MDS Array Codes With (Near) Optimal Repair Bandwidth for All Admissible Repair Degrees

Abundant high-rate
$(n, k)$
minimum storage regenerating (MSR) codes have been reported in the literature. However, most of them require contacting all the surviving nodes during a node repair process, resulting in a repair degree of
$d=n-1$
. In practical systems, it may not always be feasible to connect and download data from all surviving nodes, as some nodes may be unavailable. Therefore, there is a need for MSR code constructions with a repair degree of
$d < n-1$
. Up to now, only a few
$(n, k)$
MSR code constructions with repair degree
$d < n-1$
have been reported, some have a large sub-packetization level, a large finite field, or restrictions on the repair degree
$d$
. In this paper, we propose a new
$(n, k)$
MSR code construction that works for any repair degree
$d>k$
, and has a smaller sub-packetization level or finite field than some existing constructions. Additionally, in conjunction with a previous generic transformation to reduce the sub-packetization level, we obtain an MDS array code with a small sub-packetization level and
$(1+\epsilon)$
-optimal repair bandwidth (i.e.,
$(1+\epsilon)$
times the optimal repair bandwidth) for repair degree
$d=n-1$
. This code outperforms some existing ones in terms of either the sub-packetization level or the field size.