SDM: A Stripe-Based Data Migration Scheme to Improve the Scalability of RAID-6

The increasing expansion of data scale leads to the widespread deployment of storage systems with larger capacity and further induces the climbing probability of data loss or damage. The Maximum Distance Separable (MDS) code in RAID-6, which tolerates the concurrent failures of any two disks with minimal storage requirement, is one of the best candidates to enhance the data reliability. However, most of the existing works in this literature are more inclined to be specialized and cannot provide a satisfied performance under an all-round evaluation. Aiming at this problem, we propose an all-round MDS code named Horizontal-Vertical Code (HV Code) by taking advantage of horizontal parity and vertical parity. HV Code achieves the perfect I/O balancing and optimizes the operation of partial stripe writes to continuous data elements, while preserving the optimal encode/decode/update efficiency. Moreover, it owns a shorter parity chain which grants it a more efficient recovery for one disk failure. HV Code also behaves well for the degraded read operation and accelerates the process to reconstruct two disabled disks by executing four recovery chains in parallel. The performance evaluation demonstrates that HV Code well balances the I/O distribution and eliminates up to 27.6% and 32.4% I/O requests for partial stripe writes operation when compared with RDP Code and HDP Code. Moreover, compared to RDP Code, HDP Code, X-Code and H-Code, HV Code reduces up to 5.4%~39.8% I/O requests per element for the single disk reconstruction, decreases 6.6%~28.3% I/O requests for degraded read operations, and achieves the same efficiency of X-Code for double disk recovery by shortening 47.4%~59.7% recovery time compared with other three codes.

In large storage systems, files are often coded across several servers to improve reliability and retrieval speed. We study load balancing under the batch sampling routeing scheme for a network ofnservers storing a set of files using the maximum distance separable (MDS) code (cf. Li (2016)). Specifically, each file is stored in equally sized pieces acrossLservers such that anykpieces can reconstruct the original file. When a request for a file is received, the dispatcher routes the job into thek-shortest queues among theLfor which the corresponding server contains a piece of the file being requested. We establish a law of large numbers and a central limit theorem as the system becomes large (i.e.n→ ∞), for the setting where all interarrival and service times are exponentially distributed. For the central limit theorem, the limit process take values inℓ2, the space of square summable sequences. Due to the large size of such systems, a direct analysis of then-server system is frequently intractable. The law of large numbers and diffusion approximations established in this work provide practical tools with which to perform such analysis. The power-of-drouteing scheme, also known as the supermarket model, is a special case of the model considered here.

Maximum distance separable (MDS) codes are those codes whose minimum distance attains the Singleton bound for a given length and given dimension. It is a well-known fact that the only binary linear block codes that are MDS are the simple parity codes and the repetition codes. Known MDS codes (the Reed-Solomon codes, for instance) are defined over larger alphabets. This approach requires that (i) the encoding and decoding procedures are performed as operations over a finite field and (ii) an update in a single information bit (e.g. in storage applications) requires an update in the redundancy symbols and usually affects a number of bits in each redundancy symbol. Thus, the optimal redundancy is achieved at the expense of additional complexity in the encoding/decoding procedures as well as in the number of redundancy bits affected by an update in the information. We construct MDS codes that have the following two properties: (i) the redundancy bits are computed by simple XOR operations; and (ii) an update in an information bit affects a minimal number of redundancy bits. >

The weights in maximum distance separable (MDS) codes of length n and dimension k over the finite field GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with parameters given by the MDS conjecture are shown to contain all k weights in the range n - k + 1 to n. The proof uses the covering radius of the dual code.

