In this paper, we discuss codes for distributed storage systems with hierarchical repair properties. Specifically, we devote attention to the repair problem of the rack-aware storage model with locality, aiming to enhance the system's ability to repair a small number of erasures within each rack by locality and efficiently handling a rack erasure with a small repair bandwidth. By employing the regenerating coding technique, we construct a family of array codes with $(r,u-r+1)$-locality, where the $u$ nodes of each repair set are systematically organized into a rack. When the number of failures is less than $u - r + 1$, these failures can be repaired without counting the system bandwidth. In cases where the number of failures exceeds the locality, the failed nodes within a single rack can be recovered with optimal cross-rack bandwidth.

翻译：本文讨论了具有分层修复特性的分布式存储系统编码问题。具体而言，我们聚焦于具有局部性的机架感知存储模型的修复问题，旨在通过局部性提升每个机架内少量擦除的修复能力，同时以较小的修复带宽高效处理机架级擦除。采用再生编码技术，我们构建了一类具有$(r,u-r+1)$-局部性的阵列码，其中每个修复集合中的$u$个节点被系统性地组织到同一机架内。当故障数量小于$u-r+1$时，这些故障可在不计算系统带宽的情况下完成修复；当故障数量超过局部性阈值时，单个机架内的故障节点可通过最优跨机架带宽进行恢复。