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Arbitrary varying channels (AVC) are used to model communication settings in which a channel state may vary arbitrarily over time. Their primary objective is to circumvent statistical assumptions on channel variation. Traditional studies on AVCs optimize rate subject to the worst-case state sequence. While this approach is resilient to channel variations, it may result in low rates for state sequences that are associated with relatively good channels. This paper addresses the analysis of AVCs through the lens of competitive analysis, where solution quality is measured with respect to the optimal solution had the state sequence been known in advance. Our main result demonstrates that codes constructed by a single input distribution do not achieve optimal competitive performance over AVCs. This stands in contrast to the single-letter capacity formulae for AVCs, and it indicates, in our setting, that even though the encoder cannot predict the subsequent channel states, it benefits from varying its input distribution as time proceeds.

In multi-source multi-terminal key-dissemination, here called ``key-cast,'' introduced by the authors in [ITW2022], network nodes hold independent random bits, and one seeks a communication scheme that allows all terminal nodes to share a secret key K. The work at hand addresses positive (albeit, arbitrarily small) rate key-cast under the security requirement that no single non-terminal network node can gain information about the shared key K; this scenario is useful in cryptographic settings. Specifically, key-dissemination protocols based on secure multicast network coding are designed. The analysis presented yields two combinatorial characterizations. In each, we assume a network in which an eavesdropper may access any individual network node. The first characterization captures all networks that support positive-rate secure multicast; computing the secure-multicast capacity in the setting studied is a known open problem. The second characterizes all networks that support positive-rate secure key-cast.

Existing studies on federated learning (FL) are mostly focused on system orchestration for static snapshots of the network and making static control decisions (e.g., spectrum allocation). However, real-world wireless networks are susceptible to temporal variations of wireless channel capacity and users' datasets. In this paper, we incorporate multi-granular system dynamics (MSDs) into FL, including (M1) dynamic wireless channel capacity, captured by a set of discrete-time events, called $\mathscr{D}$-Events, and (M2) dynamic datasets of users. The latter is characterized by (M2-a) modeling the dynamics of user's dataset size via an ordinary differential equation and (M2-b) introducing dynamic model drift}, formulated via a partial differential inequality} drawing concrete analytical connections between the dynamics of users' datasets and FL accuracy. We then conduct FL orchestration under MSDs by introducing dynamic cooperative FL with dedicated MAC schedulers (DCLM), exploiting the unique features of open radio access network (O-RAN). DCLM proposes (i) a hierarchical device-to-device (D2D)-assisted model training, (ii) dynamic control decisions through dedicated O-RAN MAC schedulers, and (iii) asymmetric user selection. We provide extensive theoretical analysis to study the convergence of DCLM. We then optimize the degrees of freedom (e.g., user selection and spectrum allocation) in DCLM through a highly non-convex optimization problem. We develop a systematic approach to obtain the solution for this problem, opening the door to solving a broad variety of network-aware FL optimization problems. We show the efficiency of DCLM via numerical simulations and provide a series of future directions., Comment: 120 pages, 13 figures

Motivated by the design of low-complexity low-power coding solutions for the Gaussian relay channel, this work presents an upper bound on the minimum energy-per-bit achievable on the Gaussian relay channel using rank-1 linear relaying. Our study addresses high-dimensional relay codes and presents bounds that outperform prior known bounds using 2-dimensional schemes. A novelty of our analysis ties the optimization problem at hand to the solution of a certain differential equation which, in turn, leads to a low energy-per-bit achievable scheme., Comment: Submitted to the IEEE International Symposium on Information Theory, 2024. 13 pages, 1 figure

We consider communication over channels whose statistics are not known in full, but can be parameterized as a finite family of memoryless channels. A typical approach to address channel uncertainty is to design codes for the worst channel in the family, resulting in the well-known compound channel capacity. Although this approach is robust, it may suffer a significant loss of performance if the capacity-achieving distribution of the worst channel attains low rates over other channels. In this work, we cope with channel uncertainty through the lens of {\em competitive analysis}. The main idea is to optimize a relative metric that compares the performance of the designed code and a clairvoyant code that has access to the true channel. To allow communication rates that adapt to the channel at use, we consider rateless codes with a fixed number of message bits and random decoding times. We propose two competitive metrics: the competitive ratio between the expected rates of the two codes, and a regret defined as the difference between the expected rates. The competitive ratio, for instance, provides a percentage guarantee on the expected rate of the designed code when compared to the rate of the clairvoyant code that knows the channel at hand. Our main results are single-letter expressions for the optimal {\em competitive-ratio} and {\em regret}, expressed as a max-min or min-max optimization. Several examples illustrate the benefits of the competitive analysis approach to code design compared to the compound channel.

