Abundant high-rate$(n, k)$minimum storage regenerating (MSR) codes have been reported in the literature. However, most of them require contacting all the surviving nodes during a node repair process, resulting in a repair degree of$d=n-1$. In practical systems, it may not always be feasible to connect and download data from all surviving nodes, as some nodes may be unavailable. Therefore, there is a need for MSR code constructions with a repair degree of$d < n-1$. Up to now, only a few$(n, k)$MSR code constructions with repair degree$d < n-1$have been reported, some have a large sub-packetization level, a large finite field, or restrictions on the repair degree$d$. In this paper, we propose a new$(n, k)$MSR code construction that works for any repair degree$d>k$, and has a smaller sub-packetization level or finite field than some existing constructions. Additionally, in conjunction with a previous generic transformation to reduce the sub-packetization level, we obtain an MDS array code with a small sub-packetization level and$(1+\epsilon)$-optimal repair bandwidth (i.e.,$(1+\epsilon)$times the optimal repair bandwidth) for repair degree$d=n-1$. This code outperforms some existing ones in terms of either the sub-packetization level or the field size.

In the literature, most of the known high-rate$(n,k)$MDS array codes with the optimal repair property only support a single repair degree (i.e., the number of helper nodes contacted during a repair process)$d$, where$k\le d\le n-1$. However, in practical storage systems, the number of available nodes changes frequently. Thus, it is preferred to construct$(n,k)$MDS array codes with multiple repair degrees and the optimal repair property for all nodes. To the best of our knowledge, only two high-rate MDS array codes have such properties in the literature, which were proposed by Ye and Barg (IEEE Trans. Inform. Theory, 63(10), 2001–2014, 2017). However, their sub-packetization levels are relatively large. In this paper, we present a generic construction method that can convert some MDS array codes with a single repair degree into ones with multiple repair degrees and optimal repair property for a set of nodes, while the repair efficiency/degrees of the remaining nodes can be kept. As an application of the generic construction method, an explicit construction of high-rate MDS array code with multiple repair degrees and the optimal access property for all nodes is obtained over a small finite field by choosing the code proposed by Vajha et al. as the base code. Especially, the sub-packetization level is much smaller than that of the two codes proposed by Ye and Barg concerning the same parameters$n$and$k$.

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.

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.

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 C1with the optimal access property for all nodes is constructed over small finite field Fq, for example q = 3 for even r or q ≥ 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 C2over Fqwith q ≥ r + 2 is generated based on C1, which can effectively reduce the node capacity of C1by a factor of rr-1. However, only the first k nodes of C2have the optimal access property. Therefore, the new MSR code C3over Fqwith q ≥ r + 2 which has the optimal access property for all nodes is proposed by modifying C2. Notably, in contrast to C1, the node capacity of C3is decreased by a factor of rr-2.