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We study the rank weight hierarchy, thus in particular the rank metric, ofcyclic codes over the finite field $\mathbb F_{q^m}$, $q$ a prime power, $m\geq 2$. We establish the rank weight hierarchy for $[n,n-1]$ cyclic codes andcharacterize $[n,k]$ cyclic codes of rank metric 1 when (1) $k=1$, (2) $n$ and$q$ are coprime, and (3) the characteristic $char(\mathbb F_q)$ divides $n$.Finally, for $n$ and $q$ coprime, cyclic codes of minimal $r$-rank arecharacterized, and a refinement of the Singleton bound for the rank weight isderived.

Rank weight hierarchy of some classes of cyclic codes