Designs, groups and lattices

Bachoc, Christine

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 25-44.

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.

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.

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