Node repair is a crucial problem in erasure-code-based distributed storage systems. An important metric for repair efficiency is the I/O cost which equals the total amount of data accessed at helper nodes to repair a failed node. In this work, a general formula for computing the I/O cost of linear repair schemes is derived from a new perspective, i.e., by investigating the Hamming weight of a related linear space. Applying the formula to Reed-Solomon (RS) codes, we obtain lower bounds on the I/O cost for full-length RS codes with two and three parities. Furthermore, we build linear repair schemes for the RS codes with improved I/O cost. For full-length RS codes with two parities, our scheme meets the lower bound on the I/O cost.

翻译：节点修复是基于纠删码的分布式存储系统中的关键问题。修复效率的重要指标是I/O开销，即修复失效节点时从辅助节点访问的数据总量。本文从新的角度——通过研究相关线性空间的汉明重量——推导出计算线性修复方案I/O开销的通用公式。将该公式应用于里德-所罗门码，我们获得了具有两个和三个校验位的全长RS码I/O开销的下界。此外，我们构建了具有改进I/O开销的RS码线性修复方案。对于具有两个校验位的全长RS码，我们的方案达到了I/O开销的下界。