This paper investigates adaptive streaming codes over a three-node relayed network. In this setting, a source node transmits a sequence of message packets to a destination with help of a relay. The source-to-relay and relay-to-destination links are unreliable and introduce at most$N_{1}$and$N_{2}$packet erasures, respectively. The destination node must recover each message packet within a strict delay constraint$T$. The paper presents a new construction of streaming codes for all feasible parameters$\{N_{1}, N_{2}, T\}$. Our work improves upon the construction in Fong et al. by adapting the relaying strategy based on the erasure patterns from source to relay. Specifically, the code employs the notion of symbol estimates, which allows the relay to forward information about symbols before it can decode that symbol, and variable-rate encoding, which decreases the rate used to encode a packet as more erasures affect that packet. The codes proposed in this paper achieve rates higher than the ones proposed by Fong et al. whenever$N_{2} > N_{1}$, and achieve the same rate when$N_{2} \leq N_{1}$, in which case the rate is optimal. The paper also presents an upper bound on the achievable rate that takes into account erasures in both links in order to bound the rate in the second link. The upper bound is shown to be tighter than a trivial bound that considers only the erasures in the second link.

This paper investigates low-latency adaptive streaming codes for a three-node relay network. A source node transmits a sequence of source packets (messages) to the destination through a relay node. We focus on a particular case where the link connecting the source and relay nodes is almost reliable, but the link connecting the relay to the destination is not. The relay node can observe the erasure pattern that has occurred in the transmission between the source node and itself and adapt its relaying strategy based on that observation. Every source packet must be perfectly recovered by the destination with a strict delay T, as long as the number of erasures in the relay-to-destination link lies below some design parameter. We then characterize capacity as a function of such design parameter. The achievability scheme employs two different relaying strategies, based on whether an erasure has or has not occurred in the link from source to relay. The converse is proven by analyzing a periodic erasure pattern and lower bounding the minimum redundancy across channel packets. We show that the achievable rate can be improved compared to non-adaptive schemes previously proposed, indicating that exploiting the knowledge of the erasure pattern by the relay node is essential in achieving capacity.

In the above article[1], an upper bound on the maximum achievable rate is corrected. If we add the restriction that the packets transmitted by the relay must be independent of the erasures introduced by the first-hop channel, then no correction is needed and the converse proof need not be rectified.

We introduce a novel network-adaptive algorithm that is suitable for alleviating network packet losses for low-latency interactive communications between a source and a destination. Our network-adaptive algorithm estimates in real-time the best parameters of a recently proposed streaming code that uses forward error correction (FEC) to correct both arbitrary and burst losses, which cause a crackling noise and undesirable jitters, respectively in audio. In particular, the destination estimates appropriate coding parameters based on its observed packet loss pattern and sends them back to the source for updating the underlying code. Besides, a new explicit construction of practical low-latency streaming codes that achieve the optimal tradeoff between the capability of correcting arbitrary losses and the capability of correcting burst losses is used. Simulation evaluations based on statistical losses and real-world packet loss traces reveal the following: (i) Our proposed network-adaptive algorithm combined with our optimal streaming codes can achieve significantly higher performance compared to uncoded and non-adaptive FEC schemes over UDP (User Datagram Protocol); (ii) Our explicit streaming codes can significantly outperform traditional MDS (maximum-distance separable) streaming schemes when they are used along with our network-adaptive algorithm. In addition, we study different factors that can affect the performance of our network-adaptive algorithm.

This paper investigates low-latency streaming codes for a three-node relay network. The source transmits a sequence of messages (streaming messages) to the destination through the relay between them, where the first-hop channel from the source to the relay and the second-hop channel from the relay to the destination are subject to packet erasures. Every source message generated at a time slot must be recovered perfectly at the destination within the subsequent T time slots. In any sliding window of $ {T}+1$ time slots, we assume no more than $ {N}_{1}$ and ${N}_{2}$ erasures are introduced by the first-hop channel and second-hop channel respectively. We fully characterize the maximum achievable rate in terms of T, $ {N}_{1}$ and $ {N}_{2}$ . The achievability is proved by using a symbol-wise decode-forward strategy where the source symbols within the same message are decoded by the relay with different delays. The converse is proved by analyzing the maximum achievable rate for each channel when the erasures in the other channel are consecutive (bursty). In addition, we show that traditional message-wise decode-forward strategies, which require the source symbols within the same message to be decoded by the relay with the same delay, are sub-optimal in general.

