High-rate minimum storage regenerating (MSR) codes are known to require a large sub-packetization level, which can make meta-data management difficult and hinder implementation in practical systems. A few maximum distance separable (MDS) array code constructions have been proposed to attain a much smaller sub-packetization level by sacrificing a bit of repair bandwidth. However, to the best of our knowledge, only one construction by Guruswami et al. can support the repair of a failed node without contacting all the surviving nodes. This construction is certainly of theoretical interest but not yet practical due to its requirement for very large code parameters. In this paper, we propose a generic transformation that can convert any $(\overline{n}, \overline{k})$ MSR code with a repair degree of $\overline{d}<\overline{n}-1$ into another $(n=s\overline{n},k)$ MDS array code that supports $d<n-1$ with a small sub-packetization level and $(1+\epsilon)$-optimal repair bandwidth (i.e., $1+\epsilon$ times the optimal value) under a specific condition. We obtain three MDS array codes with small sub-packetization levels and $(1+\epsilon)$-optimal repair bandwidth by applying this transformation to three known MSR codes. All the new MDS array codes have a small repair degree of $d<n-1$ and work for both small and large code parameters.

翻译：高码率最小存储再生（MSR）码已知需要较大的子包化级别，这会产生元数据管理困难并阻碍在实际系统中的实现。已有几种最大距离可分（MDS）阵列码构造通过牺牲少量修复带宽来获得更小的子包化级别。然而，据我们所知，仅有Guruswami等人提出的一种构造能在不联系所有存活节点的情况下修复失效节点。该构造虽具有理论价值，但因需要极大的码参数而尚不实用。本文提出了一种通用转换方法，可将任意满足修复度$\overline{d}<\overline{n}-1$的$(\overline{n}, \overline{k})$ MSR码转换为$(n=s\overline{n},k)$ MDS阵列码，在特定条件下该码支持$d<n-1$，具有小子包化级别和$(1+\epsilon)$最优修复带宽（即最优值的$1+\epsilon$倍）。通过将这一转换应用于三种已知MSR码，我们获得了三种具有小子包化级别和$(1+\epsilon)$最优修复带宽的MDS阵列码。所有新型MDS阵列码均具有$d<n-1$的小修复度，且适用于小参数和大参数场景。