<p style='text-indent:20px;'>The weight distribution of the cosets of maximum distance separable (MDS) codes is considered. In 1990, P.G. Bonneau proposed a relation to obtain the full weight distribution of a coset of an MDS code with minimum distance <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula> using the known numbers of vectors of weights <inline-formula><tex-math id="M2">\begin{document}$ \le d-2 $\end{document}</tex-math></inline-formula> in this coset. In this paper, the Bonneau formula is transformed into a more structured and convenient form. The new version of the formula allows to consider effectively cosets of distinct weights <inline-formula><tex-math id="M3">\begin{document}$ W $\end{document}</tex-math></inline-formula>. (The weight <inline-formula><tex-math id="M4">\begin{document}$ W $\end{document}</tex-math></inline-formula> of a coset is the smallest Hamming weight of any vector in the coset.) For each of the considered <inline-formula><tex-math id="M5">\begin{document}$ W $\end{document}</tex-math></inline-formula> or regions of <inline-formula><tex-math id="M6">\begin{document}$ W $\end{document}</tex-math></inline-formula>, special relations more simple than the general ones are obtained. For the MDS code cosets of weight <inline-formula><tex-math id="M7">\begin{document}$ W = 1 $\end{document}</tex-math></inline-formula> and weight <inline-formula><tex-math id="M8">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula> we obtain formulas of the weight distributions depending only on the code parameters. This proves that all the cosets of weight <inline-formula><tex-math id="M9">\begin{document}$ W = 1 $\end{document}</tex-math></inline-formula> (as well as <inline-formula><tex-math id="M10">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula>) have the same weight distribution. The cosets of weight <inline-formula><tex-math id="M11">\begin{document}$ W = 2 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M12">\begin{document}$ W = d-2 $\end{document}</tex-math></inline-formula> may have different weight distributions; in this case, we proved that the distributions are symmetrical in some sense. The weight distributions of the cosets of MDS codes corresponding to arcs in the projective plane <inline-formula><tex-math id="M13">\begin{document}$ \mathrm{PG}(2,q) $\end{document}</tex-math></inline-formula> are also considered. For MDS codes of covering radius <inline-formula><tex-math id="M14">\begin{document}$ R = d-1 $\end{document}</tex-math></inline-formula> we obtain the number of the weight <inline-formula><tex-math id="M15">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula> cosets and their weight distribution that gives rise to a certain classification of the so-called deep holes. We show that any MDS code of covering radius <inline-formula><tex-math id="M16">\begin{document}$ R = d-1 $\end{document}</tex-math></inline-formula> is an almost perfect multiple covering of the farthest-off points (deep holes); moreover, it corresponds to an optimal multiple saturating set in the projective space <inline-formula><tex-math id="M17">\begin{document}$ \mathrm{PG}(N,q) $\end{document}</tex-math></inline-formula>.</p>

The algebraic setting for threshold secret sharing scheme can vary, dependent on the application. This algebraic setting can limit the number of participants of an ideal secret sharing scheme. Thus it is important to know for which thresholds one could utilize an ideal threshold sharing schemeand for which thresholds one would have to use non-ideal schemes. The implication is that more than one share may have to be dealt to some or all parties. Karnin, Greene and Hellman constructed several bounds concerning the maximal number of participants in threshold sharing scheme. There has been a number of researchers who have noted the relationship between k-arcs in projective spaces and ideal linear threshold secret schemes, as well as between MDS codes and ideal linear threshold secret sharing schemes. Further, researchers have constructed optimal bounds concerning the size of k-arcs in projective spaces, MDS codes, etc. for various finite fields. Unfortunately, the application of these results on the Karnin, Greene and Hellamn bounds has not been widely disseminated. Our contribution in this paper is revisiting and updating the Karnin, Greene, and Hellman bounds, providing optimal bounds on the number of participants in ideal linear threshold secret sharing schemes for various finite fields, and constructing these bounds using the same tools that Karnin, Greene, and Hellman introduced in their seminal paper. We provide optimal bounds for the maximal number of players for a tout of nideal linear threshold scheme when t= 3, for all possible finite fields. We also provide bounds for infinitely many tand infinitely many fields and a unifying relationship between this problem and the MDS (maximum distance separable) codes that shows that any improvement on bounds for ideal linear threshold secret sharing scheme will impact bounds on MDS codes, for which there is a number of conjectured (but open) problems.

