  This paper considers a distributed storage system, where multiple storage
nodes can be reconstructed simultaneously at a centralized location. This
centralized multi-node repair (CMR) model is a generalization of regenerating
codes that allow for bandwidth-efficient repair of a single failed node. This
work focuses on the trade-off between the amount of data stored and repair
bandwidth in this CMR model. In particular, repair bandwidth bounds are derived
for the minimum storage multi-node repair (MSMR) and the minimum bandwidth
multi-node repair (MBMR) operating points. The tightness of these bounds are
analyzed via code constructions. The MSMR point is characterized through codes
achieving this point under functional repair for general set of CMR parameters,
as well as with codes enabling exact repair for certain CMR parameters. The
MBMR point, on the other hand, is characterized with exact repair codes for all
CMR parameters for systems that satisfy a certain entropy accumulation
property. Finally, the model proposed here is utilized for the secret sharing
problem, where the codes for the multi-node repair problem is used to construct
communication efficient secret sharing schemes with the property of bandwidth
efficient share repair.
