We address the multi-node failure repair challenges for MDS array codes. Presently, two primary models are employed for multi-node repairs: the centralized model where all failed nodes are restored in a singular data center, and the cooperative model where failed nodes acquire data from auxiliary nodes and collaborate amongst themselves for the repair process.This paper focuses on the cooperative model, and we provide explicit constructions of optimal MDS array codes with $d$ helper nodes under this model. The sub-packetization level of our new codes is $(d-k+h)(d-k+1)^{\lceil n/2 \rceil}$ where $h$ is the number of failed nodes, $k$ the number of information nodes and $n$ the code length. This improves upon recent constructions given by Liu \emph{et al.} (IEEE Transactions on Information Theory, Vol. 69, 2023).

翻译：摘要：本文针对MDS阵列码的多节点故障修复问题展开研究。当前，多节点修复主要采用两种模型：集中式模型（所有故障节点在单一数据中心中恢复）和协同式模型（故障节点从辅助节点获取数据并在修复过程中相互协作）。本文聚焦于协同式模型，并在此模型下给出了具有$d$个辅助节点的最优MDS阵列码的显式构造。新码的子分组级别为 $(d-k+h)(d-k+1)^{\lceil n/2 \rceil}$，其中$h$为故障节点数，$k$为信息节点数，$n$为码长。该结果改进了Liu等人（IEEE信息论汇刊，第69卷，2023年）近期提出的构造方案。