Coded Caching via Projective Geometry: A new low subpacketization scheme

Coded caching was introduced as a technique of systematically exploiting locally available storage at the clients to increase the channel throughput via coded transmissions. Most known coded caching schemes in literature enable large gains in terms of the rate, however at the cost of subpacketization that is exponential in $K^{\frac{1}{r}}$ ($K$ being the number of clients, $r$ some positive integer). Building upon recent prior work for coded caching design via line graphs and finite-field projective geometries, we present a new scheme in this work which achieves a subexponential (in $K$) subpacketization of $q^{O((\log_{q}{K})^2)}$ and rate $\Theta\left(\frac{K}{(\log_{q} {K})^2}\right)$, for large $K$, and the cached fraction $\frac{M}{N}$ being upper bounded by a constant $\frac{2}{q^{\alpha-1}}$ (for some prime power $q$ and constant $\alpha \geq 2$) . Apart from this asymptotic improvement, we show that through some numerical comparisons that our present scheme has much lower subpacketization than previous comparable schemes, however with an increased rate and some increase in cache size required. For instance, we obtain practically relevant subpacketization levels such as $10^2 - 10^7$ for $10^2 - 10^4$ number of clients. Leveraging prior results on adapting coded caching schemes for the error-free broadcast channel to device to device (D2D) networks, we obtain a low- subpacketization scheme for D2D networks also, and give numerical comparison for the same with prior work.

In a traditional (H,r) combination network, each user connects to a unique set of r relays. However, few research efforts have considered the (H,r,u) multiaccess combination network problem wherein each unique set of r relays is connected by u users. In this paper, we focus on designing coded caching schemes for a (H,r,u) multiaccess combination network. By directly applying the well-known coding method (proposed by Zewail and Yener) for a (H,r) combination network, a coded caching scheme (called ZY scheme) for (H,r,u) multiaccess combination network is obtained. However, its subpacketization has an exponential order with the number of users which leads to high implementation complexity. In order to reduce subpacketization, a direct construction of a coded caching scheme (called the direct scheme) for (H,r,u) multiaccess combination network is proposed by means of combinational design theory, where the parameter u must be a combinatorial number. For the arbitrary parameter u, the hybrid construction of a coded caching scheme (called the hybrid scheme) for the (H,r,u) multiaccess combination network is proposed based on the direct scheme. Theoretical and numerical analysis shows that the direct scheme and the hybrid scheme have a smaller transmission load for each relay compared with the naive scheme (which is obtained by repeatedly applying the coded caching scheme for a traditional (H,r) combination network by u times) and have much lower subpacketization compared with the ZY scheme.

The network load is increasing due to the spread of content distribution services. Caching is recognized as a technique to reduce a peak network load by prefetching popular contents into memories of users. Coded caching is a new caching approach based on a carefully designed content placement in order to create coded multicasting opportunities. Coded caching schemes in single-rate networks are evaluated by the trade-off between the size of memory and the size of delivered data. Considering the network with multiple transmission rates, it is important how to operate multicast. The multicast needs to use the smallest rate among the available rates for target users. In multi-rate wireless networks, multicast scheduling schemes are evaluated by a throughput and a delay. In this paper, we discuss coded caching in the multi- rate wireless networks. We newly define a throughput and a delay for evaluating the coded caching scheme and propose a new coded caching scheme suitable for these metrics. We compare the proposed coded caching scheme with conventional coded caching schemes.

Coded caching schemes on broadcast networks with user caches help to offload traffic from peak times to off-peak times by prefetching information from the server to the users during off-peak times and thus serving the users more efficiently during peak times using coded transmissions. We consider the problem of secretive coded caching which was proposed recently, in which a user should not be able to decode any information about any file that the user has not demanded. We propose a new secretive coded caching scheme which has a lower average rate compared to the existing state-of-the-art scheme, for the same memory available at the users. The proposed scheme is based on exploiting the presence of common demands between multiple users.

In the coded caching problem apart from transmission rate, reducing the number of packets $F$ ia also meaningful for practical implementation for coded caching scheme. Yan et al. reformulate the centralized coded caching schemes as designing a corresponding placement delivery array (PDA). In this letter, we propose two concatenating constructions by concatenating some known PDAs to obtain new PDAs therefore new code caching schemes with low sub-packetization level. Further, the comparison and analysis of parameters between our new schemes and known schemes is also performed.

Coded Caching, proposed by Maddah-Ali and Niesen (MAN), has the potential to reduce network traffic by pre-storing content in the users’ local memories when the network is underutilized and transmitting coded multicast messages that simultaneously benefit many users at once during peak-hour times. This paper considers the linear function retrieval version of the original coded caching setting, where users are interested in retrieving a number of linear combinations of the data points stored at the server, as opposed to a single file. This extends the scope of the authors’ past work that only considered the class of linear functions that operate element-wise over the files. On observing that the existing cache-aided scalar linear function retrieval scheme does not work in the proposed setting, this paper designs a novel coded caching scheme that outperforms uncoded caching schemes that either use unicast transmissions or let each user recover all files in the library.

