Repairing Reed-Solomon Codes via Subspace Polynomials

Abstract

We propose new repair schemes for Reed-Solomon codes that use subspace polynomials and hence generalize previous works in the literature that employ trace polynomials. The Reed-Solomon codes are over F q and have redundancy r = n − k ≥ q m , 1 ≤ m ≤ , where n and k are the code length and dimension, respectively. In particular, for one erasure, we show that our schemes can achieve optimal repair bandwidths whenever n = q and r = q m , for all 1 ≤ m ≤ . For two erasures, our schemes use the same bandwidth per erasure as the single erasure schemes, for /m is a power of q, and for = q a , m = q b − 1 > 1 (a ≥ b ≥ 1), and for m ≥ /2 when is even and q is a power of two. I. INTRODUCTIONThe repair bandwidth is a crucial performance metric of erasure codes when deployed in distributed storage systems [2], [3]. In such systems, for an underlying finite field F, e.g. F = GF(256), a data vector in F k is transformed into a codeword vector in F n , whose components are subsequently stored at different storage nodes. When a node fails, the codedword symbol stored at that node is erased (lost). A replacement node (RN) has to recover the content stored at the failed node by downloading relevant information from the remaining operational nodes. The repair bandwidth refers to the total amount of information (in bits) that the RN has to download in order to complete the repair process. If multiple erasures occur, different RNs may also exchange information in a distributed manner, and we are interested in the bandwidth used per erasure. Alternatively, multiple erasures can be recovered by a centralized entity, which, however, is not the focus of this work.Reed-Solomon codes [4], the most practically used maximum distance separable codes [5], have been deployed in major distributed storage systems such as the Google File System II, Quantcast File System, Yahoo Object Store, Facebook f4 Storage System, Baidu Atlas Cloud Storage, Backblaze Vaults, and HDFS (see [6, Table I]). However, they perform poorly as erasure codes under the repair bandwidth metric. For instance, to repair a data chunk of size 256 MB, the default repair scheme for the Reed-Solomon code (14, 10) employed by Facebook's f4 [7]