Hengming Zhao

The OCDMA network using two Dimensional Variable-Weight Optical Orthogonal Codes (2D VWOOC) can support diverse Quality of Services (QoS) classes and multimedia services, and make the better use of bandwidth resources in fiber optical networks. To simplify practical implementation, the At Most One-Pulse Per Wavelength (AM-OPPW) restriction is often appended to a 2D VWOOC. In this paper, the upper bound on the size of AM-OPPW 2D VWOOCs is derived, semicyclic group divisible designs are introduced to construct AM-OPPW 2D VWOOCs, and an equivalence between optimal AM-OPPW 2D VWOOCs and Semicyclic Group Divisible Designs (SCGDDs) is established. Some direct and recursive constructions for SCGDDs are also presented. Consequently, several infinite families of optimal AM-OPPW 2D VWOOCs are obtained.

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.