Data migration schemes are critical to balance the load in storage clusters for performance improvement. However, as NAND flash based SSDs are widely deployed in storage systems, extending the lifespan of SSD storage clusters becomes a new challenge for data migration. Prior approaches designed for HDD storage clusters, however, are inefficient due to excessive write amplification during data migration, which significantly decrease the lifespan of SSD storage clusters. To overcome this problem, we propose EDM, an endurance aware data migration scheme with careful data placement and movement to minimize the data migrated, so as to limit the worn-out of SSDs while improving the performance. Based on the observation that performance degradation is dominated by the wear speed of an SSD, which is affected by both the storage utilization and the write intensity, two complementary data migration policies are designed to explore the trade-offs among throughput, response time during migration, and lifetime of SSD storage clusters. Moreover, we design an SSD wear model and quantitatively calculate the amount of data migrated as well as the sources and destinations of the migration, so as to reduce the write amplification caused by migration. Results on a real storage cluster using real-world traces show that EDM performs favorably versus existing HDD based migration techniques, reducing cluster-wide aggregate erase count by up to 40%. In the meantime, it improves the performance by 25% on average compared to the baseline system which achieves almost the same effectiveness of performance improvement as previous migration techniques.

In this paper, we consider a novel wireless network for content distribute storage, where devices could use device-to-device (D2D) communication to transmit data. Several storage schemes, such as maximum distance separable (MDS) code and regenerating code, are employed for content downloading and repairing as node left and data lost. On the basis of multiparameter analysis, we found that targeting on the popular file caching, smaller download locality is the most suitable to the system based on fixed storage nodes and lower repair frequency. Moreover, MBR codes do not apply to D2D distributed storage due to high download cost, and MDS code is a convenient and appropriate scheme for majority practical D2D networks.

Determining the insdel distance of linear codes is a very challenging problem. Recently, Ji et al. [‘Strict half-Singleton bound, strict direct upper bound for linear insertion-deletion codes and optimal codes’, IEEE Trans. Inform. Theory 69 (5) (2023), 2900–2910] proposed their strict half-Singleton bound for linear codes that do not include the all-one vector. A natural question asks for linear codes with both large Hamming distance and insdel distance. Almost maximum distance separable (MDS) codes are a special kind of linear code with good Hamming error-correcting capability. We present a sufficient condition for the insdel distance of almost MDS codes to be bounded by the strict half-Singleton bound. In addition, we construct a class of two-dimensional near MDS codes that achieve the strict half-Singleton bound using twisted Reed–Solomon codes. Finally, we present a construction of optimal almost MDS insdel codes from Reed–Solomon codes for large dimensions.

In this paper, we propose two novel modulation concepts based on a simple maximum distance separable (MDS) code and show that these concepts can achieve better error performance than index modulation (IM) and related schemes. In the first concept, we use amplitude and phase levels to form a simple MDS code, whereas, in the second one, in-phase and quadrature components of codeword elements are used to construct the MDS code. We depict practical schemes for using the proposed concepts with orthogonal frequency division multiplexing (OFDM). We analyze the performance in terms of the minimum Euclidean distance and bit error rate. We also show that the proposed techniques exhibit desirable properties such as efficient low-complexity detection, very simple bits-to-symbols, and symbols-to-bits mappings, and a better error performance when compared to the OFDM-IM and related schemes. More importantly, contrary to the vast majority of IM studies that focus on showing the superiority of the IM techniques against conventional modulation techniques, we show that modulation concepts based on a well-known MDS code can achieve better error performance than the IM and related schemes while exhibiting a structure as simple as these schemes.

