Repairing Reed-Solomon Codes With Multiple Erasures

Abstract

Despite their exceptional error-correcting properties, Reed-Solomon (RS) codes have been overlooked in distributed storage applications due to the common belief that they have poor repair bandwidth: A naive repair approach would require the whole file to be reconstructed in order to recover a single erased codeword symbol. In a recent work, Guruswami and Wootters (STOC'16) proposed a single-erasure repair method for RS codes that achieves the optimal repair bandwidth amongst all linear encoding schemes. Their key idea is to recover the erased symbol by collecting a sufficiently large number of its traces, each of which can be constructed from a number of traces of other symbols. As all traces belong to a subfield of the defining field of the RS code and many of them are linearly dependent, the total repair bandwidth is significantly reduced compared to that of the naive repair scheme. We extend the trace collection technique to cope with multiple erasures.

An Improved Cooperative Repair Scheme for Reed-Solomon Codes

“…The corollary follows from Lemma 2 and the facts K 1,2,3 = K 1,2 ∩ K 2,3 and |K 1,2,3 | = |K 1,2 ∩ K 2,3 | = | α2−α3 α1−α2 K ∩ K|. Note that this corollary is also displayed in [9]. Proof.…”

“…We give an improved cooperative repair scheme for Reed-Solomon codes with two or three erasures, removing all the restrictions required in Dau et al's work [9]. An interesting problem in the future is to establish a lower bound on the repair bandwidth for cooperative repair of Reed-Solomon codes.…”

“…As in the repair of two erasures, we modify the check polynomials q i (x), r i (x), i ∈ [t] by multiplying γ 1 and γ 2 respectively. Furthermore, we remove all the restrictions required in [9] by properly choosing γ 1 , γ 2 .…”

“…Recently, Dau et al [9] extend Guruswami and Wootters' scheme [3] to cooperatively repair RS codes with two or three erasures. However, for two erasures, their scheme is either restricted to the case that the characteristic of F divides the extension degree [F : B], or requires a sequential recovery of the two erasures.…”

Erasure coding for distributed storage: an overview

“…In [173], the authors present RS codes that meet the MSR point for all parameters: k < d < n − 1. Bandwidth-efficient recovery from multiple erasures in RS codes is addressed in [174] and is further extended to include general scalar MDS codes in [175]. In [176], the authors present codes that universally achieve the optimal bandwidth points for all parameters h ≤ n − k and k ≤ d ≤ n − h simultaneously.…”