  We introduce the concepts of complex Grassmannian codes and designs. Let
G(m,n) denote the set of m-dimensional subspaces of C^n: then a code is a
finite subset of G(m,n) in which few distances occur, while a design is a
finite subset of G(m,n) that polynomially approximates the entire set. Using
Delsarte's linear programming techniques, we find upper bounds for the size of
a code and lower bounds for the size of a design, and we show that association
schemes can occur when the bounds are tight. These results are motivated by the
bounds for real subspaces recently found by Bachoc, Coulangeon and Nebe, and
the bounds generalize those of Delsarte, Goethals and Seidel for codes and
designs on the complex unit sphere.
