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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]

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]

In this paper, the partial Hamming correlation properties of frequency-hopping sequences (FHSs) are discussed. The Peng–Fan bounds on sets of FHSs are generalized to the case of partial correlation. Both individual FHSs with optimal partial autocorrelation and sets of FHSs with optimal partial correlation are presented. The former has more new parameters compared with the known individual FHSs with optimal partial autocorrelation, while the later is obtained in the literature for the first time. [ABSTRACT FROM PUBLISHER]