The multiple access channel (MAC) capacity with feedback is considered under feedback models designed to tease out which factors contribute to the MAC feedback capacity benefit. Comparing the capacity of a MAC with ``perfect'' feedback, which causally delivers to the transmitters the true channel output, to that of a MAC with ``independent'' feedback, which causally delivers to the transmitters an independent instance of that same channel output, allows separation of effects like cooperation from alternative feedback benefits such as knowledge of the channel instance. Proving that the Cover-Leung (CL) achievability bound, which is known to be loose for some channels, is achievable also under (shared or distinct) independent feedback at the transmitters shows that the CL bound does not require transmitter knowledge of the channel instance. Proving that each transmitter's maximal rate under independent feedback exceeds that under perfect feedback highlights the potential power of an independent look at the channel output., Comment: 9 pages, to appear at the International Symposium on Information Theory 2023

The multicast key-dissemination problem over noiseless networks, introduced by Langberg and Effros [ITW 2022], here called the ``key-cast'' problem, captures the task of disseminating a shared secret random key to a set of terminals over a given network. Unlike traditional communication, where messages must be delivered from source to destination(s) unchanged, key-cast is more flexible since key-cast need not require source reconstruction at destination nodes. For example, the distributed keys can be mixtures of sources from which the sources themselves may be unrecoverable. The work at hand considers key dissemination in the single-source, multiple-multicast network coding setting, i.e., the ``multiple key-cast'' problem. Here, distinct keys are to be simultaneously transmitted from a single source node to multiple terminal sets, one shared random key per multicast set. Scenarios include the secure setting, in which only the source and intended destinations gain information about a given key; and the non-secure setting in which the only requirement is that the knowledge of one key does not reveal information about another. In both settings, we present combinatorial conditions for key dissemination and design corresponding multiple key-cast schemes. In addition, we compare the multiple key-cast rate with and without the restriction of source reconstruction, the former corresponding to traditional forms of communication; key-cast achieves a strict advantage in rate when source reconstruction is relaxed.

We characterize the capacity for the discrete-time arbitrarily varying channel with discrete inputs, outputs, and states when (a) the encoder and decoder do not share common randomness, (b) the input and state are subject to cost constraints, (c) the transition matrix of the channel is deterministic given the state, and (d) at each time step the adversary can only observe the current and past channel inputs when choosing the state at that time. The achievable strategy involves stochastic encoding together with list decoding and a disambiguation step. The converse uses a two-phase "babble-and-push" strategy where the adversary chooses the state randomly in the first phase, list decodes the output, and then chooses state inputs to symmetrize the channel in the second phase. These results generalize prior work on specific channels models (additive, erasure) to general discrete alphabets and models.

Group testing is one of the fundamental problems in coding theory and combinatorics in which one is to identify a subset of contaminated items from a given ground set. There has been renewed interest in group testing recently due to its applications in diagnostic virology, including pool testing for the novel coronavirus. The majority of existing works on group testing focus on the \emph{uniform} setting in which any subset of size $d$ from a ground set $V$ of size $n$ is potentially contaminated. In this work, we consider a {\em generalized} version of group testing with an arbitrary set-system of potentially contaminated sets. The generalized problem is characterized by a hypergraph $H=(V,E)$, where $V$ represents the ground set and edges $e\in E$ represent potentially contaminated sets. The problem of generalized group testing is motivated by practical settings in which not all subsets of a given size $d$ may be potentially contaminated, rather, due to social dynamics, geographical limitations, or other considerations, there exist subsets that can be readily ruled out. For example, in the context of pool testing, the edge set $E$ may consist of families, work teams, or students in a classroom, i.e., subsets likely to be mutually contaminated. The goal in studying the generalized setting is to leverage the additional knowledge characterized by $H=(V,E)$ to significantly reduce the number of required tests. The paper considers both adaptive and non-adaptive group testing and makes the following contributions. First, for the non-adaptive setting, we show that finding an optimal solution for the generalized version of group testing is NP-hard. For this setting, we present a solution that requires $O(d\log{|E|})$ tests, where $d$ is the maximum size of a set $e \in E$. Our solutions generalize those given for the traditional setting and are shown to be of order-optimal size $O(\log{|E|})$ for hypergraphs with edges that have ``large'' symmetric differences. For the adaptive setting, when edges in $E$ are of size exactly $d$, we present a solution of size $O(\log{|E|}+d\log^2{d})$ that comes close to the lower bound of $\Omega(\log{|E|} + d)$., Comment: 7 pages, 1 figure

For a multi-source multi-terminal noiseless network, the key-dissemination problem involves the task of multicasting a secret key K from the network sources to its terminals. As in secure multicast network-coding, in the key-dissemination problem the source nodes have access to independent randomness and, as the network is noiseless, the resulting key K is a function of the sources' information. However, different from traditional forms of multicast, in key-dissemination the key K need not consist of source messages, but rather may be any function of the information generated at the sources, as long as it is shared by all terminals. Allowing the shared key K to be a mixture of source information grants a flexibility to the communication process which gives rise to the potential of increased key-rates when compared to traditional secure multicast. The multicast key-capacity is the supremum of achievable key-rates, subject to the security requirement that the shared key is not revealed to an eavesdropper with predefined eavesdropping capabilities. The key-dissemination problem (termed also, secret key-agreement) has seen significant studies over the past decades in memoryless network structures. In this work, we initiate the study of key-dissemination in the context of noiseless networks, i.e., network coding. In this context, we study similarities and differences between traditional secure-multicast and the more lenient task of key-dissemination.

