Toward Lower Repair Bandwidth and Optimal Repair Complexity of Piggybacking Codes With Small Sub-Packetization

As a special class of array codes, piggybacking codes are maximum distance separable (MDS) codes that can achieve low repair bandwidth for single-node erasure. An
$(n,k,m)$
piggybacking code contains
k
data nodes and
$r=n-k$
parity nodes, each node stores
m
symbols. In this paper, we propose a new piggybacking design by jointly designing piggyback functions for both data node repair and parity node repair to reduce the repair bandwidth. Our piggybacking codes can support flexible sub-packetization
m
with
$2\leq m\leq r$
. When
$m=r$
, we derive a new lower bound of repair bandwidth based on our piggybacking structure and show that this lower bound is lower than the corresponding lower bounds derived from the existing piggybacking codes for
$5 < r\ll k$
. When
$m < r$
, we show that our piggybacking codes have lower repair bandwidth than the existing piggybacking codes for all the evaluated high-code-rate parameters with
$10\leq r\leq 20$
,
$k=100$
and
$m\leq 5$
. Moreover, we derive the lower bound of repair complexity defined as the number of multiplication operations required in repairing single-node erasure under general piggybacking framework and show that our codes can achieve the repair complexity lower bound, i.e., our codes have optimal repair complexity.