An @math maximum distance separable (MDS) code has optimal repair access if the minimum number of symbols accessed from @math surviving nodes is achieved, where @math . Existing results show that the sub-packetization @math of an @math high code rate (i.e., @math ) MDS code with optimal repair access is at least @math . In this paper, we propose a class of multi-layer transformed MDS codes such that the sub-packetization is @math , where @math , and the repair access is optimal for any single node. We show that the sub-packetization of the proposed multi-layer transformed MDS codes is strictly less than the existing known lower bound when @math , achieving by restricting the choice of @math specific helper nodes in repairing a failed node. We further propose multi-layer transformed EVENODD codes that have optimal repair access for any single node and lower sub-packetization than the existing binary MDS array codes with optimal repair access for any single node. With our multi-layer transformation, we can design new MDS codes that have the properties of low computational complexity, optimal repair access for any single node, and relatively small sub-packetization, all of which are critical for maintaining the reliability of distributed storage systems.
There are other practical concerns in distributed storage systems, such as how to mitigate the computational complexity. Binary MDS array codes are a special class of MDS codes that have low computational complexity, since the encoding and decoding procedures only involve XOR operations. Typical examples of binary MDS array codes are EVENODD @cite_8 @cite_6 , X-code @cite_12 and RDP @cite_33 @cite_10 . Some efficient decoding methods of binary MDS array codes are given in @cite_9 @cite_19 @cite_14 @cite_23 @cite_24 . There have been also many studies @cite_29 @cite_0 @cite_30 @cite_9 @cite_26 on the optimal repair bandwidth of binary MDS array codes.
@cite_30 @cite_14 @cite_26 @cite_33 @cite_8 @cite_9 @cite_29 @cite_6 @cite_24 @cite_19 @cite_0 @cite_23 @cite_10 @cite_12
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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. Binary maximum distance separable (MDS) array codes are a special class of erasure codes for distributed storage that not only provide fault tolerance with minimum storage redundancy, but also achieve low computational complexity. They are constructed by encoding k information columns into r parity columns, in which each element in a column is a bit, such that any k out of the k + r columns suffice to recover all information bits. In addition to providing fault tolerance, it is critical to improve repair performance. Specifically, if a single column fails, our goal is to minimize the repair bandwidth by downloading the least amount of bits from d non-failed columns, where k ≤ d ≤ k + r − 1. However, existing binary MDS codes that achieve high data rates (i.e., k (k + r) > 1 2) and minimum repair bandwidth only support double fault tolerance (i.e., r = 2), which is insufficient for failure-prone distributed storage environments in practice. This paper fills the void by proposing an explicit construction of triple-fault-tolerant (i.e., r = 3) binary MDS array codes that achieve asymptotically minimum repair bandwidth for d = k + 1. Binary maximum distance separable (MDS) array codes contain @math information columns and @math parity columns in which each entry is a bit that can tolerate @math arbitrary erasures. When a column in an MDS code fails, it has been proven that we must download at least half of the content from each helper column if @math columns are selected as the helper columns. If the lower bound is achieved such that the @math helper columns can be selected from any @math surviving columns, then the repair is an optimal repair. Otherwise, if the lower bound is achieved with @math specific helper columns, the repair is a weak-optimal repair. This paper proposes a class of binary MDS array codes with @math and @math that asymptotically achieve weak-optimal repair of an information column with @math helper columns. We show that there exist many encoding matrices such that the corresponding binary MDS array codes can asymptotically achieve weak-optimal repair for repairing any information column. Consider a binary maximum distance separate (MDS) array code composed of an @math array of bits with @math information columns and @math parity columns, such that any @math out of @math columns suffice to reconstruct the @math information columns. Our goal is to provide optimal repair access for binary MDS array codes, meaning that the bandwidth triggered to repair any single failed information or parity column is minimized. In this paper, we propose a generic transformation framework for binary MDS array codes, using EVENODD codes as a motivating example, to support optimal repair access for @math , where @math denotes the number of non-failed columns that are connected for repair. In addition, we show that the efficient decoding property of the original EVENODD codes is maintained after transformation. Furthermore, we show how our transformation framework applies to general binary MDS array codes and enables them to support optimal repair access.