This work addresses the cooperation facilitator (CF) model, in which network nodes coordinate through a rate limited communication device. For independent multiple-access channel (MAC) encoders, the CF model is known to show significant rate benefits, even when the rate of cooperation is negligible. Specifically, the benefit in MAC sum-rate, as a function of the cooperation rate $C_{CF}$, sometimes has an infinite slope at $C_{CF}=0$. This work studies the question of whether cooperation through a CF can yield similar infinite-slope benefits when applied to internal network encoders in which dependence among MAC transmitters can be established without the help of the CF. Towards this end, this work studies the CF model when applied to relay nodes of a single-source, single-terminal, diamond network consisting of a broadcast channel followed by a MAC. In the relay channel with orthogonal receiver components, careful generalization of the partial-decode-forward/compress-forward lower bound to the CF model yields sufficient conditions for an infinite-slope benefit. Additional results include derivation of a family of diamond networks for which the infinite-slope rate-benefit derives directly from the properties of the corresponding MAC component when studied in isolation., Comment: Full version of a paper submitted to ISIT 2022. 9 pages, 3 figures

This paper focuses on error-correcting codes that can handle a predefined set of specific error patterns. The need for such codes arises in many settings of practical interest, including wireless communication and flash memory systems. In many such settings, a smaller field size is achievable than that offered by MDS and other standard codes. We establish a connection between the minimum alphabet size for this generalized setting and the combinatorial properties of a hypergraph that represents the prespecified collection of error patterns. We also show a connection between error and erasure correcting codes in this specialized setting. This allows us to establish bounds on the minimum alphabet size and show an advantage of non-linear codes over linear codes in a generalized setting. We also consider a variation of the problem which allows a small probability of decoding error and relate it to an approximate version of hypergraph coloring.

The work at hand presents a finite-blocklength analysis of the multiple access channel (MAC) sum-rate under the cooperation facilitator (CF) model. The CF model, in which independent encoders coordinate through an intermediary node, is known to show significant rate benefits, even when the rate of cooperation is limited. We continue this line of study for cooperation rates which are sub-linear in the blocklength $n$. Roughly speaking, our results show that if the facilitator transmits $\log{K}$ bits, there is a sum-rate benefit of order $\sqrt{\log{K}/n}$. This result extends across a wide range of $K$: even a single bit of cooperation is shown to provide a sum-rate benefit of order $1/\sqrt{n}$., Comment: Full version of a paper submitted to ISIT 2021

The traditional notion of capacity studied in the context of memoryless network communication builds on the concept of block-codes and requires that, for sufficiently large blocklength n, all receiver nodes simultaneously decode their required information after n channel uses. In this work, we generalize the traditional capacity region by exploring communication rates achievable when some receivers are required to decode their information before others, at different predetermined times; referred here as the "time-rate" region. Through a reduction to the standard notion of capacity, we present an inner-bound on the time-rate region. The time-rate region has been previously studied and characterized for the memoryless broadcast channel (with a sole common message) under the name "static broadcasting".

The edge-removal problem asks whether the removal of a $\lambda$-capacity edge from a given network can decrease the communication rate between source-terminal pairs by more than $\lambda$. In this short manuscript, we prove that for undirected networks, removing a $\lambda$-capacity edge decreases the rate by $O(\lambda)$. Through previously known reductive arguments, here newly applied to undirected networks, our result implies that the zero-error capacity region of an undirected network equals its vanishing-error capacity region. Whether it is possible to prove similar results for directed networks remains an open question.

A Maximum Distance Separable code over an alphabet $F$ is defined via an encoding function $C:F^k \rightarrow F^n$ that allows to retrieve a message $m \in F^k$ from the codeword $C(m)$ even after erasing any $n-k$ of its symbols. The minimum possible alphabet size of general (non-linear) MDS codes for given parameters $n$ and $k$ is unknown and forms one of the central open problems in coding theory. The paper initiates the study of the alphabet size of codes in a generalized setting where the coding scheme is required to handle a pre-specified subset of all possible erasure patterns, naturally represented by an $n$-vertex $k$-uniform hypergraph. We relate the minimum possible alphabet size of such codes to the strong chromatic number of the hypergraph and analyze the tightness of the obtained bounds for both the linear and non-linear settings. We further consider variations of the problem which allow a small probability of decoding error., Comment: 7 pages, 1 figure