This paper considers multiplexing two sequences of messages with two different decoding delays over a packet erasure channel. In each time slot, the source constructs a packet based on the current and previous messages and transmits the packet, which may be erased when the packet travels from the source to the destination. The destination must perfectly recover every source message in the first sequence subject to a decoding delay Tv, and every source message in the second sequence subject to a shorter decoding delay Tu≤ Tv,. We assume that the channel loss model introduces a burst erasure of a fixed length B on the discrete timeline. Under this channel loss assumption, the capacity region for the case where Tv, ≤ Tu+B was previously solved. In this paper, we fully characterize the capacity region for the remaining case Tv, ) Tu+B. The key step in the achievability proof is achieving the non-trivial corner point of the capacity region through using a multiplexed streaming code constructed by superimposing two single-stream codes. The main idea in the converse proof is obtaining a genie-aided bound when the channel is subject to a periodic erasure pattern where each period consists of a length-B burst erasure followed by a length-Tunoiseless duration.

In this paper, we consider transmitting a sequence of messages (a streaming source) over a packet erasure channel, where every source message must be recovered perfectly at the destination subject to a fixed decoding delay. Recently, the capacity of such a channel was established. However, the codes shown to achieve the capacity are either non-explicit constructions (proven to exist) or explicit constructions requiring large field size that scales exponentially with the delay. This work presents an explicit rate-optimal construction for all channel and delay parameters over a field size that scales only quadratically with the delay.

This paper presents a new construction of error correcting codes which achieves optimal recovery of a streaming source over a packet erasure channel. The channel model considered is the sliding-window erasure model, with burst and arbitrary losses, introduced by Badr et al. We present a simple construction, when the rate of the code is at least 1/2, which achieves optimal error correction in this setup. Our proposed construction is explicit and systematic. It uses off-the-shelf maximum distance separable (MDS) codes and maximum rank distance (MRD) Gabidulin block codes as constituent codes and combines them in a simple manner. This is in contrast to other recent works, where the construction involves a careful design of the generator or parity check matrix from first principles. The field size requirement which depends on the constituent MDS and MRD codes is also analyzed.

We introduce a novel network-adaptive algorithm that is suitable for alleviating network packet losses for low-latency interactive communications between a source and a destination. Network packet losses happen in a bursty manner as well as an arbitrary manner, where the former is usually due to network congestion and the latter can be caused by unreliable wireless links. Our network-adaptive algorithm estimates in real time the best parameters of a recently proposed streaming code that corrects both arbitrary losses (which cause crackling noise in audio) and burst losses (which cause undesirable jitters and pauses in audio) using forward error correction (FEC). The network-adaptive algorithm updates the coding parameters in real time as follows: The destination estimates appropriate coding parameters based on its observed packet loss pattern and then the parameters are fed back to the source for updating the underlying code. In addition, a new explicit construction of practical low-latency streaming codes that achieve the optimal tradeoff between the capability of correcting arbitrary losses and the capability of correcting burst losses is provided. Simulation evaluations based on real-world packet loss traces reveal that our proposed network-adaptive algorithm combined with our optimal streaming codes achieves significantly higher reliability compared to uncoded and non-adaptive FEC schemes over UDP (User Datagram Protocol).

An additive white Gaussian noise energy-harvesting channel with an infinite-sized battery is considered. The energy arrival process is modeled as a sequence of independent and identically distributed random variables. The channel capacity 1/2 log(1 + P) is achievable by the so-called best-effort and save-and-transmit schemes where P denotes the battery recharge rate. This paper analyzes the save-and-transmit scheme whose transmit power is strictly less than P and the best-effort scheme as a special case of save-and-transmit without a saving phase. In the finite blocklength regime, we obtain new nonasymptotic achievable rates for these schemes that approach the capacity with gaps vanishing at rates proportional to 1/√n and ((log n)/n)1/2respectively where n denotes the blocklength. The proof technique involves analyzing the escape probability of a Markov process. When P is sufficiently large, we show that allowing the transmit power to back off from P can improve the performance for save-and-transmit. The results are extended to a block energy arrival model where the length of each energy block L grows sublinearly in n. We show that the save-and-transmit and best-effort schemes achieve coding rates that approach the capacity with gaps vanishing at rates proportional to √(L/n) and (max{log n, L}/n)1/2, respectively.

