## 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 24 is that of Venkov's original paper constructed by Batut and Venkov and published in "L'Enseignement Mathématique" in 2001. At the date of December 31st, 2016, no new lattice in these dimensions has been found. However, in the preprint (arXiv:math1805.01196, 3. May 2018; see also 17. May 2019), S. Hu and G. Nebe, have onstructed a new infinite series of strongly perfect lattices, including two in dimension16.

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):
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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