In this paper, we prove topology dependent bounds on the number of rounds needed to compute Functional Aggregate Queries (FAQs) studied by Abo Khamis et al. [PODS 2016] in a synchronous distributed network under the model considered by Chattopadhyay et al. [FOCS 2014, SODA 2017]. Unlike the recent work on computing database queries in the Massively Parallel Computation model, in the model of Chattopadhyay et al., nodes can communicate only via private point-to-point channels and we are interested in bounds that work over an {\em arbitrary} communication topology. This is the first work to consider more practically motivated problems in this distributed model. For the sake of exposition, we focus on two special problems in this paper: Boolean Conjunctive Query (BCQ) and computing variable/factor marginals in Probabilistic Graphical Models (PGMs). We obtain tight bounds on the number of rounds needed to compute such queries as long as the underlying hypergraph of the query is $O(1)$-degenerate and has $O(1)$-arity. In particular, the $O(1)$-degeneracy condition covers most well-studied queries that are efficiently computable in the centralized computation model like queries with constant treewidth. These tight bounds depend on a new notion of `width' (namely internal-node-width) for Generalized Hypertree Decompositions (GHDs) of acyclic hypergraphs, which minimizes the number of internal nodes in a sub-class of GHDs. To the best of our knowledge, this width has not been studied explicitly in the theoretical database literature. Finally, we consider the problem of computing the product of a vector with a chain of matrices and prove tight bounds on its round complexity (over the finite field of two elements) using a novel min-entropy based argument., Comment: A conference version was presented at PODS 2019

Group network codes are a generalization of linear codes that have seen several studies over the last decade. When studying network codes, operations performed at internal network nodes called local encoding functions, are of significant interest. While local encoding functions of linear codes are well understood (and of operational significance), no similar operational definition exists for group network codes. To bridge this gap, we study the connections between group network codes and a family of codes called Coordinate-Wise-Linear (CWL) codes. CWL codes generalize linear codes and, in addition, can be defined locally (i.e., operationally). In this work, we study the connection between CWL codes and group codes from both a local and global encoding perspective. We show that Abelian group codes can be expressed as CWL codes and, as a result, they inherit an operational definition.

In communication networks, cooperative strategies are coding schemes where network nodes work together to improve network performance metrics such as the total rate delivered across the network. This work studies encoder cooperation in the setting of a discrete multiple access channel (MAC) with two encoders and a single decoder. A network node, here called the cooperation facilitator (CF), that is connected to both encoders via rate-limited links, enables the cooperation strategy. Previous work by the authors presents two classes of MACs: (i) one class where the average-error sum-capacity has an infinite derivative in the limit where CF output link capacities approach zero, and (ii) a second class of MACs where the maximal-error sum-capacity is not continuous at the point where the output link capacities of the CF equal zero. This work contrasts the power of the CF in the maximal- and average-error cases, showing that a constant number of bits communicated over the CF output link can yield a positive gain in the maximal-error sum-capacity, while a far greater number of bits, even numbers that grow sublinearly in the blocklength, can never yield a non-negligible gain in the average-error sum-capacity., Comment: 30 pages, 1 figure. This article supersedes arXiv:1607.07024 and arXiv:1801.03655

It is common in the study of secure multicast network coding in the presence of an eavesdropper that has access to $z$ network links, to assume that the source node is the only node that generates random keys. In this setting, the secure multicast rate is well understood. Computing the secure multicast rate, or even the secure unicast rate, in the more general setting in which all network nodes may generate (independent) random keys is known to be as difficult as computing the (non-secure) capacity of multiple-unicast network coding instances --- a well known open problem. This work treats an intermediate model of secure unicast in which only one node can generate random keys, however that node need not be the source node. The secure communication rate for this setting is characterized again with an eavesdropper that has access to $z$ network links.

The edge removal problem studies the loss in network coding rates that results when a network communication edge is removed from a given network. It is known, for example, that in networks restricted to linear coding schemes and networks restricted to Abelian group codes, removing an edge e* with capacity Re* reduces the achievable rate on each source by no more than Re*. In this work, we seek to uncover larger families of encoding functions for which the edge removal statement holds. We take a local perspective: instead of requiring that all network encoding functions satisfy certain restrictions (e.g., linearity), we limit only the function carried on the removed edge e*. Our central results give sufficient conditions on the function carried by edge e* in the code used to achieve a particular rate vector under which we can demonstrate the achievability of a related rate vector once e* is removed.

We consider the problem of communication over a channel with a causal jamming adversary subject to quadratic constraints. A sender Alice wishes to communicate a message to a receiver Bob by transmitting a real-valued length-$n$ codeword $\mathbf{x}=x_1,...,x_n$ through a communication channel. Alice and Bob do not share common randomness. Knowing Alice's encoding strategy, an adversarial jammer James chooses a real-valued length-n noise sequence $\mathbf{s}=s_1,..,s_n$ in a causal manner, i.e., each $s_t (1<=t<=n)$ can only depend on $x_1,...,x_t$. Bob receives $\mathbf{y}$, the sum of Alice's transmission $\mathbf{x}$ and James' jamming vector $\mathbf{s}$, and is required to reliably estimate Alice's message from this sum. In addition, Alice and James's transmission powers are restricted by quadratic constraints $P>0$ and $N>0$. In this work, we characterize the channel capacity for such a channel as the limit superior of the optimal values of a series of optimizations. Upper and lower bounds on the optimal values are provided both analytically and numerically. Interestingly, unlike many communication problems, in this causal setting Alice's optimal codebook may not have a uniform power allocation - for certain SNR, a codebook with a two-level uniform power allocation results in a strictly higher rate than a codebook with a uniform power allocation would., Comment: 80 pages, ISIT 2018

