In this paper, we explore a practical system setting where a rack-aware storage system consists of racks, each containing a few parity checks, referred to as a rack-aware system with locality. To minimize cross-rack bandwidth in this system, we organize the repair sets of locally repairable codes into racks and investigate the problem of repairing erasures in locally repairable codes beyond the code locality. We devise two repair schemes to reduce the repair bandwidth for Tamo-Barg codes under the rack-aware model by setting each repair set as a rack. We then establish a cut-set bound for locally repairable codes under the rack-aware model with locality. Using this bound we show that our second repair scheme is optimal. Furthermore, we consider the partial-repair problem for locally repairable codes under the rack-aware model with locality, and introduce both repair schemes and bounds for this scenario.

翻译：本文探索了一种实际的系统设置，其中机架感知存储系统由多个机架组成，每个机架包含少量奇偶校验，这种系统被称为具有局部性的机架感知系统。为最小化该系统中的跨机架带宽，我们将局部可修复码的修复集组织到机架中，并研究超越码局部性的局部可修复码擦除修复问题。我们设计了两种修复方案，通过在机架感知模型下将每个修复集设定为一个机架来降低Tamo-Barg码的修复带宽。随后，我们建立了具有局部性的机架感知模型下局部可修复码的割集界。利用该界限，我们证明了第二种修复方案的最优性。此外，我们考虑了具有局部性的机架感知模型下局部可修复码的部分修复问题，并针对该场景提出了相应的修复方案与界限。