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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]

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]

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]

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]