As erasure codes have been widely adopted in most large-scale data storage systems and applications, implementations of high-performance erasure codes have been improved significantly in recent years, especially by employing Intel’s Streaming SIMD Extensions (SSE) instructions. Augmenting the survey work in Plank et al. (Fast, 9:253–65, 2009) conducted almost a decade ago, this paper compares practical performance of three open-source or public domain erasure coding libraries, namely Jerasure and Intel’s ISA-L for RS code, and a STAR code implementation. The goal of this paper is to provide data storage practitioner a guideline when they choose a proper erasure code for storage applications and systems that need high performance in encoding and decoding operations in the order of GBs/s. Additionally, this paper identifies a practical technique that can further improve decoding performance of RS code greatly for both Jerasure and ISA-L for the most frequent disk failure pattern, i.e., one disk failure.

The diffusion layer of the SPN block ciphers is usually built on the basis of the MDS (Maximum Distance Separable) matrices which is the matrix of the maximum distance separable code (MDS code). MDS codes have long been studied in error correcting code theory and have applications not only in coding theory but also in the design of block ciphers and hash functions. Thanks to that important role, there have been many studies on methods of building MDS matrices. In particular, the recursive MDS matrices and the symmetric recursive MDS matrices have particularly important applications because they are very efficient for execution. In this paper, we will give an estimate of the number of MDS matrices, recursive MDS matrices and symmetric recursive MDS matrices built from Reed-Solomon codes. This result is meaningful in determining the efficiency from this method of building matrices based on the Reed-Solomon codes. From there, this method can be applied to find out many MDS matrices, secure and efficient symmetric recursive MDS matrices for execution to apply in current block ciphers. Furthermore, recursive MDS matrices can be efficiently implemented using Linear Feedback Shift Registers (LFSR), making them well suited for lightweight cryptographic algorithms, so suitable for limited resources application.

In this letter, we propose a construction of nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes based on a cyclic maximum distance separable (MDS) code. The parity-check matrices are significantly rank deficient square matrices and their Tanner graphs have a girth of at least 6. The minimum distances of the codes are very respectable as far as LDPC codes are concerned. Based on plurality voting and iterative mechanism, a low-complexity nonbinary massage-passing decoding algorithm is proposed. It only requires finite field operations, integer additions and integer comparisons. Simulation results show that the decoding algorithm is fit for the proposed codes, providing efficient trade-offs between performance and decoding complexity, which suggests that the coding scheme may find some applications in communication or storage systems with high-speed and low-power consumption requirements.

Linear codes are considered. A code is characterized by the length n, the dimension k, and the minimum distance d. An (n,k,d) code over the finite field GF(q) is said to be maximum distance separable (MDS) if d=n-k+1. A. Krishna and D.V. Sarwate (1990) investigated the existence of pseudo-cyclic MDS codes over GF(q) of length n, where n divides q-1 or q+1. It is shown that the pseudo-cyclic MDS codes constructed by Krishna and Sarwate are generalised Reed-Solomon codes. Pseudo-cyclic codes are studied over GF(q), where n and q=p/sup h/ are not relatively prime. It is proven that pseudo-cyclic (n,k) MDS codes modulo (x/sup n/-a) over GF(q) exist, if and only if n= p. Furthermore, any pseudo-cyclic (p,k) code modulo (x/sup p/-a) over GF(q) turns out to be MDS and generalized Reed-Solomon. It is explicitly proven that some classes of pseudo-cyclic (n,k) MDS codes over GF(q) are generalized Reed-Solomon codes. Furthermore, pseudo-cyclic (q+1,4) MDS codes over GF(q), q even, are completely classified. >

Mobile edge caching, that prefetches the frequently requested contents at edge nodes, is envisaged to play the leading role in reducing network traffic and latency. Limited storage space at each edge node necessitates efficient caching strategies of what and how to store in advance, and storing segmented content based on maximum distance separable (MDS) code enables to utilize the limited storage space more efficiently. This paper designs a strategy of probabilistic content caching based on MDS code to minimize the help from a macro base station and thus to lessen backhaul traffic. Desirable length of an MDS coded segment and the probability of caching an MDS coded segment are determined to maximize the amount of information obtained from neighboring edge nodes. Numerical results demonstrate that the proposed caching strategy enable to lessen backhaul load effectively.