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.

翻译：文献中已报道了大量高码率（n,k）最小存储再生（MSR）码。然而，其中大部分在节点修复过程中需要联系所有存活节点，即修复度d=n-1。但在实际系统中，由于某些节点可能不可用，连接并下载所有存活节点的数据并不总是可行的。因此，修复度d<n-1的MSR码构造也备受期待。截至目前，仅有少数修复度d<n-1的（n,k）MSR码构造被报道，其中一些存在较大的子分组化水平、较大的有限域或对修复度d的限制。本文提出了一种新的（n,k）MSR码构造，该构造适用于任意修复度d>k，且与现有构造相比具有更小的子分组化水平或有限域。此外，结合一种先前提出的降低子分组化水平的通用变换，我们还针对修复度d=n-1获得了一种具有小子分组化水平和(1+ε)最优修复带宽（即最优修复带宽的(1+ε)倍）的MDS阵列码，其在子分组化水平或域大小方面均优于现有方案。