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For high-rate linear systematic maximum distance separable (MDS) codes, most early constructions could initially optimally repair all the systematic nodes but not all the parity nodes. Fortunately, this issue was first solved by Li et al. in (IEEE Trans. Inform. Theory, 64(9), 6257-6267, 2018), where a transformation that can convert any nonbinary MDS array code into another one with desired properties was proposed. However, the transformation does not work for binary MDS array codes. In this paper, we address this issue by proposing another generic transformation that can convert any $[n, k]$ binary MDS array code into a new one, which endows any $r=n-k\ge 2$ chosen nodes with optimal repair bandwidth and optimal rebuilding access properties, and at the same time, preserves the normalized repair bandwidth/rebuilding access for the remaining $k$ nodes under some conditions. As two immediate applications, we show that 1) by applying the transformation multiple times, any binary MDS array code can be converted into one with optimal rebuilding access for all nodes, 2) any binary MDS array code with optimal repair bandwidth or optimal rebuilding access for the systematic nodes can be converted into one with the corresponding optimality property for all nodes. [ABSTRACT FROM AUTHOR]

We propose a generic transformation that can convert any nonbinary $(n=k{+}r,k)$ maximum distance separable (MDS) code into another $(n,k)$ MDS code over the same field such that: 1) some arbitrarily chosen $r$ nodes have the optimal repair bandwidth and the optimal rebuilding access; 2) for the remaining $k$ nodes, the normalized repair bandwidth and the normalized rebuilding access (over the file size) are preserved; and 3) the sub-packetization level is increased only by a factor of $r$. Two immediate applications of this generic transformation are then presented. The first application is that we can transform any nonbinary MDS code with the optimal repair bandwidth or the optimal rebuilding access for the systematic nodes only, into a new MDS code which possesses the corresponding repair optimality for all nodes. The second application is that by applying the transformation multiple times, any nonbinary $(n,k)$ scalar MDS code can be converted into an $(n,k)$ MDS code with the optimal repair bandwidth and the optimal rebuilding access for all nodes, or only a subset of nodes, whose sub-packetization level is also optimal. [ABSTRACT FROM AUTHOR]

In this paper, we reinterpret the $(k+2,k)$ zigzag code in coding matrix and then propose an optimal exact repair strategy for its parity nodes, whose repair disk I/O approaches a lower bound derived in this paper. [ABSTRACT FROM PUBLISHER]

Piggybacking is an efficient method to decrease the repair bandwidth of maximum distance separable codes. In this paper, in order to reduce the repair bandwidth of parity nodes of the known minimum storage regenerating (MSR) codes with high rate, which is usually the whole amount of the original data, i.e., the maximal, a new systematic piggybacking design is proposed through an in-depth analysis of the design of piggybacking. As a result, new MSR codes are obtained with almost optimal repair bandwidth of parity nodes while retaining the optimal repair bandwidth of systematic nodes. Furthermore, MSR codes with balanced download during node repair process are presented based on the new piggybacking design. [ABSTRACT FROM PUBLISHER]

The newly presented $(k+m,k)$ Hadamard minimum storage regenerating (MSR) codes are a class of high rate storage codes with optimal repair property for single node failure. In this paper, we propose a new simple optimal repair strategy for $(k+m,k)$ Hadamard MSR codes, which can considerably reduce the computation compared with the original one during the node repair. [ABSTRACT FROM PUBLISHER]

In this paper, we present a generic framework for constructing systematic minimum storage regenerating codes with two parity nodes based on the invariant subspace technique. Codes constructed in our framework not only contain some best known codes as special cases, but also include some new codes with key properties, such as the optimal access property and the optimal update property. In particular, for a given storage capacity of an individual node, one of the new codes has the largest number of systematic nodes and two of the new codes have the largest number of systematic nodes with the optimal update property. [ABSTRACT FROM AUTHOR]

Recently, some high-rate maximum distance separable (MDS) array codes were designed to optimally repair a single failed node by connecting all the surviving nodes. However, in practical systems, sometimes not all the surviving nodes are available. To facilitate the practical storage system, a few constructions of $(n,k)$ MDS array codes with the property that any single failed node can be optimally repaired by accessing any $d$ surviving nodes (i.e., minimum-storage regenerating (MSR) codes) have been proposed, where $d\in [k+1:n-1)$. However, all high-rate MDS array codes with this property either have large sub-packetization levels or are not explicit for all the parameters. To address these issues, we propose a generic transformation that can convert any $(n',k')$ MDS array/scalar code to another $(n=n'-\delta,k=k'-\delta)$ MDS array code with the optimal repair property and optimal access property for an arbitrary set of two nodes, while the repair efficiency of the remaining $n-2$ nodes can be kept, where $2\le \delta \le n'-k'$. By recursively applying the generic transformation to an MDS scalar code multiple times, we get a high-rate MDS array code with the optimal repair property and the optimal access property for all nodes, which outperforms previous known high-rate MDS array codes in terms of either the sub-packetization level or the flexibility of the parameters. [ABSTRACT FROM AUTHOR]

