Dianhua Wu

In an (n, k, d) rack-aware storage model, the system consists of n nodes uniformly distributed across n successive racks, such that each rack contains u nodes of equal capacity and the reconstructive degree satisfies k = ku + v where k = ⌊k/u⌋, 0 ≤ v ≤ u − 1. Suppose there are h ≥ 1 failed nodes in a rack (called the host rack). Then together with its surviving nodes, the host rack downloads recovery data from d helper racks and repairs its failed nodes. In this paper, we focus on the rack-aware minimum storage regenerating (MSR) codes for repairing h failed nodes within the same rack. By using the coupled-layer construction with the alignment technique, we construct a class of rack-aware MSR codes for all k+1 ≤ d ≤ n−1 which achieve the small sub-packetization l = s^⌈n/s⌉ where the field size q increases linearly with n and s = d − k + 1. In addition, these codes achieve the optimal repair bandwidth for 1 ≤ h ≤ u−v, and the asymptotically optimal repair bandwidth for u − v + 1 ≤ h ≤ u. In particular, they achieve the optimal access when h = u−v. It is worth noting that the existing rack-aware MSR codes which achieve the same sub-packetization l = s^⌈n/s⌉ are only known for the special case of d = n − 1, h = 1, and the field size is much larger than ours.

The rack-aware storage model improves repair efficiency by exploiting locality within racks to minimize cross-rack traffic in a distributed storage system. While the partially cooperative repair model presents a solution for multiple node failures that reduces the need to exchange data with all other host racks (defined as racks containing failed nodes), thus enhancing system flexibility. In this paper, we focus on rack-aware minimum storage partially cooperative regenerating (MSPCR) codes for repairing multiple node failures. We first derive the lower bound on the repair bandwidth for rack-aware MSPCR codes using extremal combinatorics, and then explicitly construct the first class of (asymptotically) optimal repair schemes for rack-aware MSPCR codes with a sub-packetization level of (s + h − δ)sⁿ, which is smaller than that of the known rack-aware minimumstorage cooperative regenerating (MSCR) codes when δ ≥ 2. By utilizing the grouping technique, we explicitly construct the second class of (asymptotically) optimal repair schemes for rack-aware MSPCR codes with a sub-packetization level of 2ⁿ. In particular, when δ = 1, our second codes reduce to rack-aware MSCR codes, while achieving an (h + 1)-fold reduction in sub-packetization level compared to the known rack-aware MSCR codes.

A design is said to be super‐simple if the intersection of any two blocks has at most two elements. A design with index is said to be completely reducible, if its blocks can be partitioned into nonempty collections , such that each together with the point set forms a design with index unity. In this paper, it is proved that there exists a completely reducible super‐simple (CRSS) balanced incomplete block design (‐BIBD for short) if and only if and or . A ‐ary constant weight code (CWC) of length with weight and distance is denoted as a code. The maximum size of a code is denoted as , and the codes achieving this size are called optimal. CRSS designs with index are closely related to ‐ary CWCs. By using the results of CRSS ‐BIBDs, s are determined for all .

Coded distributed computing (CDC) introduced by Li et al. can greatly reduce the communication load for MapReduce computing systems. In the cascaded CDC with $K$ workers, $N$ input files and $Q$ output functions, each input file will be mapped by $r$ workers and each output function will be computed by $s$ workers such that coding techniques can be applied to create multicast opportunities. The main drawback of most existing CDC schemes is that they require the original data to be split into a large number of input files that grows exponentially with $K$ , which would significantly increase the coding complexity and degrade the system performance. In this paper, we first use a classical combinatorial structure $t$ -design, for any integer $t\geq 2$ , to develop a low-complexity and communication-efficient CDC with $r=s$ . Our scheme has much smaller $N$ and $Q$ than the existing schemes under the same parameters $K$ , $r$ , and $s$ ; and achieves smaller communication loads compared with the state-of-the-art schemes when $K$ is relatively large. Remarkably, unlike the previous schemes that realize on large operation fields, our scheme operates in one-shot communication on the minimum binary field $\mathbb{F}_2$ . With a derived lower bound on the communication load under one-shot linear delivery, we show that the $t$ -design scheme is asymptotically optimal. Furthermore, we show that our construction method can incorporate the other combinatorial structures that have a similar property to $t$ -design. For instance, we use $t$ -GDD to obtain another one-shot asymptotically optimal CDC scheme over $\mathbb{F}_2$ that has different parameters from $t$ -design. Finally, we show that our construction method can also be used to construct CDC schemes with $r\neq s$ that have small file number and output function number.

A design is said to be super‐simple if the intersection of any two blocks has at most two elements. A design with index is said to be ‐decomposable, if its blocks can be partitioned into nonempty collections , , such that each with the point set forms a design with index . In this paper, it is proved that there exists a ‐decomposable super‐simple ‐BIBD (balanced incomplete block design) if and only if and .