As an attempt to bridge the gap between the probabilistic world of classical information theory and the combinatorial world of zero-error information theory, this paper studies the performance of randomly generated codebooks over discrete memoryless channels under a zero-error list-decoding constraint. This study allows the application of tools from one area to the other. Furthermore, it leads to an information-theoretic formulation of the birthday problem, which is concerned with the probability that in a given population, a fixed number of people have the same birthday. Due to the lack of a closed-form expression for this probability when the distribution of birthdays is not uniform, the resulting expression is not simple to analyze; in the information-theoretic formulation, however, the asymptotic behavior of this probability can be characterized exactly for all distributions., Comment: Extended version of paper presented at ISIT 2017 in Aachen

In communication networks, cooperative strategies are coding schemes where network nodes work together to improve network performance metrics such as sum-rate. This work studies encoder cooperation in the setting of a discrete multiple access channel with two encoders and a single decoder. A node in the network that is connected to both encoders via rate-limited links, referred to as the cooperation facilitator (CF), enables the cooperation strategy. Previously, the authors presented a class of multiple access channels where the average-error sum-capacity has an infinite derivative in the limit where CF output link capacities approach zero. The authors also demonstrated that for some channels, the maximal-error sum-capacity is not continuous at the point where the output link capacities of the CF equal zero. This work shows that the the average-error sum-capacity is continuous when CF output link capacities converge to zero; that is, the infinite derivative of the average-error sum-capacity is not a result of its discontinuity as in the maximal-error case., Comment: 20 pages, 1 figure. To be submitted to ISIT '18

In many communication networks, the availability of channel state information at various nodes provides an opportunity for network nodes to work together, or "cooperate." This work studies the benefit of cooperation in the multiple access channel with a cooperation facilitator, distributed state information at the encoders, and full state information available at the decoder. Under various causality constraints, sufficient conditions are obtained such that encoder cooperation through the facilitator results in a gain in sum-capacity that has infinite slope in the information rate shared with the encoders. This result extends the prior work of the authors on cooperation in networks where none of the nodes have access to state information., Comment: Extended version of paper presented at ISIT 2017 in Aachen. 20 pages, 1 figure

We prove that there exist non-linear binary cyclic codes that attain the Gilbert-Varshamov bound.

In this work we consider a communication problem in which a sender, Alice, wishes to communicate with a receiver, Bob, over a channel controlled by an adversarial jammer, James, who is {\em myopic}. Roughly speaking, for blocklength $n$, the codeword $X^n$ transmitted by Alice is corrupted by James who must base his adversarial decisions (of which locations of $X^n$ to corrupt and how to corrupt them) not on the codeword $X^n$ but on $Z^n$, an image of $X^n$ through a noisy memoryless channel. More specifically, our communication model may be described by two channels. A memoryless channel $p(z|x)$ from Alice to James, and an {\it Arbitrarily Varying Channel} from Alice to Bob, $p(y|x,s)$ governed by a state $X^n$ determined by James. In standard adversarial channels the states $S^n$ may depend on the codeword $X^n$, but in our setting $S^n$ depends only on James's view $Z^n$. The myopic channel captures a broad range of channels and bridges between the standard models of memoryless and adversarial (zero-error) channels. In this work we present upper and lower bounds on the capacity of myopic channels. For a number of special cases of interest we show that our bounds are tight. We extend our results to the setting of {\em secure} communication in which we require that the transmitted message remain secret from James. For example, we show that if (i) James may flip at most a $p$ fraction of the bits communicated between Alice and Bob, and (ii) James views $X^n$ through a binary symmetric channel with parameter $q$, then once James is "sufficiently myopic" (in this case, when $q>p$), then the optimal communication rate is that of an adversary who is "blind" (that is, an adversary that does not see $X^n$ at all), which is $1-H(p)$ for standard communication, and $H(q)-H(p)$ for secure communication. A similar phenomenon exists for our general model of communication., Comment: 18 pages

We prove tight network topology dependent bounds on the round complexity of computing well studied $k$-party functions such as set disjointness and element distinctness. Unlike the usual case in the CONGEST model in distributed computing, we fix the function and then vary the underlying network topology. This complements the recent such results on total communication that have received some attention. We also present some applications to distributed graph computation problems. Our main contribution is a proof technique that allows us to reduce the problem on a general graph topology to a relevant two-party communication complexity problem. However, unlike many previous works that also used the same high level strategy, we do not reason about a two-party communication problem that is induced by a cut in the graph. To `stitch' back the various lower bounds from the two party communication problems, we use the notion of timed graph that has seen prior use in network coding. Our reductions use some tools from Steiner tree packing and multi-commodity flow problems that have a delay constraint., Comment: 31 pages, 2 figures

