  So far, there is no polynomial-time list decoding algorithm (beyond half the
minimum distance) for Gabidulin codes. These codes can be seen as the
rank-metric equivalent of Reed--Solomon codes. In this paper, we provide bounds
on the list size of rank-metric codes in order to understand whether
polynomial-time list decoding is possible or whether it works only with
exponential time complexity. Three bounds on the list size are proven. The
first one is a lower exponential bound for Gabidulin codes and shows that for
these codes no polynomial-time list decoding beyond the Johnson radius exists.
Second, an exponential upper bound is derived, which holds for any rank-metric
code of length $n$ and minimum rank distance $d$. The third bound proves that
there exists a rank-metric code over $\Fqm$ of length $n \leq m$ such that the
list size is exponential in the length for any radius greater than half the
minimum rank distance. This implies that there cannot exist a polynomial upper
bound depending only on $n$ and $d$ similar to the Johnson bound in Hamming
metric. All three rank-metric bounds reveal significant differences to bounds
for codes in Hamming metric.
