A distributed storage system stores data across multiple nodes, with the primary objective of enabling efficient data recovery even in the event of node failures. The main goal of an exact repair scheme is to recover the data from a failed node by accessing and downloading information from the rest of the nodes. In a groundbreaking paper, ~\cite{GW} developed an exact repair scheme for a distributed storage system that is based on Reed-Solomon codes, which depend on single-variable polynomials. In these notes, we extend the repair scheme to the family of distributed storage systems based on Reed-Muller codes, which are linear codes based on multivariate polynomials. The repair scheme we propose repairs any single node failure and multiple node failures, provided the positions satisfy certain conditions.

翻译：分布式存储系统将数据存储于多个节点之中，其主要目标是在节点发生故障时仍能实现高效的数据恢复。精确修复方案的核心目标是通过访问并下载其余节点的信息，从故障节点中恢复数据。在一篇开创性论文中，~\cite{GW} 提出了一种基于 Reed-Solomon 码的分布式存储系统精确修复方案，该方案依赖于单变量多项式。本文中，我们将该修复方案扩展至基于 Reed-Muller 码的分布式存储系统族，这类线性码建立在多元多项式基础上。我们所提出的修复方案能够修复任何单节点故障及多节点故障，只要故障位置满足特定条件。