## NOTE: This database has moved to Gabriele Nebe's web site in Aachen.

Keywords: unimodular lattices, tables, minimal norm, quadratic forms

Part of the Catalogue of Lattices, which is a joint project of Gabriele Nebe, University of Ulm (nebe@mathematik.uni-ulm.de) and Neil J. A. Sloane, AT&T Labs-Research (njas@research.att.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 3a-4 N - 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 and P_48n At least 3 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 M N - 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 -

home | people | projects | research areas | resources |