Flag orbit codes and their expansion to Stiefel codes

We discuss group orbits codes in homogeneous spaces for the unitary group, known as flag manifolds. The distances used to describe the codes arise from embedding the flag manifolds into Euclidean hyperspheres, providing a generalization of the spherical embedding of Grassmann manifolds equipped with the so-called chordal distance. Flag orbits are constructed by acting with a unitary representation of a finite group. In the construction, the center of the finite group has no effect, and thus it is sufficient to consider its inner automorphism group. Accordingly, some explicit constructions from projective unitary representations of finite groups in 2 and 4 dimensions are described. We conclude with examples of codes on the Stiefel manifold constructed as orbits of the linear representation of the projective groups, and thus expansion of the flag codes considered.