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In general coding theory, we often assume that error is observed in transferring or storing encoded symbols, while the process of encoding itself is error-free. Motivated by recent applications of coding theory, in this paper, we consider the case where the process of encoding is distributed and prone to error. We introduce the problem of distributed encoding, comprised of a set of $K \in \mathbb {N}$ isolated source nodes and $N \in \mathbb {N}$ encoding nodes. Each source node has one symbol from a finite field, which is sent to each of the encoding nodes. Each encoding node stores an encoded symbol from the same field, as a function of the received symbols. However, some of the source nodes are controlled by the adversary and may send different symbols to different encoding nodes. Depending on the number of the adversarial nodes, denoted by $\beta \in \mathbb {N}$ , and the cardinality of the set of symbols that each one generates, denoted by $v \in \mathbb {N}$ , the process of decoding from the encoded symbols could be impossible. Assume that a decoder connects to an arbitrary subset of $t \in \mathbb {N}$ encoding nodes and wants to decode the symbols of the honest nodes correctly, without necessarily identifying the sets of honest and adversarial nodes. An important characteristic of a distributed encoding system is $t^{*} \in \mathbb {N}$ , the minimum of such $t$ , which is a function of $K$ , $N$ , $\beta $ , and $v$. In this paper, we study the distributed linear encoding system, i.e. one in which the encoding nodes use linear coding. We show that $t^{*}_{\textrm {Linear}}=K+2\beta (v-1)$ , if $N\ge K+2\beta (v-1)$ , and $t^{*}_{\textrm {Linear}}=N$ , if $N\le K+2\beta (v-1)$. In order to achieve $t^{*}_{\textrm {Linear}}$ , we use random linear coding and show that in any feasible solution that the decoder finds, the messages of the honest nodes are decoded correctly. In order to prove the converse of the fundamental limit, we show that when the adversary behaves in a particular way, it can always confuse the decoder between two feasible solutions that differ in the message of at least one honest node. [ABSTRACT FROM AUTHOR]

In this paper, we consider $K$ –user interference channels with $M$ antennas per node and with backhaul collaboration in one side (among the transmitters or among the receivers), for $M,K\in \mathbb {N}$ , and investigate the tradeoff between the rate in the channel versus the communication load in the backhaul. In this investigation, each node is equipped with $M$ antennas and we focus on a first order approximation result, where the rate of the wireless channel is measured by the degrees of freedom (DoF) per user, and the load of the backhaul is measured by the entropy of backhaul messages per user normalized by $\log $ of transmit power, at high power regimes. This tradeoff is fully characterized for the case of even values of $K$ and approximately characterized for the case of odd values of $K$ , with vanishing approximation gap as $K$ grows. To achieve DoF of $M$ per user, this result establishes the asymptotic optimality of the most straightforward scheme, called central processing, in which the messages are collected at one of the nodes, centrally processed, and forwarded back to each node. In addition, this result shows that the gain of the schemes, relying on distributed processing, through pairwise communication among the nodes (e.g., cooperative alignment) does not scale with the size of the network. For the converse, we develop a new outer-bound on the tradeoff based on splitting the set of collaborative nodes (transmitters or receivers) into two subsets and assuming full cooperation within each group. We further present a sufficient condition on the wireless channel connectivity, which although more relaxed, guarantees the validity of the above tradeoff. Finally, we show that verifying this condition takes a polynomial time in the network size. [ABSTRACT FROM AUTHOR]

We study the capacity region of the two-user binary fading (or erasure) interference channel, where the transmitters have no knowledge of the channel state information. We develop new inner bounds and outer bounds for this problem. We identify three regimes based on the channel parameters: weak, moderate, and strong interference regimes. Interestingly, this is similar to the generalized degrees of freedom of the two-user Gaussian interference channel, where transmitters have perfect channel knowledge. We show that for the weak interference regime, treating interference as erasure is optimal while for the strong interference regime, decoding interference is optimal. For the moderate interference regime, we provide new inner and outer bounds. The inner bound is based on a modification of the Han–Kobayashi scheme for the erasure channel, enhanced by time-sharing. We study the gap between our inner bound and our outer bounds for the moderate interference regime and compare our results to that of the Gaussian interference channel. Deriving our new outer bounds has three main steps. We first create a contracted channel that has fewer states compared with the original channel, in order to make the analysis tractable. We then prove the correlation lemma that shows an outer bound on the capacity region of the contracted channel and also serves as an outer bound for the original channel. Finally, using the conditional entropy leakage lemma, we derive our outer bound on the capacity region of the contracted channel. [ABSTRACT FROM AUTHOR]

