Repairing Reed-Solomon Codes

We study the performance of Reed–Solomon (RS) codes for the <italic>exact repair problem</italic> in distributed storage. Our main result is that, in some parameter regimes, Reed–Solomon codes are optimal regenerating codes, among maximum distance separable (MDS) codes with linear repair schemes. Moreover, we give a characterization of MDS codes with linear repair schemes, which holds in any parameter regime, and which can be used to give non-trivial repair schemes for RS codes in other settings. More precisely, we show that for <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-dimensional RS codes whose evaluation points are a finite field of size <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, there are exact repair schemes with bandwidth <inline-formula> <tex-math notation="LaTeX">$(n-1)\log ((n-1)/(n-k))$ </tex-math></inline-formula> bits, and that this is optimal for any MDS code with a linear repair scheme. In contrast, the naive (commonly implemented) repair algorithm for this RS code has bandwidth <inline-formula> <tex-math notation="LaTeX">$k\log (n)$ </tex-math></inline-formula> bits. When the entire field is used as evaluation points, the number of nodes <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is much larger than the number of bits per node (which is <inline-formula> <tex-math notation="LaTeX">$O(\log (n))$ </tex-math></inline-formula>), and so this result holds only when the degree of sub-packetization is small. However, our method applies in any parameter regime, and to illustrate this for high levels of sub-packetization, we give an improved repair scheme for a specific (14,10)-RS code used in the facebook hadoop analytics cluster.