ON A GENERALISATION OF SPHERICAL CODES AND

ABSTRACT. We introduced with C. Bachoc and G. Nebe anotion of design (resp. code) in Grassmannian spaces. This extends the classical notions of spherical design (resp. code), which in this setting corre-sponds to the Grassmannian space $\mathcal{G}_{1,n} $ , the set of lines in $\mathbb{R}^{n} $. This paper will survey on various results and applications, in particular to the study of Rankin invariants of lattices. We also give some bounds for tlzc size of designs and codes which were obtained in collaboration with E. Bannai. 1. INTRODUCTION. This asurvey on recent work with C. Bachoc, E. Bannai and G. Nebe. To start with, let us recall the classical notions of spherical designs and codes (see [5]). We endow the sphere $S^{n-1} $ with its canonical $O(n)$-invariant measure $dx $ , normalized so as $\int_{S^{n-1}}dx=1 $