On Secure Coded Caching via Combinatorial Method

Coded caching is an efficient way to reduce network traffic congestion during peak hours by storing some content at the user's local cache memory without knowledge of later demands. The goal of coded caching design is to minimize the transmission rate and the subpacketization. In practice the demand for each user is sensitive since one can get the other users' preferences when it gets the other users' demands. The first coded caching scheme with private demands was proposed by Wan et al. However the transmission rate and the subpacketization of their scheme increase with the file number stored in the library. In this paper we consider the following secure coded caching: prevent the wiretappers from obtaining any information about the files in the server and protect the demands from all the users in the delivery phase. We firstly introduce a combinatorial structure called secure placement delivery array (SPDA in short) to realize a coded caching scheme for our security setting. Then we obtain three classes of secure schemes by constructing SPDAs, where one of them is optimal. It is worth noting that the transmission rates and the subpacketizations of our schemes are independent to the file number. Furthermore, comparing with the previously known schemes with the same security setting, our schemes have significantly advantages on the subpacketizations and for some parameters have the advantage on the transmission rates.Comment: 13 page

A Novel Recursive Construction for Coded Caching Schemes

As a strategy to further reduce the transmission pressure during the peak traffic times in wireless network, coded caching has been widely studied recently. And several coded caching schemes are constructed focusing on the two core problems in practice, i.e., the rate transmitted during the peak traffic times and the packet number of each file divided during the off peak traffic times. It is well known that there exits a tradeoff between the rate and the packet number. In this paper, a novel recursive construction is proposed. As an application, several new schemes are obtained. Comparing with previously known schemes, new schemes could further reduce packet number by increasing little rate. And for some parameters in coded caching systems, the packet number of our new schemes are smaller than that of schemes generated by memory sharing method which is widely used in the field of caching. By the way our new schemes include all the results constructed by Tang et al., (IEEE ISIT, 2790-2794, 2017) as special cases.Comment: 10 page

Coded Caching Schemes with Linear Subpacketizations

In coded caching system we prefer to design a coded caching scheme with low subpacketization and small transmission rate (i.e., the low implementation complexity and the efficient transmission during the peak traffic times). Placement delivery arrays (PDA) can be used to design code caching schemes. In this paper we propose a framework of constructing PDAs via Hamming distance. As an application, two classes of coded caching schemes with linear subpacketizations and small transmission rates are obtained.Comment: 14 page

Strongly Separable Codes

Binary $t$-frameproof codes ($t$-FPCs) are used in multimedia fingerprinting schemes where the identification of authorized users taking part in the averaging collusion attack is required. In this paper, a binary strongly $\bar{t}$-separable code ($\bar{t}$-SSC) is introduced to improve such a scheme based on a binary $t$-FPC. A binary $\bar{t}$-SSC has the same traceability as a binary $t$-FPC but has more codewords than a binary $t$-FPC. A composition construction for binary $\bar{t}$-SSCs from $q$-ary $\bar{t}$-SSCs is described, which stimulates the research on $q$-ary $\bar{t}$-SSCs with short length. Several infinite series of optimal $q$-ary $\bar{2}$-SSCs of length $2$ are derived from the fact that a $q$-ary $\bar{2}$-SSC of length $2$ is equivalent to a $q$-ary $\bar{2}$-separable code of length $2$. Combinatorial properties of $q$-ary $\bar{2}$-SSCs of length $3$ are investigated, and a construction for $q$-ary $\bar{2}$-SSCs of length $3$ is provided. These $\bar{2}$-SSCs of length $3$ have more than $12.5\%$ codewords than $2$-FPCs of length $3$ could have.Comment: 11 pages, submitted to Designs, Codes and Cryptography. arXiv admin note: text overlap with arXiv:1411.684

On the Placement Delivery Array Design for Coded Caching Scheme in D2D Networks

The coded caching scheme is an efficient technique as a solution to reduce the wireless network burden during the peak times in a Device-to-Device (D2D in short) communications. In a coded caching scheme, each file block should be divided into $F$ packets. It is meaningful to design a coded caching scheme with the rate and $F$ as small as possible, especially in the practice for D2D network. In this paper we first characterize coded caching scheme for D2D network by a simple array called D2D placement delivery array (DPDA in shot). Consequently some coded caching scheme for D2D network can be discussed by means of an appropriate DPDA. Secondly we derive the lower bounds on the rate and $F$ of a DPDA. According these two lower bounds, we show that the previously known determined scheme proposed by Ji et al., (IEEE Trans. Inform. Theory, 62(2): 849-869,2016) reaches our lower bound on the rate, but does not meet the lower bound on $F$ for some parameters. Finally for these parameters, we construct three classes of DPDAs which meet our two lower bounds. Based on these DPDAs, three classes of coded caching scheme with low rate and lower $F$ are obtained for D2D network.Comment: 20 page

