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.

Variable-weight optical orthogonal code (OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. In this paper, the upper bound on the size of variable-weight OOCs is improved, a cyclic$t\hbox{-}(v, W, \lambda , Q)$packing is introduced to construct a variable-weight OOC, an upper bound for the number of blocks of$t\hbox{-}(v, W, \lambda , Q)$packings is obtained, and an equivalence between optimal cyclic packing and optimal variable-weight optical orthogonal code is established. Recursive constructions for optimal$2\hbox{-}{\rm CP}(W, 1, Q;v)$s are also presented. By using skew starters and these constructions, infinite classes of optimal$(v, W, 1, \{1/2, 1/2\})$-OOCs are obtained for$W=\{3, 4\}$, and$\{4, 5\}$.