Maximum distance separable (MDS) codes facilitate the achievement of elevated levels of fault tolerance in storage systems while incurring minimal redundancy overhead. Reed-Solomon (RS) codes are typical MDS codes with the sub-packetization level being one, however, they require large repair bandwidth defined as the total amount of symbols downloaded from other surviving nodes during single-node failure/repair. In this paper, we present the {\em set transformation}, which can transform any MDS code into set transformed code such that (i) the sub-packetization level is flexible and ranges from 2 to $(n-k)^{\lfloor\frac{n}{n-k}\rfloor}$ in which $n$ is the number of nodes and $k$ is the number of data nodes, (ii) the new code is MDS code, (iii) the new code has lower repair bandwidth for any single-node failure. We show that our set transformed codes have both lower repair bandwidth and lower field size than the existing related MDS array codes, such as elastic transformed codes \cite{10228984}. Specifically, our set transformed codes have $2\%-6.6\%$ repair bandwidth reduction compared with elastic transformed codes \cite{10228984} for the evaluated typical parameters.

翻译：最大距离可分（MDS）码能够在存储系统中实现高容错性，同时仅引入极小的冗余开销。Reed-Solomon（RS）码是典型的MDS码，其子分组级别为1，但在单节点故障/修复过程中，需要下载大量来自其他存活节点的符号，即修复带宽较大。本文提出了一种“集合变换”方法，可将任意MDS码转换为集合变换码，其特性如下：（i）子分组级别灵活可变，取值范围为2到$(n-k)^{\lfloor\frac{n}{n-k}\rfloor}$，其中$n$为节点数，$k$为数据节点数；（ii）新码仍为MDS码；（iii）对于任意单节点故障，新码的修复带宽更低。我们证明，与现有相关MDS阵列码（如弹性变换码\cite{10228984}）相比，集合变换码在修复带宽和域大小方面均具有优势。具体而言，对于评估的典型参数，我们的集合变换码相比弹性变换码\cite{10228984}，修复带宽降低$2\%-6.6\%$。