Rank weight hierarchy of some classes of cyclic codes

We study the rank weight hierarchy, thus in particular the rank metric, of cyclic codes over the finite field F_qm, q a prime power, m ≤ 2. We establish the rank weight hierarchy for [n, n - 1] cyclic codes and characterize [n, k] cyclic codes of rank metric 1 when (1) k = 1, (2) n and q are coprime, and (3) the characteristic char(F_q) divides n. Finally, for n and q coprime, cyclic codes of minimal r-rank are characterized, and a refinement of the Singleton bound for the rank weight is derived.