An additive white Gaussian noise (AWGN) energy-harvesting (EH) channel is considered where the transmitter is equipped with an infinite-sized battery which stores energy harvested from the environment. The energy arrival process is modeled as a sequence of independent and identically distributed (i.i.d.) random variables. The capacity of this channel is known and is achievable by the so-called best-effort and save-and-transmit schemes. This paper investigates the best-effort scheme in the finite blocklength regime and establishes the first nonasymptotic achievable rate for it. The first-order term of the nonasymptotic achievable rate equals the capacity, and the second-order term is proportional to -√{logn/n}-where n denotes the blocklength. The proof technique involves analyzing the escape probability of a Markov process. In addition, we use this new proof technique to analyze the save-and-transmit and obtain a new non-asymptotic achievable rate for it, whose first-order and second-order terms achieve the capacity and the scaling -1/√n respectively. For all sufficiently large signal-to-noise ratios (SNRs), our new achievable rate outperforms the existing ones.

This paper considers transmitting a sequence of messages (streaming messages) over a packet erasure channel. In each time slot, the source constructs a packet based on the current and the previous messages and transmits the packet, which may be erased when the packet travels from the source to the destination. Every source message must be recovered perfectly at the destination subject to a fixed decoding delay. We assume that the channel loss model introduces either one burst erasure or multiple arbitrary erasures in any fixed-sized sliding window. Under this channel loss assumption, we fully characterize the maximum achievable rate by constructing streaming codes that achieve the optimal rate. In addition, our construction of optimal streaming codes implies the full characterization of the maximum achievable rate for convolutional codes with any given column distance, column span, and decoding delay. Numerical results demonstrate that the optimal streaming codes outperform existing streaming codes of comparable complexity over some instances of the Gilbert-Elliott channel and the Fritchman channel.

This paper investigates the achievable rates of an additive white Gaussian noise energy-harvesting (EH) channel with an infinite battery. The EH process is characterized by a sequence of blocks of harvested energy, which is known causally at the source. The harvested energy remains constant within a block while the harvested energy across different blocks is characterized by a sequence of independent and identically distributed random variables. The blocks have length L , which can be interpreted as the coherence time of the energy-arrival process. If L is a constant or grows sublinearly in the blocklength n , we fully characterize the first-order term in the asymptotic expansion of the maximum transmission rate subject to a fixed tolerable error probability ε. The first-order term is known as the ε-capacity. In addition, we obtain lower and upper bounds on the second-order term in the asymptotic expansion, which reveal that the second order term is proportional to -(L/n)1/2for any ε less than 1/2. The lower bound is obtained through analyzing the save-and-transmit strategy. If L grows linearly in n, we obtain lower and upper bounds on the ε-capacity, which coincide whenever the cumulative distribution function of the EH random variable is continuous and strictly increasing. In order to achieve the lower bound, we have proposed a novel adaptive save-and-transmit strategy, which chooses different save-and-transmit codes across different blocks according to the energy variation across the blocks.

This paper considers a multimessage network where each node may send a message to any other node in the network. Under the discrete memoryless model, we prove the strong converse theorem for any network with tight cut-set bound, i.e., whose cut-set bound is achievable. Our result implies that for any network with tight cut-set bound and any fixed rate vector that resides outside the capacity region, the average error probabilities of any sequence of length-n codes operated at the rate vector must tend to 1 as n grows. The proof is based on the method of types. The proof techniques are inspired by the work of Csiszar and Korner in 1982 which fully characterized the reliability function of any discrete memoryless channel (DMC) with feedback for rates above capacity.

This paper investigates the achievable rates of an additive white Gaussian noise (AWGN) energy-harvesting (EH) channel with an infinite battery under the assumption that the error probabilities do not vanish as the blocklength increases. The EH process is characterized by a sequence of blocks of harvested energy. The harvested energy remains constant within a block while the harvested energy across different blocks is characterized by a sequence of independent and identically distributed (i.i.d.) random variables. The blocks have length L, which can be interpreted as the coherence time of the energy arrival process. If L is a constant or grows sublinearly in the blocklength n, we fully characterize the first-order coding rate. In addition, we obtain lower and upper bounds on the second-order coding rate, which are proportional to −√L/n for any fixed error probability < 1 /2. If L grows linearly in n, we obtain lower and upper bounds on the first-order coding rate, which coincide whenever the EH random variable is continuous. Our results suggest that correlation in the energy-arrival process decreases the effective blocklength by a factor of L.

