New characterization of full weight spectrum one-orbit cyclic subspace codes

Castello $\mathit{\text{et}}\mathit{\text{al}}$. [J. Comb. Theory Ser. A, 212, 106005 (2025)] provided a complete classification for full weight spectrum (FWS) one-orbit cyclic subspace codes. In this paper, we determine the weight distributions of a family of FWS codes and exhibit some equivalence classes of FWS codes under certain conditions. Furthermore, we provide a complete classification for $r$-FWS codes.