  Gabidulin codes, serving as the rank-metric counterpart of Reed-Solomon
codes, constitute an important class of maximum rank distance (MRD) codes.
However, unlike the fruitful positive results about the list decoding of
Reed-Solomon codes, results concerning the list decodability of Gabidulin codes
in the rank metric are all negative so far. For example, in contrast to
Reed-Solomon codes, which are always list decodable up to the Johnson bound in
the Hamming metric, Raviv and Wachter-Zeh (IEEE TIT, 2016 and 2017) constructed
a class of Gabidulin codes that are not even combinatorially list decodable
beyond the unique decoding radius in the rank metric. Proving the existence of
Gabidulin codes with good combinatorial list decodability in the rank metric
has remained a long-standing open problem.
  In this paper, we resolve the aforementioned open problem by showing that,
with high probability, random Gabidulin codes over sufficiently large alphabets
attain the optimal generalized Singleton bound for list decoding in the rank
metric. In particular, they achieve list decoding capacity in the rank metric.
  Our work is significantly influenced by the recent breakthroughs in the
combinatorial list decodability of Reed-Solomon codes, especially the work by
Brakensiek, Gopi, and Makam (STOC 2023). Our major technical contributions,
which may hold independent interest, consist of the following: (1) We initiate
the study of ``higher order MRD codes'' and provide a novel unified theory,
which runs parallel to the theory of ``higher order MDS codes'' developed by
BGM. (2) We prove a natural analog of the GM-MDS theorem, proven by Lovett
(FOCS 2018) and Yildiz and Hassibi (IEEE TIT, 2019), which we call the GM-MRD
theorem. In particular, our GM-MRD theorem for Gabidulin codes are strictly
stronger than the GM-MDS theorem for Gabidulin codes, proven by Yildiz and
Hassibi (IEEE TIT, 2019).
