This paper investigates the coded caching problem in a multi-access multiple-input single-output (MAMISO) network with the combinatorial topology. The considered system consists of a server containing $N$ files, $Λ$ cache nodes, and $K$ cache-less users, where each user can access a unique subset of $r$ cache nodes. The server is equipped with $L$ transmit antennas. Our objective is to design a caching scheme that simultaneously achieves a high sum Degree of Freedom (sum-DoF) and low subpacketization complexity. To address this challenge, we formulate the design of multi-antenna placement delivery arrays (MAPDA) as a $0$--$1$ knapsack problem to maximize the achievable DoF, thereby transforming the complex combinatorial caching structure into a tractable optimization framework that yields efficient cache placement and flexible delivery strategies. Theoretical and numerical analyses demonstrate that: for networks with combinatorial topologies, the proposed scheme achieves a higher sum-DoF than existing schemes. Under identical cache size constraints, the subpacketization level remains comparable to existing linear subpacketization schemes. Moreover, under specific system conditions, the proposed scheme attains the theoretical maximum sum-DoF of $\min\{L+KM/N, K\}$ while achieving further reductions subpacketization. For particular combinatorial structures, we further derive optimized constructions that achieve even higher sum-DoF with lower subpacketization. ```

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.

Coded distributed computing (CDC), introduced by Li et al., effectively reduces communication load in MapReduce systems. In cascaded CDC with$K$nodes,$N$input files, and$Q$output functions, each input file is mapped by$r \geq 1$nodes, and each output function is computed by$s>1$nodes, enabling coding for multicast opportunities. However, existing CDC schemes often require splitting data into exponentially many files or functions as$K$grows, increasing complexity and degrading performance. This paper addresses the case of$K / s \in \mathbb{N}$, proposing a low-complexity CDC scheme through carefully designing the strategies of data placement and output function assignment. The proposed scheme offers key advantages:$\mathbf{1}$) multicast gains of$(r+s-1)(1-1 / s)$and approximately$r+s-1$for large$s$, with better communication load than the well-known Li et al.'s scheme; 2) reduce input and output file requirements; and 3) binary field$\mathbb{F}_{2}$operations implement in a one-shot manner, enabling immediate decoding. We also derive a new information-theoretic bound under the proposed strategies, showing that the communication load is order-optimal within a factor of 2 and approximately optimal when$K$is sufficiently large for a given$r$.

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.

Let W = {w1, . . . , wr} be a set of positive integers, λca positive integer, Λa= (λa(1), . . . λa(r)) an r-tuple of positive integers, and Q = (q1, . . . qr) an r-tuple of positive rational numbers whose sum is 1. In 1996, Yang introduced variable-weight optical orthogonal code, (n, W, Λa, λc, Q)-OOC, for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. Some work had been done on the constructions of optimal (n, W, Λa, 1, Q)-OOCs with unequal auto-correlation constraints for W = {3, 4} and {3, 5}, while little is known on optimal (n, W, Λa, 1, Q)-OOCs for |W| ≥ 3. In this paper, we focus our main attentions on (n, {3, 4, 5}, Λa, 1, Q)-OOCs with Λa∈ {(2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)}. Tight upper bounds on the maximum code size of (n, {3, 4, 5}, Λa, 1, Q)-OOCs are obtained, and infinite classes of optimal (n, {3, 4, 5}, Λa, 1, Q)-OOCs are constructed.

Zero-difference balanced (ZDB) functions were introduced recently by Ding for the construction of optimal constant-composition codes, and optimal and perfect difference systems of sets. They are closely related to partitioned difference families. In this paper, we present generic constructions of ZDB functions from functions with difference-balanced property. In particular, two classes of ZDB functions with new and flexible parameters are reported. Employing these new ZDB functions, we obtain at the same time optimal (1) constant-composition codes, (2) constant-weight codes, and (3) perfect difference systems of sets, all with new and flexible parameters.

Seven infinite classes of relative difference families with variable block sizes are presented explicitly. In particular, a balanced (gv,g,K,1)-DF withg=Σk∈K[(k2-k)/2] is explicitly given for: (i)K={3,4,5} and everyvcoprime to 6; (ii)K={3,4,6}, {3,5,6} or {3,4,5,6} and everyvcoprime to 30. As far as the authors are aware, these difference families can be viewed as the first explicit constructions of infinite classes of optimal variable-weight optical orthogonal codes with more than two weights. It is observed, however, that there are infinitely many values ofvfor which an optimal (v,W,1,Q) -OOC exists, whatever the set of weightsWand the weight distribution sequenceQare.

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, four general constructions for optimal cyclic packings and optimal variable-weight OOCs are presented. Many new infinite classes of optimal (v, W,1,Q)-OOCs are obtained. New infinite classes of optimal (v, W,1,Q)-OOCs for |W| ≥ 4 are easily obtained by the constructions.

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$\{4, 5\}$.

Variable-weight Optical orthogonal code (OOC) was introduced by G-C Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirement. In this paper, a construction for optimal variable-weight OOCs via cyclic difference families is given. Several new constructions for cyclic difference families are also given. By using these constructions, new optimal (n,W, 1,Q)-OOCs for 2 ≤ |W| ≤ 4 are constructed.