Three constructions of perfect authentication codes from projective geometry over finite fields

In this paper, a centralized coded caching scheme is developed for wireless backhaul networks with heterogeneous channel conditions, i.e., the wireless channels between microBSs and a macroBS have different link rates and packet loss rates. To address these heterogeneous channel conditions, a joint random linear network coding and nested coded modulation (RLNCNCM) encoding scheme is designed for the delivery phase. Based on RLNC-NCM encoding, broadcast opportunities can be fully utilized for each coded subfile transmission. To support RLNC-NCM transmission, unequal file partition and cache size allocation are conducted in the placement phase. It is shown that a microBS with weaker channel condition needs to be allocated with more cache size. The theoretical achievable transmission time of the RLNC-NCM coded caching scheme is also derived. Moreover, numerical results show that the coded caching scheme developed in this paper significantly reduces the transmission time as compared to the existing schemes.

This paper studies device to device (D2D) coded-caching with information theoretic security guarantees. A broadcast network consisting of a server, which has a library of files, and end users equipped with cache memories, is considered. Information theoretic security guarantees for confidentiality are imposed upon the files. The server populates the end user caches, after which D2D communications enable the delivery of the requested files. Accordingly, we require that a user must not have access to files it did not request, i.e., secure caching. First, a centralized coded caching scheme is provided by jointly optimizing the cache placement and delivery policies. Next, a decentralized coded caching scheme is developed that does not require the knowledge of the number of active users during the caching phase. Both schemes utilize non-perfect secret sharing and one-time pad keying, to guarantee secure caching. Furthermore, the proposed schemes provide secure delivery as a side benefit, i.e., any external entity which overhears the transmitted signals during the delivery phase cannot obtain any information about the database files. The proposed schemes provide the achievable upper bound on the minimum delivery sum rate. Lower bounds on the required transmission sum rate are also derived using cut-set arguments indicating the multiplicative gap between the lower and upper bounds. Numerical results indicate that the gap vanishes with increasing memory size. Overall, the work demonstrates the effectiveness of D2D communications in cache-aided systems even when confidentiality constraints are imposed at the participating nodes and against external eavesdroppers.

This paper studies a multi-tier cache-aided relaying network, where the destination <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> is randomly located in the network and it requests files from the source <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> through the help of cache-aided base station (BS) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> relays. In this system, the multi-tier architecture imposes a significant impact on the system collaborative caching and file delivery, which brings a big challenge to the system performance evaluation and optimization. To address this problem, we first evaluate the system performance by deriving analytical outage probability expression, through fully taking into account the random location of the destination and different file delivery modes related to the file caching status. We then perform the asymptotic analysis on the system outage probability when the signal-to-noise ratio (SNR) is high, to enclose some important and meaningful insights on the network. We further optimize the caching strategies among the relays and BS, to improve the network outage probability. Simulations are performed to show the effectiveness of the derived analytical and asymptotic outage probability for the proposed caching strategy. In particular, the proposed caching is superior to the conventional caching strategies such as the most popular content (MPC) and equal probability caching (EPC) strategies.

Let PG be the ‐dimensional projective space over and let be a simple graph of order for some . A design over is a collection of graphs (blocks) isomorphic to with the following properties: the vertex set of every block is a subspace of PG; every two distinct points of PG are adjacent in exactly blocks. This new definition covers, in particular, the well‐known concept of a design over corresponding to the case that is complete. In this study of a foundational nature we illustrate how difference methods allow us to get concrete nontrivial examples of ‐decompositions over or for which is a cycle, a path, a prism, a generalized Petersen graph, or a Moebius ladder. In particular, we will discuss in detail the special and very hard case that is complete and , that is, the Steiner 2‐designs over a finite field. Also, we briefly touch the new topic of near resolvable designs over . This study has led us to some (probably new) collateral problems concerning difference sets. Supported by multiple examples, we conjecture the existence of infinite families of Γ‐decompositions over a finite field that can be obtained by suitably labeling the vertices of Γ with the elements of a Singer difference set.

The maximum number of mutually orthogonal Sudoku Latin squares (MOSLS) of order $$n=m^2$$n=m2 is $$n-m$$n-m. In this paper, we construct for $$n=q^2$$n=q2, q a prime power, a set of $$q^2-q-1$$q2-q-1 MOSLS of order $$q^2$$q2 that cannot be extended to a set of $$q^2-q$$q2-q MOSLS. This contrasts to the theory of ordinary Latin squares of order n, where each set of $$n-2$$n-2 mutually orthogonal Latin Squares (MOLS) can be extended to a set of $$n-1$$n-1 MOLS (which is best possible). For this proof, we construct a particular maximal partial spread of size $$q^2-q+1$$q2-q+1 in $$\mathrm {PG}(3,q)$$PG(3,q) and use a connection between Sudoku Latin squares and projective geometry, established by Bailey, Cameron and Connelly.

Coded caching has received great attention as a promising method to alleviate the network traffic during peak-traffic times. Placement delivery array (PDA) was proposed to characterize the placement and delivery phases in a centralized coded caching system. In this paper, from the viewpoint of combinational design we construct a new class of PDA, and further reduce the subpacketization compared to the existing coded caching schemes by means of maximum distance separable (MDS) code.

In recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or, in the ω-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence of pseudolinearity and local modularity.The proof necessitated an extension of the model-theoretic framework to includealmost hyperimaginaries, and the study ofpolygroups.

Constructions of ( q, k, 1) difference families with q a prime power and k = 4,5