Designs, groups and lattices
Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, p. 25-44

The notion of designs in Grassmannian spaces was introduced by the author and R. Coulangeon, G. Nebe, in [3]. After having recalled some basic properties of these objects and the connections with the theory of lattices, we prove that the sequence of Barnes-Wall lattices hold 6-Grassmannian designs. We also discuss the connections between the notion of Grassmannian design and the notion of design associated with the symmetric space of the totally isotropic subspaces in a binary quadratic space, which is revealed in a certain construction involving the Clifford group.

La notion de designs dans les espaces Grassmanniens a été introduite par l’auteur et R. Coulangeon, G. Nebe dans [3]. Après avoir rappelé les premières propriétés de ces objets et les relations avec la théorie des réseaux, nous montrons que la famille des réseaux de Barnes-Wall contient des 6-designs grassmanniens. Nous discutons également des relations entre cette notion de designs et les designs associés à l’espace symétrique formé des espaces totalement isotropes d’un espace quadratique binaire, qui sont mises en évidence par une certaine construction utilisant le groupe de Clifford.

@article{JTNB_2005__17_1_25_0,
     author = {Bachoc, Christine},
     title = {Designs, groups and lattices},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {1},
     year = {2005},
     pages = {25-44},
     doi = {10.5802/jtnb.474},
     mrnumber = {2152208},
     zbl = {1074.05023},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2005__17_1_25_0}
}
Bachoc, Christine. Designs, groups and lattices. Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 25-44. doi : 10.5802/jtnb.474. http://www.numdam.org/item/JTNB_2005__17_1_25_0/

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