In this work we study the capacity advantage achieved by adding a single bit of communication -- not a link of capacity 1 but a single bit over all time -- to a memoryless network. Specifically, we present a memoryless network in which adding a single bit of communication strictly increases the capacity region., Comment: 6 pages

We consider the problem of communicating a message $m$ in the presence of a malicious jamming adversary (Calvin), who can erase an arbitrary set of up to $pn$ bits, out of $n$ transmitted bits $(x_1,\ldots,x_n)$. The capacity of such a channel when Calvin is exactly causal, i.e. Calvin's decision of whether or not to erase bit $x_i$ depends on his observations $(x_1,\ldots,x_i)$ was recently characterized to be $1-2p$. In this work we show two (perhaps) surprising phenomena. Firstly, we demonstrate via a novel code construction that if Calvin is delayed by even a single bit, i.e. Calvin's decision of whether or not to erase bit $x_i$ depends only on $(x_1,\ldots,x_{i-1})$ (and is independent of the "current bit" $x_i$) then the capacity increases to $1-p$ when the encoder is allowed to be stochastic. Secondly, we show via a novel jamming strategy for Calvin that, in the single-bit-delay setting, if the encoding is deterministic (i.e. the transmitted codeword is a deterministic function of the message $m$) then no rate asymptotically larger than $1-2p$ is possible with vanishing probability of error, hence stochastic encoding (using private randomness at the encoder) is essential to achieve the capacity of $1-p$ against a one-bit-delayed Calvin., Comment: 21 pages, 4 figures, extended draft of submission to ISIT 2016

In the $q$-ary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword $\mathbf{x} =(x_1,\ldots,x_n) \in \{0,1,\ldots,q-1\}^n$ symbol by symbol via a channel limited to at most $pn$ errors and/or $p^{*} n$ erasures. The channel is "online" in the sense that at the $i$th step of communication the channel decides whether to corrupt the $i$th symbol or not based on its view so far, i.e., its decision depends only on the transmitted symbols $(x_1,\ldots,x_i)$. This is in contrast to the classical adversarial channel in which the corruption is chosen by a channel that has a full knowledge on the sent codeword $\mathbf{x}$. In this work we study the capacity of $q$-ary online channels for a combined corruption model, in which the channel may impose at most $pn$ {\em errors} and at most $p^{*} n$ {\em erasures} on the transmitted codeword. The online channel (in both the error and erasure case) has seen a number of recent studies which present both upper and lower bounds on its capacity. In this work, we give a full characterization of the capacity as a function of $q,p$, and $p^{*}$., Comment: This is a new version of the binary case, which can be found at arXiv:1412.6376

Cooperation strategies allow communication devices to work together to improve network capacity. Consider a network consisting of a $k$-user multiple access channel (MAC) and a node that is connected to all $k$ encoders via rate-limited bidirectional links, referred to as the "cooperation facilitator" (CF). Define the cooperation benefit as the sum-capacity gain resulting from the communication between the encoders and the CF and the cooperation rate as the total rate the CF shares with the encoders. This work demonstrates the existence of a class of $k$-user MACs where the ratio of the cooperation benefit to cooperation rate tends to infinity as the cooperation rate tends to zero. Examples of channels in this class include the binary erasure MAC for $k=2$ and the $k$-user Gaussian MAC for any $k\geq 2$., Comment: 46 pages, 4 figures. Submitted to the IEEE Transactions on Information Theory. This article supersedes arXiv:1504.04432 and arXiv:1508.03349

In network cooperation strategies, nodes work together with the aim of increasing transmission rates or reliability. This paper demonstrates that enabling cooperation between the transmitters of a two-user multiple access channel, via a cooperation facilitator that has access to both messages, always results in a network whose maximal- and average-error sum-capacities are the same---even when those capacities differ in the absence of cooperation and the information shared with the encoders is negligible. From this result, it follows that if a multiple access channel with no transmitter cooperation has different maximal- and average-error sum-capacities, then the maximal-error sum-capacity of the network consisting of this channel and a cooperation facilitator is not continuous with respect to the output edge capacities of the facilitator. This shows that there exist networks where sharing even a negligible number of bits per channel use with the encoders yields a non-negligible benefit., Comment: 27 pages, 3 figures. Submitted to the IEEE Transactions on Information Theory

The multivariate covering lemma states that given a collection of $k$ codebooks, each of sufficiently large cardinality and independently generated according to one of the marginals of a joint distribution, one can always choose one codeword from each codebook such that the resulting $k$-tuple of codewords is jointly typical with respect to the joint distribution. We give a proof of this lemma for weakly typical sets. This allows achievability proofs that rely on the covering lemma to go through for continuous channels (e.g., Gaussian) without the need for quantization. The covering lemma and its converse are widely used in information theory, including in rate-distortion theory and in achievability results for multi-user channels., Comment: 10 pages