This paper investigates the asymptotic expansion of the maximum coding rate of a parallel Gaussian channel with feedback under the following setting: A peak power constraint is imposed on every transmitted codeword, and the average error probabilities of decoding the transmitted message are non-vanishing as the blocklength increases. This paper proves an upper bound on the first-and second-order asymptotics. Combined with existing achievability results, our result implies that the presence of feedback does not improve the first-and second-order asymptotics.

This paper revisits the Gaussian degraded relay channel, where the link that carries information from the source to the destination is a physically degraded version of the link that carries information from the source to the relay. The source and the relay are subject to expected power constraints. The ε-capacity of the channel is characterized and it is strictly larger than the capacity for any ε > 0, which implies that the channel does not possess the strong converse property. The proof of the achievability part is based on several key ideas: block Markov coding, which is used in the classical decode-forward strategy, power control for Gaussian channels under expected power constraints, and a careful scaling between the block size and the total number of block uses. The converse part is proved by first establishing two non-asymptotic lower bounds on the error probability, which are derived from the type-II errors of some binary hypothesis tests. Subsequently, each lower bound is simplified by conditioning on an event related to the power of some linear combination of the codewords transmitted by the source and the relay. Lower and upper bounds on the second-order term of the optimal coding rate are also obtained.

This paper investigates the asymptotic expansion for the maximum rate of fixed-length codes over a parallel Gaussian channel with feedback under the following setting: a peak power constraint is imposed on every transmitted codeword, and the average error probabilities of decoding the transmitted message are non-vanishing as the blocklength increases. The main contribution of this paper is a self-contained proof of an upper bound on the first- and second-order asymptotics of the parallel Gaussian channel with feedback. The proof techniques involve developing an information spectrum bound followed by using Curtiss' theorem to show that a sum of dependent random variables associated with the information spectrum bound converges in distribution to a sum of independent random variables, thus facilitating the use of the usual central limit theorem. Combined with existing achievability results, our result implies that the presence of feedback does not improve the first- and second-order asymptotics.

In this paper, we consider single-and multi-user Gaussian channels with feedback under expected power constraints and with non-vanishing error probabilities. In the first of two contributions, we study asymptotic expansions for the additive white Gaussian noise (AWGN) channel with feedback under the average error probability formalism. By drawing ideas from Gallager and Nakiboǧlu's work for the direct part and the meta-converse for the converse part, we establish the e-capacity and show that it depends on e in general and so the strong converse fails to hold. Furthermore, we provide bounds on the second-order term in the asymptotic expansion. We show that for any positive integer L, the second-order term is bounded between a term proportional to - ln(L) n (where ln(L)(·) is the L-fold nested logarithm function) and a term proportional to +(n ln n)1/2, where n is the blocklength. The lower bound on the second-order term shows that feedback does provide an improvement in the maximal achievable rate over the case where no feedback is available. In our second contribution, we establish the e-capacity region for the AWGN multiple access channel with feedback under the expected power constraint by combining ideas from hypothesis testing, information spectrum analysis, Ozarow's coding scheme, and power control.

We prove that 2-user Gaussian broadcast channels admit the strong converse. This implies that for every sequence of block codes with an asymptotic maximal error probability smaller than one, the limit points of the corresponding sequence of rate pairs must lie within the capacity region derived by Cover and Bergmans. The main mathematical tool required for our analysis is a logarithmic Sobolev inequality known as the Gaussian Poincaré inequality.

This paper investigates the information-theoretic limits of the additive white Gaussian noise (AWGN) energy-harvesting (EH) channel in the finite blocklength regime. The EH process is characterized by a sequence of i.i.d. random variables with finite variances. We use the save-and-transmit strategy proposed by Ozel and Ulukus (2012) together with Shannon's non-asymptotic achievability bound to obtain a lower bound on the achievable rate for the AWGN EH channel. The first-order term of the lower bound on the achievable rate is equal to C and the second-order (backoff from capacity) term is proportional to equation, where n denotes the blocklength and C denotes the capacity of the EH channel, which is the same as the capacity without the EH constraints. The constant of proportionality of the backoff term is found and qualitative interpretations are provided.

