Coded distributed computing (CDC) is an efficient method to reduce the communication load in general distributed computing frameworks such as MapReduce. In these systems, one usually needs to split the data set into disjoint files and design several output functions to complete a computational task. Li et al. provided some CDC schemes achieving optimal communication load. However, as the number of computing nodes increases, the numbers of input files and output functions of such schemes grow too fast to be applied in practice. In this paper, several infinite families of cascaded CDC schemes, where each output function is computed multiple times, are constructed by taking advantage of symmetric designs. Most importantly, the numbers of input files and output functions of each new scheme are linear with the number of computing nodes and the communication load of each new scheme approximates to that of the CDC scheme derived by Li et al. when the number of nodes becomes large.

In coded caching system, we prefer to design a scheme with the rate R and the packet number F of each file split as small as possible since the efficiency of transmission in the peak traffic times increases with the decreasing of R and the realizing complexity increases with the increasing of F. Up to now, almost all of the previously known schemes can be realized by the combinatorial structure which is called placement delivery array (PDA). In this paper, we also study the schemes by means of PDAs. We first show that given the minimum rate, the scheme proposed by Maddah-Ali and Niesen (MN scheme) has the minimum packet number which is too large in practice. From the view point of combinatorial design, two variant MN schemes, which can significantly reduce the packet number by increasing some rate, are obtained. Especially one of these schemes has better performance than the scheme generated by the well known grouping method.

Coded caching, which could significantly reduce the maximum amount of transmission rate during the peak traffic times in wireless network, has been widely studied recently. Apart from the transmission rate, sub-packetization F reflecting the implementation complexity, is also concerned in coded caching. The grouping method proposed by Shanmugam et al. is wellknown and widely used to reduce the sub-packetization level of the coded caching problem. In this paper, we propose a concatenating construction method for coded caching schemes, which generalizes the grouping method. Moreover, we demonstrate the advantage of our method in reducing the transmission rate over the grouping method. In particular, some new explicit schemes are obtained from previously known schemes. From one of these schemes, we can derive all the results by Tang and Ramamoorthy as special cases. Furthermore, the analysis and comparison of these new schemes are also performed.

Coded caching scheme recently has become quite popular in the wireless network, since the maximum transmission amount R reduces effectively during the peak-traffic times. To realize a coded caching scheme, each file must be divided into F packets, which usually increases the computation complexity of a coded caching scheme. So we prefer to design a scheme with R and F as small as possible in practice. However, there exists a tradeoff between R and F. In this paper, we generalize the schemes constructed by Shangguan et al. (IEEE TRANSACTIONS ON INFORMATION THEORY, 64, 5755-5766, 2018) and Yan et al. (IEEE TRANSACTIONS ON INFORMATION THEORY 63, 5821-5833, 2017), respectively. These two classes of schemes have a wider range of application due to the more flexible memory size than the original ones. By comparing with the previous known deterministic schemes, our new schemes have advantages on R or F.

Let $$\mathbb{C }$$ be a code of length $$n$$ over an alphabet of $$q$$ letters. The descendant code $$\mathsf{desc}(\mathbb C _0)$$ of $$\mathbb C _0 = \{\mathbf{c}^1, \mathbf{c}^2, \ldots , \mathbf{c}^t\} \subseteq \mathbb{C }$$ is defined to be the set of words $$\mathbf{x} = (x_1, x_2, \ldots ,x_n)$$ such that $$x_i \in \{c^1_i, c^2_i, \ldots , c^t_i\}$$ for all $$i=1, \ldots , n$$ . $$\mathbb{C }$$ is a $$\overline{t}$$ -separable code if for any two distinct $$\mathbb{C }_1, \mathbb{C }_2 \subseteq \mathbb{C }$$ such that $$|\mathbb{C }_1| \le t$$ , $$|\mathbb{C }_2| \le t$$ , we always have $$\mathsf{desc}(\mathbb{C }_1) \ne \mathsf{desc}(\mathbb{C }_2)$$ . The study of separable codes is motivated by questions about multimedia fingerprinting for protecting copyrighted multimedia data. Let $$M(\overline{t},n,q)$$ be the maximal possible size of such a separable code. In this paper, we provide an improved upper bound for $$M(\overline{2},2,q)$$ by a graph theoretical approach, and a new lower bound for $$M(\overline{2},2,q)$$ by deleting suitable points and lines from a projective plane, which coincides with the improved upper bound in some places. This corresponds to the bounds of maximum size of bipartite graphs with girth $$6$$ and a construction of such maximal bipartite graphs.