Volume of ball and Hamming-type bounds for Stiefel manifold with Euclidean distance

We compute the volume of the Stiefel manifold induced by the canonical embedding of the manifold as a surface in a Euclidean hypersphere and taking the corresponding Euclidean/chordal distance. Exploiting a power series expansion of the volume element, the volume of a small metric ball under the chordal distance is evaluated. Evaluating the volume of a metric ball is critical to derive Hamming-type bounds. Using a spherical embedding argument, we provide results generalizing previously known bounds on codes in the Grassmann manifold and the unitary group.