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, i.e., the repair degree is d=n-1. However, connecting and downloading data from all the surviving nodes is not always feasible in practical systems, as some nodes may be unavailable. Therefore, constructions of MSR codes with repair degree $d<n-1$ are also desired. 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, which works for any repair degree d>k, and has a smaller sub-packetization level or finite field than existing constructions. In addition, in conjunction with a previous generic transformation to reduce the sub-packetization level, 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) is also obtained for repair degree d=n-1, which outperforms existing ones in terms of either the sub-packetization level or the field size.