We study the rack-aware storage system where all storage nodes are organized in racks and within each rack the nodes can communicate freely without taxing the system bandwidth. Rack-aware regenerating codes (RRCs) were proposed for minimizing the repair bandwidth for single erasures. In the initial setting of RRCs, the repair of a single node requires the participation of all the remaining nodes in the rack containing the failed node as well as a large number of helper racks containing no failures. Consequently, the repair may be infeasible in front of multiple node failures. In this work, a relaxed repair model that can tolerate multiple node failures by simultaneously reducing the intra-rack connections and cross-rack connections is proposed. A tradeoff between the storage and repair bandwidth under the relaxed repair model is derived, and parameters of the two extreme points on the tradeoff curve are characterized for the minimum storage and minimum bandwidth respectively. Moreover, two codes corresponding to the extreme points are explicitly constructed over the fields of size comparable to the code length and with the lowest sub-packetization. Finally, for the convenience of practical use, systematic encoding processes for the two codes are also established.

中文翻译：

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具有多种擦除容差的机架感知重新生成代码


我们研究机架感知存储系统，其中所有存储节点都组织在机架中，并且每个机架内的节点可以自由通信，而不会对系统带宽造成负担。提出了机架感知再生码（RRC）来最小化单次擦除的修复带宽。在RRC的初始设置中，单个节点的修复需要包含故障节点的机架中的所有剩余节点以及大量未包含故障的辅助机架的参与。因此，在多个节点发生故障的情况下修复可能是不可行的。在这项工作中，提出了一种宽松的修复模型，可以通过同时减少机架内连接和跨机架连接来容忍多个节点故障。推导了松弛修复模型下存储和修复带宽之间的权衡，并分别表征了权衡曲线上两个极值点的参数为最小存储和最小带宽。此外，在大小与代码长度相当并且具有最低子分组化的字段上显式构造对应于极值点的两个代码。最后，为了方便实际使用，还建立了两种编码的系统编码流程。