## 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.
1. #### 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 8-dimensional lattices.
• Eutactic and semi-eutactic perfect, \$8\$-dimensional lattices, after Cordian RIENER , "On extreme forms in dimension 8", J. Th. Nombres Bordeaux 18 (2006), 677-682. 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}, semi-eutactic; P_7^{18}, P_7^{29}, only weakly eutactic.
• Dimensions 2 to 8: perfect and dual-extreme lattices. Text-file (dualextremeperf.txt)

2. #### Tables on perfect lattices in dimensions 2 to 8.

Two global tables for the 10916 perfect, 8-dimensional lattices, following one table for dimensions 2 to 7.
3. 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 2-component 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 2-components 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 semi-eutactic, 2 means semi-eutactic 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].

6. #### Strongly eutactic and strongly perfect lattices. The two PARI-GP files can be loaded under PARI-GP; 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 semi-eut. 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 16-dimensional 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,...

7. #### Varia

• Some lattices of Roland Bacher (Nov. 28th, 2008) after Invent. Math. 130 (1997), 153--158:
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 ; GP-file.

• 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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