In the literature, all the known high-rate MDS codes with the optimal repair bandwidth possess a significantly large sub-packetization level, which may prevent the codes to be implemented in practical systems. To build MDS codes with small sub-packetization level, existing constructions and theoretical bounds imply that one may sacrifice the optimality of the repair bandwidth. Partly motivated by the work of Tamo et al. (IEEE Trans. Inform. Theory, 59(3), 1597–1616, 2013), in this paper, we present a transformation that can greatly reduce the sub-packetization level of MDS codes with the optimal repair bandwidth with respect to the same code length n. As applications of the transformation, four high-rate MDS codes having both small sub-packetization level and near-optimal repair bandwidth can be obtained, where three of them are explicit and the required field sizes are around or even smaller than the code length ${n}$. Additionally, we propose another explicit MDS code which has a similar structure as that of the first resultant code obtained by the generic transformation, but can be built on a smaller finite field. [ABSTRACT FROM AUTHOR]

As a kind of erasure code, piggybacking has been applied in practice since it can significantly reduce the repair bandwidth of distributed storage systems. Currently, several efficient piggybacking designs have been proposed. In this paper, we propose a more efficient “one-to-one” piggybacking design (OOP) to further reduce the repair bandwidth. Different from the existing piggybacking designs, OOP adopts a simple encoding principle that one parity node only piggybacks symbols from one substripe. Particularly, OOP takes into account the efficient repair of systematic nodes and parity nodes simultaneously. It is shown that for OOP design, the optimal number of substripes is $(\sqrt {r-1}+r-1)$ , the average repair bandwidth ratio of systematic nodes can be as low as $\frac {2\sqrt {r-1}+1}{2\sqrt {r-1}+r}$ , and the average repair bandwidth ratio of parity nodes reaches $\left({\frac {\sqrt {r-1}+1}{r}+\frac {(r-1)^{2}-\sqrt {(r-1)^{3}}}{rk}}\right)$. In contrast to the existing piggybacking designs, OOP can further reduce the repair bandwidth of both system nodes and parity nodes. [ABSTRACT FROM AUTHOR]

Up to now, many $(k+r,k,N)$ minimum-storage regenerating (MSR) codes with $k$ information nodes, $r$ parity nodes, and node capacity $N$ have been proposed. However, most of them are constructed over a relatively large finite field. In this paper, we propose three high-rate MSR codes over small finite fields. First, the new MSR code $\mathcal {C}_{1}$ with the optimal access property for all nodes is constructed over small finite field $\mathbb {F}_{q}$ , for example $q=3$ for even $r$ or $q\ge r+1$ for odd $r$ , which is much smaller than that of the known one given by Ye and Barg. Further, considering to reduce the node capacity, another new MSR code $\mathcal {C}_{2}$ over $\mathbb {F}_{q}$ with $q\ge r+2$ is generated based on $\mathcal {C}_{1}$ , which can effectively reduce the node capacity of $\mathcal {C}_{1}$ by a factor of $r^{r-1}$. However, only the first $k$ nodes of $\mathcal {C}_{2}$ have the optimal access property. Therefore, the new MSR code $\mathcal {C}_{3}$ over $\mathbb {F}_{q}$ with $q\ge r+2$ which has the optimal access property for all nodes is proposed by modifying $\mathcal {C}_{2}$. Notably, in contrast to $\mathcal {C}_{1}$ , the node capacity of $\mathcal {C}_{3}$ is decreased by a factor of $r^{r-2}$. [ABSTRACT FROM AUTHOR]

We study the Singleton-type bound that provides an upper limit on the minimum distance of locally repairable codes. We present an improved bound by carefully analyzing the combinatorial structure of the repair sets. Thus, we show the previous bound is unachievable for certain parameters. We then also provide explicit constructions of optimal codes which show that for certain parameters the new bound is sharp. Additionally, as a byproduct, some previously known codes are shown to attain the new bound and are thus proved to be optimal. [ABSTRACT FROM AUTHOR]

We construct maximally recoverable codes (corresponding to partial MDS codes) which are based on linearized Reed-Solomon codes. The new codes have a smaller field size requirement compared with known constructions. For certain asymptotic regimes, the constructed codes have order-optimal alphabet size, asymptotically matching the known lower bound. [ABSTRACT FROM AUTHOR]

In this paper, locally repairable codes which have optimal minimum Hamming distance with respect to the bound presented by Prakash et al. are considered. New upper bounds on the length of such optimal codes are derived. The new bounds apply to more general cases, and have weaker requirements compared with the known ones. In this sense, they both improve and generalize previously known bounds. Further, optimal codes are constructed, whose length is order-optimal with respect to the new upper bounds. Notably, the length of the codes is super-linear in the alphabet size. [ABSTRACT FROM AUTHOR]

Locally repairable codes are desirable for distributed storage systems to improve the repair efficiency. In this paper, a connection between locally repairable codes with multiple disjoint repair sets and packings is derived under the condition that each repair set contains exactly one check symbol. Particularly, conditions under which an optimal locally repairable code corresponds to a packing are also characterized. As an application of this connection, some optimal locally repairable codes can be obtained by packings. Specifically, two constructions of locally repairable codes are proposed which not only generalize some known explicit constructions but also give optimal locally repairable codes with flexible new parameters. [ABSTRACT FROM AUTHOR]