**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 10^{20} 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 | - |