  In a modern distributed storage system, storage nodes are organized in racks,
and the cross-rack communication dominates the system bandwidth. In We study
the rack-aware storage system where all storage nodes are organized in racks
and within each rack the nodes can communicate freely without taxing the system
bandwidth. Rack-aware regenerating codes (RRCs) were proposed for minimizing
the repair bandwidth for single erasures. In the initial setting of RRCs, the
repair of a single node requires the participation of all the remaining nodes
in the rack containing the failed node as well as a large number of helper
racks containing no failures. Consequently, the repair may be infeasible in
front of multiple node failures. In this work, a relaxed repair model that can
tolerate multiple node failures by simultaneously reducing the intra-rack
connections and cross-rack connections is proposed. A tradeoff between the
storage and repair bandwidth under the relaxed repair model is derived, and
parameters of the two extreme points on the tradeoff curve are characterized
for the minimum storage and minimum bandwidth respectively. Moreover, two codes
corresponding to the extreme points are explicitly constructed over the fields
of size comparable to the code length and with the lowest sub-packetization.
Finally, for the convenience of practical use, systematic encoding processes
for the two codes are also established.
