| 64 | 6 | L8,2.otimes.L_32,2 | Found by G. Nebe |
| 65 | M | N | - |
| 66 | M | N | - |
| 67 | M | N | - |
| 68 | 6 | HKO68, dim68min6 | Found by Harada-Kitazume-Ozeki and by Gaborit |
| 69 | M | N | - |
| 70 | M | N | - |
| 71 | M | N | - |
| 72 | 8 | Gamma72 | found by G. Nebe |
| 73 | M | N | - |
| 74 | M | N | - |
| 75 | M | N | - |
| 76 | M | N | - |
| 77 | M | N | - |
| 78 | M | N | - |
| 79 | M | N | - |
| 80 | 8 | L_80, M_80 | At least 2 lattices |
| 81 | M | N | - |
| D | M | N | - |
| D | M | N | - |
| D | M | N | - |
| D | M | N | - |
| D | M | N | - |
| D | M | N | - |