J. Martinet Last update: December 1st, 2010

UPDATE FOR SLIDES OF GRAS’S CONGRESS (COMBINATORICS AND LATTICES)

\underline{page 3.} A.-M. Bergé & J. Martinet has appeared : Monatshefte Math. 140 (2003), 179–195. J. Martinet & B. Venkov has appeared : Algebra i Analiz (Saint-Petersburg) 16, 3 (2004), 99–142.

pages 8 and 9. The classification is known for n = 6 and could be completed for n = 7, using P. Elbaz-Vincent, H. Gangl, C. Soulé, Perfect forms and the cohomology of modular groups, submitted; preprint at arXiv:math/1001.0789v1. Some numerical data are available on my homepage

http://www.math.u-bordeaux.fr/ martinet/ ,

“Catalogue of perfect lattices”, nu. 5.

\underline{page 12.} The classification for n = 12 and a partial classification for n = 14 have been obtained by G, Nebe & B. Venkov:

(1) Low dimensional strongly perfect lattices. I: The 12-dimensional case, L’Enseignement Mathématique 51 (2005), 129–163 ;

(2) Low dimensional strongly perfect lattices. III: Dual strongly perfect lattices of dimension 14 , Int. J. Number Theory 6 (2010), 387–409. 7- (= 6-) designs: add to JM

(3) On lattices whose minimal vectors form a 6-design, European J. Combin. 30 (2009), 716–724.

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pages 13 and 14. More results on modular lattices can be read in R. Scharlau, R. Schulze-Pillot, Extremal lattices, in Algorithmic Algebra and Number Theory, B.H. Matzat, G.-M. Greul, G. Hiss ed., Springer-Verlag, Heidelberg (1999), 139–170. G. Nebe has recently constructed an extremal unimodular lattice of dimension 72: An even unimodular 72-dimensional lattice of minimum 8, preprint, Aachen (August 11th, 2010), 10 pp.

\underline{page 15.} ”Probably ...”: this has been solved in Christine Bachoc, Designs, groups and lattices, J. Théorie Nombres Bordeaux 17 (2005), 25–44.

pages 16 and 17. The tables are due to Batut and Venkov. No new strongly perfect lattices in dimensions n ≤ 24 have been found since Venkov’s paper on spherical designs was written.