  Codes in the sum-rank metric have received many attentions in recent years,
since they have wide applications in the multishot network coding, the
space-time coding and the distributed storage. Fundamental bounds, some
explicit or probabilistic constructions of sum-rank codes and their decoding
algorithms have been developed in previous papers. In this paper, we construct
covering codes in the sum-rank metric from covering codes in the Hamming
metric. Then some upper bounds on sizes of covering codes in the sum-rank
metric are presented. Block length functions of covering codes in the sum-rank
metric are also introduced and studied. As applications of our upper bounds on
covering codes in the sum-rank metric and block length functions, several
strong Singleton-like bounds on sum-rank codes are proposed and proved. These
strong Singleton-like bounds are much stronger than the Singleton-like bound
for sum-rank codes, when block lengths are larger and minimum sum-rank
distances are small. An upper bound on sizes of list-decodable codes in the
sum-rank metric is also given, which leads to an asymptotic bound on
list-decodability of sum-rank codes. We also give upper bounds on block lengths
of general MSRD codes.
