A repair scheme for a distributed storage system based on multivariate polynomials

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.