A catalogue of Perfect Lattices (Prepared jointly with C. Batut)
(Maintained by the second author alone since September 2010.)
See also
A Catalogue of Lattices ,
written by Gabriele Nebe and Neil Sloane.

About perfect lattices.
(The second file is a text file.)

Perfection: An introductory paper about perfect lattices.
dvi ,
postscript ,
pdf .

Statistics in dimension 8
: Statistics on perfect 8dimensional lattices.

Eutactic and semieutactic perfect, $8$dimensional lattices,
after Cordian RIENER ,
"On extreme forms in dimension 8",
J. Th. Nombres Bordeaux 18 (2006), 677682.
dvi ,
postscript ,
pdf .

Numerical data for the paper above:
read_me ,
eutax8 .

Dimensions 2 to 7.
All lattices are extreme except four:
P_6^4 and P_7^{26}, semieutactic; P_7^{18}, P_7^{29}, only weakly eutactic.

Dimensions 2 to 8: perfect and dualextreme lattices.
Textfile (dualextremeperf.txt)

Tables on perfect lattices in dimensions 2 to 8.
Two global tables for the 10916 perfect,
8dimensional lattices, following one table
for dimensions 2 to 7.
Next the
correspondence between the notation above and the old notation
[1175 Laihem lattices lh(i), 53 Baril lattices bari(i),
9542 Napias lattices nap(i), and 146 Batut lattices batu(i)],
both for perfection and eutaxy properties.

The file contains 6 vectors new2oldp8d{k},
k=2,...,7;
new2oldp8d{k}[i] is a 2component vector [m,j], m=1,2,3,4, and we have
p8d{k}[i]=lh(j) if m=1, bari(j) if m=2, nap(j) if m=3, batu(j) if m=4.

The file contains 4 vectors ,
old2newlh, bari, nap, batu, of 2components vectors [m,j], m=2,...,7,
and, say, if old2newnap[i]=[k,j], then p8d{k}[j]=nap(i).

For the sake of generality we depart from Riener's notation: 0 means
for not weakly eutactic (though 0 does not occur for perfect lattices),
1 means weakly eutactic but not semieutactic, 2 means semieutactic
but not eutactic, and 3 means eutactic. The file
contains six files eutp8d{k} ; eutp8d{k}[i] (=1, 2, or 3) gives
the eutactic character of p8d{k}[i].

Tables of various (very often perfect) lattices in PARIGP format.
(Creation: Nov. 2004.)
These tables can be loaded under PARIGP;
the comments can be read with any editor.

Voronoi graphs in dimensions 2 to 7; minimal classes in dimensions 2 to 4.
(Dec. 2004; modif. Nov. 11th, 2013;
files on minimal classes expanded from January 16th, 2017 onwards).
The three PARIGP files can be downloaded under PARIGP;
the comments can be read with an editor (e.g., emacs).

Strongly eutactic and strongly perfect lattices.
The two PARIGP files can be loaded under PARIGP;
the comments can be read with an editor (e.g., emacs).

Lattices and spherical designs (last update: March 6., 2014).
dvi ,
postscript ,
pdf .

Strongly (semi)eutactic lattices
(dimensions 2 to 6: complete except possibly semieut. in dim. 6;
7,8,9,10: examples).

Known strongly perfect lattices in dimensions 2 to 26.
N.B. This list in dimensions up to 23 is that of Venkov's original paper
constructed by Batut and Venkov and published
in "L'Enseignement Mathématique" in 2001, except for two
16dimensional lattices found in 2018 by S. Hu and G. Nebe, as part
of a series of lattices dual to oneanother in dimensions 4^m, m=2,3,...

Varia

Some lattices of Roland Bacher
(Nov. 28th, 2008)
after Invent. Math. 130 (1997), 153158:
Lattices in dimensions 28 to 25.

Unimodular lattices of minimum 3 in dimensions 23 to 28
(May 14th, 2010)
after Bacher and Venkov (plus two higher dimensional examples);
put in bases of minmal vectors on November 11th, 2023:
User's guide
dvi ,
postscript ,
pdf ;
GPfile.
Sept. 26th,2020.
I just hear from Bill Allombert that in a joint work
with Gaëtan Chenevier, the classification of unimodular lattices
of minimum 3 has been pushed to dimension 29. There are 10092 such lattices,
9987 of general type and 105 of exceptional type.
Data can now be downloaded at
http://gaetan.chenevier.perso.math.cnrs.fr/pub.html .
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