Rack-Aware Regenerating Codes with Fewer Helper Racks

We consider the rack-aware storage system where
$n$
nodes are organized in
$\bar{n}$
racks each containing
$u$
nodes, and any
$k$
nodes can retrieve the stored file. Moreover, any single node erasure can be recovered by downloading data from
$\bar{d}$
helper racks as well as the remaining
$u-1$
nodes in the same rack. Previous work mostly focuses on minimizing the cross-rack repair bandwidth under the condition
$\bar{d}\geq\bar{k}$
, where
$\bar{k}=\lfloor\frac{k}{u}\rfloor$
. However,
$\bar{d}\geq\bar{k}$
is not an intrinsic condition for the rack-aware storage model. Reducing
$\bar{d}$
can improve the repair efficiency in practice and bring more flexibility into the repair process. We establish a tradeoff between the storage overhead and cross-rack repair bandwidth for the more interesting case
$\bar{d} < \bar{k}$
, and explicitly construct codes with parameters lying on the tradeoff curve respectively at the minimum storage point and minimum bandwidth point. The codes are scalar or have sub-packetization
$\bar{d}$
, and operate over finite fields of size comparable to
$n$
. Moreover, they remove the restriction of MBR codes having rate less than
$\frac{1}{2}$
and that of high-rate MSR codes having exponential sub-packetization level.