A secret sharing scheme is a method to store information securely and reliably. Particularly, in a threshold secret sharing scheme, a secret is encoded into $n$ shares, such that any set of at least $t_1$ shares suffice to decode the secret, and any set of at most $t_2 < t_1$ shares reveal no information about the secret. Assuming that each party holds a share and a user wishes to decode the secret by receiving information from a set of parties; the question we study is how to minimize the amount of communication between the user and the parties. We show that the necessary amount of communication, termed "decoding bandwidth", decreases as the number of parties that participate in decoding increases. We prove a tight lower bound on the decoding bandwidth, and construct secret sharing schemes achieving the bound. Particularly, we design a scheme that achieves the optimal decoding bandwidth when $d$ parties participate in decoding, universally for all $t_1 \le d \le n$. The scheme is based on Shamir's secret sharing scheme and preserves its simplicity and efficiency. In addition, we consider secure distributed storage where the proposed communication efficient secret sharing schemes further improve disk access complexity during decoding., Comment: submitted to the IEEE Transactions on Information Theory. New references and a new construction added

We show that solving a multiple-unicast network coding problem can be reduced to solving a single-unicast network error correction problem, where an adversary may jam at most a single edge in the network. Specifically, we present an efficient reduction that maps a multiple-unicast network coding instance to a network error correction instance while preserving feasibility. The reduction holds for both the zero probability of error model and the vanishing probability of error model. Previous reductions are restricted to the zero-error case. As an application of the reduction, we present a constructive example showing that the single-unicast network error correction capacity may not be achievable, a result of separate interest., Comment: ISIT 2015. arXiv admin note: text overlap with arXiv:1410.1905

In a cooperative coding scheme, network nodes work together to achieve higher transmission rates. To obtain a better understanding of cooperation, we consider a model in which two transmitters send rate-limited descriptions of their messages to a "cooperation facilitator", a node that sends back rate-limited descriptions of the pair to each transmitter. This model includes the conferencing encoders model and a prior model from the current authors as special cases. We show that except for a special class of multiple access channels, the gain in sum-capacity resulting from cooperation under this model is quite large. Adding a cooperation facilitator to any such channel results in a network that does not satisfy the edge removal property. An important special case is the Gaussian multiple access channel, for which we explicitly characterize the sum-rate cooperation gain., Comment: 15 pages, 3 figures. To be presented at ISIT 2015

In the binary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword $\mathbf{x} =(x_1,\ldots,x_n) \in \{0,1\}^n$ bit by bit via a channel limited to at most $pn$ corruptions. The channel is "online" in the sense that at the $i$th step of communication the channel decides whether to corrupt the $i$th bit or not based on its view so far, i.e., its decision depends only on the transmitted bits $(x_1,\ldots,x_i)$. This is in contrast to the classical adversarial channel in which the error is chosen by a channel that has a full knowledge on the sent codeword $\mathbf{x}$. In this work we study the capacity of binary online channels for two corruption models: the {\em bit-flip} model in which the channel may flip at most $pn$ of the bits of the transmitted codeword, and the {\em erasure} model in which the channel may erase at most $pn$ bits of the transmitted codeword. Specifically, for both error models we give a full characterization of the capacity as a function of $p$. The online channel (in both the bit-flip and erasure case) has seen a number of recent studies which present both upper and lower bounds on its capacity. In this work, we present and analyze a coding scheme that improves on the previously suggested lower bounds and matches the previously suggested upper bounds thus implying a tight characterization.

We propose efficient coding schemes for two communication settings: 1. asymmetric channels, and 2. channels with an informed encoder. These settings are important in non-volatile memories, as well as optical and broadcast communication. The schemes are based on non-linear polar codes, and they build on and improve recent work on these settings. In asymmetric channels, we tackle the exponential storage requirement of previously known schemes, that resulted from the use of large Boolean functions. We propose an improved scheme, that achieves the capacity of asymmetric channels with polynomial computational complexity and storage requirement. The proposed non-linear scheme is then generalized to the setting of channel coding with an informed encoder, using a multicoding technique. We consider specific instances of the scheme for flash memories, that incorporate error-correction capabilities together with rewriting. Since the considered codes are non-linear, they eliminate the requirement of previously known schemes (called polar write-once-memory codes) for shared randomness between the encoder and the decoder. Finally, we mention that the multicoding scheme is also useful for broadcast communication in Marton's region, improving upon previous schemes for this setting., Comment: Submitted to IEEE Transactions on Information Theory. Partially presented at ISIT 2014

We study the problem of communicating over a single-source single-terminal network in the presence of an adversary that may jam a single link of the network. If any one of the edges can be jammed, the capacity of such networks is well understood and follows directly from the connection between the minimum cut and maximum flow in single-source single- terminal networks. In this work we consider networks in which some edges cannot be jammed, and show that determining the network communication capacity is at least as hard as solving the multiple-unicast network coding problem for the error-free case. The latter problem is a long standing open problem.

