Classify optimal finite subgroup codes in O(d) and U(d) for general d

Determine, for general dimension d, which finite subgroups of the orthogonal group O(d) and the unitary group U(d) form optimal codes in the Stiefel manifold StF(d,d) under the chordal (Frobenius) distance metric.

Background

The paper studies optimal codes in Stiefel manifolds with chordal distance and shows several constructions that are optimal by meeting Rankin-type bounds. In low dimensions, the authors observed that many optimal codes in O(d) and U(d) arise as finite subgroups (e.g., cyclic or dihedral subgroups for O(2), and optimizers reported for U(2)).

This motivates a general classification problem: when r=d, StF(d,d) equals O(d) (over R) or U(d) (over C), and codewords are orthogonal or unitary matrices. The open question asks which finite subgroups in these groups yield globally optimal Stiefel codes for arbitrary d.