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.

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，其中大多数要求在节点修复过程中联系所有存活节点，导致修复度为d=n-1。在实际系统中，由于部分节点可能不可用，连接并从所有存活节点下载数据并非总是可行。因此，需要构建修复度d<n-1的MSR码。截至目前，仅有少数几种修复度d<n-1的(n,k)MSR码结构被报道，其中一些存在较大的子包化水平、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码构造方法，该方法适用于任意修复度d>k，且相比于现有某些构造具有更小的子包化水平或有限域。此外，结合先前用于降低子包化水平的通用变换，我们获得了一种具有小子包化水平且在修复度d=n-1下具备$(1+\epsilon)$-最优修复带宽（即最优修复带宽的$(1+\epsilon)$倍）的MDS阵列码。该码在子包化水平或域大小方面优于某些现有方案。