We consider a network consisting of a file server connected through a shared link to a number of users, each equipped with a cache. Knowing the popularity distribution of the files, the goal is to optimally populate the caches, such as to minimize the expected load of the shared link. For a single cache, it is well known that storing the most popular files is optimal in this setting. However, we show here that this is no longer the case for multiple caches. Indeed, caching only the most popular files can be highly suboptimal. Instead, a fundamentally different approach is needed, in which the cache contents are used as side information for coded communication over the shared link. We propose such a coded caching scheme and prove that it is close to optimal. [ABSTRACT FROM AUTHOR]

Although interference alignment (IA) can theoretically achieve the optimal degrees of freedom (DoFs) in the $K$ -user Gaussian interference channel, its direct application comes at the prohibitive cost of precoding over exponentially many signaling dimensions. On the other hand, it is known that practical one-shot IA precoding (i.e., linear schemes without symbol expansion) provides a vanishing DoFs gain in large fully connected networks with generic channel coefficients. In our previous work, we introduced the concept of cellular IA for a network topology induced by hexagonal cells with sectors and nearest-neighbor interference. Assuming that neighboring sectors can exchange decoded messages (and not received signal samples) in the uplink, we showed that linear one-shot IA precoding over $M$ transmit/receive antennas can achieve the optimal $M/2$ DoFs per user. In this paper, we extend this framework to networks with omni-directional (non-sectorized) cells and consider a limited practical scenario where users have 2 antennas, and base-stations have 2, 3, or 4 antennas. We provide linear one-shot IA schemes for the $2\times 2$ , $2\times 3$ , and $2\times 4$ cases, and show the achievability of 3/4, 1, and 7/6 DoFs per user, respectively. DoFs converses for one-shot schemes require the solution of a discrete optimization problem over a number of variables that grows with the network size. We develop a new approach to transform such optimization problem into a tractable linear program with significantly fewer variables. This approach is used to show that 3/4 DoFs per user are indeed optimal for one-shot schemes over large (extended) cellular network with $2\times 2$ links. [ABSTRACT FROM AUTHOR]

Interference alignment promises that, in Gaussian interference channels, each link can support half of a degree of freedom (DoF) per pair of transmit–receive antennas. However, in general, this result requires to precode the data bearing signals over a signal space of asymptotically large diversity, e.g., over an infinite number of dimensions for time-frequency varying fading channels, or over an infinite number of rationally independent signal levels, in the case of time-frequency invariant channels. In this paper, we consider a wireless cellular system scenario where the promised optimal DoFs are achieved with linear precoding in one-shot (i.e., over a single time-frequency slot). We focus on the uplink of a symmetric cellular system, where each cell is split into three sectors with orthogonal intrasector multiple access. In our model, interference is local, i.e., it is due to transmitters in neighboring cells only. We consider a noniterative local cooperation scheme where base stations pass to their neighbors their decoded messages such that interference from already decoded messages can be canceled. Therefore, for a given decoding order, the interference between sectors is described by a directed locally connected graph. The problem consists of maximizing the per-sector DoFs over all possible decoding orders and precoding schemes. In particular, we provide a decoding order and a one-shot interference alignment scheme able to achieve optimal per-sector DoFs, up to an additive gap due to boundary effects, that vanishes as the size of the network becomes large. Then, we extend our treatment by considering the case of intersector interference with joint processing of the three sector at each cell site. In order to avoid signaling schemes relying on the strength of interference, we further introduce the notion of topologically robust schemes, which are able to guarantee a minimum rate (or DoFs) irrespectively of the strength of the interfering links. Toward this end, we design a different decoding order and alignment scheme, which is topologically robust and still achieves the same optimum DoFs. Finally, we provide a new scheme for the downlink, based on local base station cooperation, where base stations pass to their neighbors a quantized version of their dirty-paper coded signals. For the proposed downlink scheme, we can prove a DoFs duality result showing that, for an appropriate choice of the precoding order and of the alignment beamforming vectors, it can achieve the same per-sector DoFs of the corresponding uplink schemes. [ABSTRACT FROM PUBLISHER]