We obtain novel index coding schemes and show that they provably outperform all previously known graph theoretic bounds proposed so far. Further, we establish a rather strong negative result: all known graph theoretic bounds are within a logarithmic factor from the chromatic number. This is in striking contrast to minrank since prior work has shown that it can outperform the chromatic number by a polynomial factor in some cases. The conclusion is that all known graph theoretic bounds are not much stronger than the chromatic number., Comment: 8 pages, 2 figures. Submitted to ISIT 2014

We consider the problem of secure communication over a network in the presence of wiretappers. We give a new cut-set bound on secrecy capacity which takes into account the contribution of both forward and backward edges crossing the cut, and the connectivity between their endpoints in the rest of the network. We show the bound is tight on a class of networks, which demonstrates that it is not possible to find a tighter bound by considering only cut set edges and their connectivity.

In this paper, we propose a new cooperation model for discrete memoryless multiple access channels. Unlike in prior cooperation models (e.g., conferencing encoders), where the transmitters cooperate directly, in this model the transmitters cooperate through a larger network. We show that under this indirect cooperation model, there exist channels for which the increase in sum-capacity resulting from cooperation is significantly larger than the rate shared by the transmitters to establish the cooperation. This result contrasts both with results on the benefit of cooperation under prior models and results in the network coding literature, where attempts to find examples in which similar small network modifications yield large capacity benefits have to date been unsuccessful., Comment: Extended version of a paper submitted to ISIT 2014 (10 pages, 3 figures)

Traditional studies of multi-source, multi-terminal interference channels typically allow a vanishing probability of error in communication. Motivated by the study of network coding, this work addresses the task of quantifying the loss in rate when insisting on zero error communication in the context of interference channels.

We consider the problem of secure unicast transmission between two nodes in a directed graph, where an adversary eavesdrops/jams a subset of nodes. This adversarial setting is in contrast to traditional ones where the adversary controls a subset of links. In particular, we study, in the main, the class of routing-only schemes (as opposed to those allowing coding inside the network). Routing-only schemes usually have low implementation complexity, yet a characterization of the rates achievable by such schemes was open prior to this work. We first propose an LP based solution for secure communication against eavesdropping, and show that it is information-theoretically rate-optimal among all routing-only schemes. The idea behind our design is to balance information flow in the network so that no subset of nodes observe "too much" information. Interestingly, we show that the rates achieved by our routing-only scheme are always at least as good as, and sometimes better, than those achieved by "na\"ive" network coding schemes (i.e. the rate-optimal scheme designed for the traditional scenario where the adversary controls links in a network rather than nodes.) We also demonstrate non-trivial network coding schemes that achieve rates at least as high as (and again sometimes better than) those achieved by our routing schemes, but leave open the question of characterizing the optimal rate-region of the problem under all possible coding schemes. We then extend these routing-only schemes to the adversarial node-jamming scenarios and show similar results. During the journey of our investigation, we also develop a new technique that has the potential to derive non-trivial bounds for general secure-communication schemes.

We present a novel upper bound for the optimal index coding rate. Our bound uses a graph theoretic quantity called the local chromatic number. We show how a good local coloring can be used to create a good index code. The local coloring is used as an alignment guide to assign index coding vectors from a general position MDS code. We further show that a natural LP relaxation yields an even stronger index code. Our bounds provably outperform the state of the art on index coding but at most by a constant factor., Comment: 14 Pages, 3 Figures; A conference version submitted to ISIT 2013; typos corrected

We show that the network coding and index coding problems are equivalent. This equivalence holds in the general setting which includes linear and non-linear codes. Specifically, we present an efficient reduction that maps a network coding instance to an index coding one while preserving feasibility. Previous connections were restricted to the linear case.

Given a graph $G=(V,E)$, an integer $k$, and a function $f_G:V^k \times V^k \to {0,1}$, the $k^{th}$ graph product of $G$ w.r.t $f_G$ is the graph with vertex set $V^k$, and an edge between two vertices $x=(x_1,...,x_k)$ and $y=(y_1,...,y_k)$ iff $f_G(x,y)=1$. Graph products are a basic combinatorial object, widely studied and used in different areas such as hardness of approximation, information theory, etc. We study graph products for functions $f_G$ of the form $f_G(x,y)=1$ iff there are at least $t$ indices $i \in [k]$ s.t. $(x_i,y_i)\in E$, where $t \in [k]$ is a fixed parameter in $f_G$. This framework generalizes the well-known graph tensor-product (obtained for $t=k$) and the graph or-product (obtained for $t=1$). The property that interests us is the edge distribution in such graphs. We show that if $G$ has a spectral gap, then the number of edges connecting "large-enough" sets in $G^k$ is "well-behaved", namely, it is close to the expected value, had the sets been random. We extend our results to bi-partite graph products as well. For a bi-partite graph $G=(X,Y,E)$, the $k^{th}$ bi-partite graph product of $G$ w.r.t $f_G$ is the bi-partite graph with vertex sets $X^k$ and $Y^k$ and edges between $x \in X^k$ and $y \in Y^k$ iff $f_G(x,y)=1$. Finally, for both types of graph products, optimality is asserted using the "Converse to the Expander Mixing Lemma" obtained by Bilu and Linial in 2006. A byproduct of our proof technique is a new explicit construction of a family of co-spectral graphs.