Find optimal Stiefel codes realized as group orbits

Identify additional optimal codes in Stiefel manifolds StF(d,r) that can be realized as orbits of finite groups under the natural symmetries given by left multiplication by linear isometries and right multiplication by phased permutations.

Background

Beyond the special cases where Stiefel manifolds have a group structure (r=d), Stiefel manifolds admit rich symmetry actions: left multiplication by linear isometries and right actions by phased permutations. The authors used a subgroup of these symmetries to construct orthoplex codes in every complex Stiefel manifold (Theorem 18).

The open problem asks whether more optimal Stiefel codes can be obtained as finite group orbits under these symmetries, extending the orbit-construction paradigm beyond the examples given.