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（MDS）码因其能在给定存储开销下提供最优容错能力，被广泛应用于分布式存储系统。Dimakis等人的开创性工作首次为MDS码单个故障节点的修复带宽建立了一个下界，即割集界。达到该下界的MDS码称为最小存储再生（MSR）码。大量关于MSR码的构造和理论分析表明，它们通常需要指数级大的子分组化水平，从而导致显著的磁盘I/O开销。为了缓解这一问题，许多研究探索了子分组化水平与修复带宽之间的权衡，以次优的修复带宽为代价实现了降低子分组化水平。尽管取得了这些进展，但对于具有固定子分组化水平的MDS码，确定其单节点故障的最小修复带宽这一根本问题仍然悬而未决。在本文中，我们针对两个校验节点（$n-k=2$）和子分组化$\ell=2$的情况解决了这一挑战。在这些参数下，我们建立了修复方案与射影线\(\mathbb{P}^1\)上点集之间的对应关系，然后利用\(\text{PGL}_2(\Fq)\)的锐3-传递作用推导出修复带宽的下界。此外，我们将此下界推广至修复I/O，并构造了两类达到这些下界的显式MDS阵列码，提供了具有可证明修复效率的实用码设计方案。