Repairing Reed-Solomon Codes Over $GF(2^\ell)$

Abstract

We provide a detailed algorithm to repair a single node failure in an $(n,k)$ Reed-Solomon code over $ {\textit{GF}}(2^\ell)$ with repair bandwidth $\frac {\ell }{\textit {a}}(\textit {n}-1)( {\textit a-s})$ , for any integers $a,s$ such that $ {a}|\ell, 2^{ {a}} \ge {n}+1, 2^{ {s}}\leq {\textit n-k}$ . We present the constructions of necessary lookup tables for the repair. The storage overhead and the repair complexity of our algorithm are also analyzed. The algorithm can be applied to the $(14,10)$ Reed-Solomon codes over ${\textit{GF}}(2^{8})$ , which is a modification of the code in Facebook’s f4 system, and reaches the lowest repair bandwidth among the existing schemes to the best of our knowledge. The algorithm can be generalized to other codes, including the ones based on Yahoo Object Store and Baidu’s Atlas.