On the Placement Delivery Array Design in Centralized Coded Caching Scheme

Caching is a promising solution to satisfy the ever increasing demands for the multi-media traffics. In caching networks, coded caching is a recently proposed technique that achieves significant performance gains over the uncoded caching schemes. However, to implement the coded caching schemes, each file has to be split into $F$ packets, which usually increases exponentially with the number of users $K$. Thus, designing caching schemes that decrease the order of $F$ is meaningful for practical implementations. In this paper, by reviewing the Ali-Niesen caching scheme, the placement delivery array (PDA) design problem is firstly formulated to characterize the placement issue and the delivery issue with a single array. Moreover, we show that, through designing appropriate PDA, new centralized coded caching schemes can be discovered. Secondly, it is shown that the Ali-Niesen scheme corresponds to a special class of PDA, which realizes the best coding gain with the least $F$. Thirdly, we present a new construction of PDA for the centralized caching system, wherein the cache size of each user $M$ (identical cache size is assumed at all users) and the number of files $N$ satisfies $M/N=1/q$ or ${(q-1)}/{q}$ ($q$ is an integer such that $q\geq 2$). The new construction can decrease the required $F$ from the order $O\left(e^{K\cdot\left(\frac{M}{N}\ln \frac{N}{M} +(1-\frac{M}{N})\ln \frac{N}{N-M}\right)}\right)$ of Ali-Niesen scheme to $O\left(e^{K\cdot\frac{M}{N}\ln \frac{N}{M}}\right)$ or $O\left(e^{K\cdot(1-\frac{M}{N})\ln\frac{N}{N-M}}\right)$ respectively, while the coding gain loss is only $1$.Comment: 21 pages, 2 figure

Optimal Locally Repairable Systematic Codes Based on Packings

Locally repairable codes are desirable for distributed storage systems to improve the repair efficiency. In this paper, we first build a bridge between locally repairable code and packing. As an application of this bridge, some optimal locally repairable codes can be obtained by packings, which gives optimal locally repairable codes with flexible parameters.Comment: 13 page

Linear Coded Caching Scheme for Centralized Networks

Coded caching systems have been widely studied to reduce the data transmission during the peak traffic time. In practice, two important parameters of a coded caching system should be considered, i.e., the rate which is the maximum amount of the data transmission during the peak traffic time, and the subpacketization level, the number of divided packets of each file when we implement a coded caching scheme. We prefer to design a scheme with rate and packet number as small as possible since they reflect the transmission efficiency and complexity of the caching scheme, respectively. In this paper, we first characterize a coded caching scheme from the viewpoint of linear algebra and show that designing a linear coded caching scheme is equivalent to constructing three classes of matrices satisfying some rank conditions. Then based on the invariant linear subspaces and combinatorial design theory, several classes of new coded caching schemes over $\mathbb{F}_2$ are obtained by constructing these three classes of matrices. It turns out that the rate of our new rate is the same as the scheme construct by Yan et al. (IEEE Trans. Inf. Theory 63, 5821-5833, 2017), but the packet number is significantly reduced. A concatenating construction then is used for flexible number of users. Finally by means of these matrices, we show that the minimum storage regenerating codes can also be used to construct coded caching schemes.Comment: 23 page

Constructions of Coded Caching Schemes with Flexible Memory Size

Coded caching scheme recently has become quite popular in the wireless network due to its effectively reducing the transmission amount (denote such an amount by $R$) during peak traffic times. However 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 construct a caching scheme that decreases the order of $F$ for practical implementations. In this paper, we construct four classes of new schemes where two classes can significantly reduce the value of $F$ by increasing a little $R$ comparing with the well known scheme proposed by Maddah-Ali and Niesen, and $F$ in the other two classes grows sub-exponentially with $K$ by sacrificing more $R$. It is worth noting that a tradeoff between $R$ and $F$, which is a hot topic in the field of caching scheme, is proposed by our constructions. In addition, our constructions include all the results constructed by Yan et al., (IEEE Trans. Inf. Theory 63, 5821-5833, 2017) and some main results obtained by Shangguan et al., (arXiv preprint arXiv:1608.03989v1) as the special cases.Comment: 18 page

Bounds and Constructions for 3‾\overline{3}-Separable Codes with Length 33

Separable codes were introduced to provide protection against illegal redistribution of copyrighted multimedia material. Let $\mathcal{C}$ be a code of length $n$ over an alphabet of $q$ letters. The descendant code ${\sf desc}(\mathcal{C}_0)$ of $\mathcal{C}_0 = \{{\bf c}_1, {\bf c}_2, \ldots, {\bf c}_t\} \subseteq {\mathcal{C}}$ is defined to be the set of words ${\bf x} = (x_1, x_2, \ldots,x_n)^T$ such that $x_i \in \{c_{1,i}, c_{2,i}, \ldots, c_{t,i}\}$ for all $i=1, \ldots, n$, where ${\bf c}_j=(c_{j,1},c_{j,2},\ldots,c_{j,n})^T$. $\mathcal{C}$ is a $\overline{t}$-separable code if for any two distinct $\mathcal{C}_1, \mathcal{C}_2 \subseteq \mathcal{C}$ with $|\mathcal{C}_1| \le t$, $|\mathcal{C}_2| \le t$, we always have ${\sf desc}(\mathcal{C}_1) \neq {\sf desc}(\mathcal{C}_2)$. Let $M(\overline{t},n,q)$ denote the maximal possible size of such a separable code. In this paper, an upper bound on $M(\overline{3},3,q)$ is derived by considering an optimization problem related to a partial Latin square, and then two constructions for $\overline{3}$-SC$(3,M,q)$s are provided by means of perfect hash families and Steiner triple systems.Comment: 19 page