Codes and designs in Grassmannian spaces

AbstractThe notion of t-design in a Grassmannian space Gm,n was introduced by the first and last authors and G. Nebe in a previous paper. In the present work, we give a general lower bound for the size of such designs. The method is inspired by Delsarte, Goethals and Seidel work in the case of spherical designs. This leads us to introduce a notion of f-code in Grassmannian spaces, for which we obtain upper bounds, as well as a kind of duality tight-designs/tight-codes. The bounds are in terms of the dimensions of the irreducible representations of the orthogonal group O(n) occurring in the decomposition of the space L2(Gm,n°) of square integrable functions on Gm,n°, the set of oriented Grassmanianns

A new Euclidean tight 6-design

We give a new example of Euclidean tight 6-design in $\mathbb R^{22}$.Comment: 9 page

Cubature formulas, geometrical designs, reproducing kernels, and Markov operators

Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way, several known results for spheres in Euclidean spaces, involving cubature formulas for polynomial functions and spherical designs, are shown to generalize to large classes of finite measure spaces $(\Omega,\sigma)$ and appropriate spaces of functions inside $L^2(\Omega,\sigma)$. The last section points out how spherical designs are related to a class of reflection groups which are (in general dense) subgroups of orthogonal groups