Array codes have been widely used in communication and storage systems. To reduce computational complexity, one important property of the array codes is that only XOR operation is used in the encoding and decoding process. In this work, we present a novel family of maximal-distance separable (MDS) array codes based on Cauchy matrix, which can correct up to any number of failures. We also propose an efficient decoding method for the new codes to recover the failures. We show that the encoding decoding complexities of the proposed approach are lower than those of existing Cauchy MDS array codes, such as Rabin-Like codes and CRS codes. Thus, the proposed MDS array codes are attractive for distributed storage systems.
Row-diagonal parity (RDP) code proposed in @cite_25 and EVENODD code in @cite_8 can tolerate two arbitrary disk erasures. Due to increasing capacities of hard disks and requirement of low bit error rates, the protection offered by double parities will soon be inadequate. The issue of reliability is more pronounced in solid-state drives (SSD), which have significant wear-out rate when the frequencies of disk writes are high. Indeed, triple-parity RAID (Redundant Arrays of Inexpensive Disks) has already been advocated in storage technology @cite_29 . Construction of array codes recovering multiple disk erasures is relatively rare, in compare to array codes recovering double erasures. We name the existing MDS array codes in @cite_25 @cite_8 @cite_27 @cite_22 @cite_0 @cite_23 @cite_9 @cite_3 @cite_10 as , as their constructions are based on Vandermonde matrices.
@cite_22 @cite_8 @cite_29 @cite_9 @cite_3 @cite_0 @cite_27 @cite_23 @cite_10 @cite_25
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Array codes are used widely in data storage systems such as redundant array of independent disks. The row-diagonal parity (RDP) codes and EVENODD codes are two popular double-parity array codes. The increasing capacity of hard disks demands better fault tolerance by using array codes with three or more parity disks. Although many extensions of RDP and EVENODD codes have been proposed, their main drawback is high decoding complexity. In this paper, we propose a unified form of RDP and EVENODD codes under which RDP codes can be treated as shortened EVENODD codes. Moreover, an efficient decoding algorithm based on an LU factorization of a Vandermonde matrix is proposed. The LU decoding method is applicable to all the erasure patterns of RDP and EVENODD codes with three parity columns. It is also applicable to the erasure decoding of RDP and EVENODD codes with more than three parity columns when the number of continuous surviving parity columns is no less than the number of erased information columns and the first parity column has not failed. The proposed efficient decoding algorithm is also applicable to other Vandermonde array codes, with less decoding complexity than that of the existing method. The current parallel storage systems use thousands of inexpensive disks to meet the storage requirement of applications. Data redundancy and or coding are used to enhance data availability, for instance, Row-diagonal parity (RDP) and EVENODD codes, which are widely used in RAID-6 storage systems, provide data availability with up to two disk failures. To reduce the probability of data unavailability, whenever a single disk fails, disk recovery will be carried out. We find that the conventional recovery schemes of RDP and EVENODD codes for a single failed disk only use one parity disk. However, there are two parity disks in the system, and both can be used for single disk failure recovery. In this article, we propose a hybrid recovery approach that uses both parities for single disk failure recovery, and we design efficient recovery schemes for RDP code (RDOR-RDP) and EVENODD code (RDOR-EVENODD). Our recovery scheme has the following attractive properties: (1) “read optimality” in the sense that our scheme issues the smallest number of disk reads to recover a single failed disk and it reduces approximately 1 4 of disk reads compared with conventional schemes; (2) “load balancing property” in that all surviving disks will be subjected to the same (or almost the same) amount of additional workload in rebuilding the failed disk. We carry out performance evaluation to quantify the merits of RDOR-RDP and RDOR-EVENODD on some widely used disks with DiskSim. The offline experimental results show that RDOR-RDP and RDOR-EVENODD outperform the conventional recovery schemes of RDP and EVENODD codes in terms of total recovery time and recovery workload on individual surviving disk. However, the improvements are less than the theoretical value (approximately 25p), as RDOR-RDP and RDOR-EVENODD change the disk access pattern from purely sequential to a more random one compared with their conventional schemes. Binary maximum distance separable (MDS) array codes are widely used in storage systems. EVENODD code and row-diagonal parity (RDP) code are two well-known binary MDS array codes with two parity columns. The Blaum–Bruck–Vardy (BBV) code is an extension of the double-erasure-correcting EVENODD code. It is known that the BBV code is always MDS for three parity columns, and sufficient conditions for up to eight parity columns to be MDS are also known. However, the MDS condition for more than eight parity columns is an open problem since then. In this letter, we study the MDS condition of BBV code and give a sufficient MDS condition of the BBV code with more than eight parity columns. Row-Diagonal Parity (RDP) is a new algorithm for protecting against double disk failures. It stores all data unencoded, and uses only exclusive-or operations to compute parity. RDP is provably optimal in computational complexity, both during construction and reconstruction. Like other algorithms, it is optimal in the amount of redundant information stored and accessed. RDP works within a single stripe of blocks of sizes normally used by file systems, databases and disk arrays. It can be utilized in a fixed (RAID-4) or rotated (RAID-5) parity placement style. It is possible to extend the algorithm to encompass multiple RAID-4 or RAID-5 disk arrays in a single RDP disk array. It is possible to add disks to an existing RDP array without recalculating parity or moving data. Implementation results show that RDP performance can be made nearly equal to single parity RAID-4 and RAID-5 performance.