Codes and designs in Grassmannian spaces

The notion of t-design in a Grassmannian space G"m","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 L^2(G"m","n^o) of square integrable functions on G"m","n^o, the set of oriented Grassmanianns.