Unimodular Lattices, Together With A Table of the Best Such Lattices

 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.

Unimodular Lattices

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

Table of Highest Minimal Norm of Unimodular Lattices

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 -

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