Maximum distance separable (MDS) codes are widely used in distributed storage systems as they provide optimal fault tolerance for a given amount of storage overhead. The seminal work of Dimakis~\emph{et al.} first established a lower bound on the repair bandwidth for a single failed node of MDS codes, known as the \emph{cut-set bound}. MDS codes that achieve this bound are called minimum storage regenerating (MSR) codes. Numerous constructions and theoretical analyses of MSR codes reveal that they typically require exponentially large sub-packetization levels, leading to significant disk I/O overhead. To mitigate this issue, many studies explore the trade-offs between the sub-packetization level and repair bandwidth, achieving reduced sub-packetization at the cost of suboptimal repair bandwidth. Despite these advances, the fundamental question of determining the minimum repair bandwidth for a single failure of MDS codes with fixed sub-packetization remains open. In this paper, we address this challenge for the case of two parity nodes ($n-k=2$) and sub-packetization $\ell=2$. Under these parameters, we establish a correspondence between repair schemes and point sets on the projective line $\mathbb{P}^1$, and then derive a lower bound on repair bandwidth utilizing the sharply 3-transitive action of $\text{PGL}_2(\Fq)$. Furthermore, we extend this lower bound to the repair I/O, and construct two classes of explicit MDS array codes that achieve these bounds, offering practical code designs with provable repair efficiency.

翻译：最大距离可分码广泛应用于分布式存储系统，因其在给定存储开销下能提供最优容错能力。Dimakis等开创性工作首次建立了单节点失效下MDS码修复带宽的下界，即所谓的割集界。达到该界的MDS码称为最小存储再生码。对MSR码的大量构造与理论分析表明，该类码通常需要指数级增长的子分组化级别，导致显著的磁盘I/O开销。为缓解这一问题，许多研究探索了子分组化级别与修复带宽之间的权衡，通过牺牲修复带宽的次优性来降低子分组化程度。尽管取得了这些进展，但在固定子分组化条件下确定单节点失效的最小修复带宽这一基本问题仍未解决。本文针对两个校验节点（$n-k=2$）且子分组化级别$\ell=2$的情形，解决了该挑战。在此参数设定下，我们建立了修复方案与射影直线$\mathbb{P}^1$上点集间的对应关系，进而利用$\text{PGL}_2(\Fq)$的锐3-传递作用推导出修复带宽的下界。此外，我们将该下界扩展至修复I/O，并构造了两类显式MDS阵列码以实现这些界，从而提供了具有可证明修复效率的实用码设计。