In this paper, we analyze the asymptotic expansion for additive white Gaussian noise (AWGN) channels with feedback under an expected power constraint and the average error probability formalism. We show that the ε-capacity depends on ε in general and so the strong converse fails to hold. Furthermore, we provide bounds on the second-order term in the asymptotic expansion. We show that the second-order term is bounded between −ln ln n and a term that is proportional to +√n ln n. The lower bound on the second-order term shows that feedback does provide an improvement in the maximal achievable rate over the case where no feedback is available.

This paper investigates the scaling exponent of polar codes for binary-input energy-harvesting (EH) channels with infinite-capacity batteries. The EH process is characterized by a sequence of independent and identically distributed random variables with finite variances. The scaling exponent μ of polar codes for a binary-input memoryless channel (BMC) qY|X with capacity C(qY|X) characterizes the closest gap between the capacity and the non-asymptotic achievable rates in the following way. For a fixed average error probability e ∈ (0, 1), the closest gap between the capacity C(qY|X) and a non-asymptotic achievable rate Rn for a length-n polar code scales as n-1/μ, i.e., min{|C(qY|X) - Rn|} = O(n-1/μ). It has been shown that the scaling exponent μ for any binary-input memoryless symmetric channel with C(qY|X) ∈ (0, 1) lies between 3.579 and 4.714, where the upper bound 4.714 was shown by an explicit construction of polar codes. Our main result shows that 4.714 remains to be a valid upper bound on the scaling exponent for any binary-input EH channel, i.e., a BMC subject to additional EH constraints. Our result thus implies that the EH constraints do not worsen the rate of convergence to capacity if polar codes are employed. An auxiliary contribution of this paper is that the upper bound on μ holds for binary-input memoryless asymmetric channels.

This paper investigates the information-theoretic limits of energy-harvesting (EH) channels in the finite blocklength regime. The EH process is characterized by a sequence of i.i.d. random variables with finite variances. We use the save-andtransmit strategy proposed by Ozel and Ulukus (2012) together with Shannon's non-asymptotic achievability bound to obtain lower bounds on the achievable rates for both additive white Gaussian noise channels and discrete memoryless channels under EH constraints. The first-order terms of the lower bounds of the achievable rates are equal to C and the second-order (backoff from capacity) terms are proportional to -√log n/n, where n denotes the blocklength and C denotes the capacity of the EH channel, which is the same as the capacity without the EH constraints. The constant of proportionality of the backoff term is found and qualitative interpretations are provided.

In a network, a node is said to incur a delay if its encoding of each transmitted symbol involves only its received symbols obtained before the time slot in which the transmitted symbol is sent (hence the transmitted symbol sent in a time slot cannot depend on the received symbol obtained in the same time slot). A node is said to incur no delay if its received symbol obtained in a time slot is available for encoding its transmitted symbol sent in the same time slot. Under the classical model, every node in the network incurs a delay. In this paper, we investigate the multimessage multicast network (MMN) under a generalized-delay model which allows some nodes to incur no delay. We obtain the capacity regions for three classes of MMNs with zero-delay nodes, namely, the deterministic network dominated by product distributions, the MMN consisting of independent DMCs, and the wireless erasure network. In addition, we show that for any MMN belonging to one of the above three classes, the set of achievable rate tuples under the generalized-delay model and under the classical model are the same, which implies that the set of achievable rate tuples for the MMN does not depend on the delay amounts incurred by the nodes in the network.

We consider a communication network consisting of nodes and directed edges that connect the nodes. The network may contain cycles. The communications are slotted where the duration of each time slot is equal to the maximum propagation delay experienced by the edges. The edges with negligible delays are allowed to be operated before the other edges in each time slot. For any pair of adjacent edges (ℓ, i) and (i, j), where (ℓ, i) terminates at node i and (i, j) originates from node i, we say (ℓ, i) incurs zero delay on (i, j) if (ℓ, i) is operated before (i, j); otherwise, we say (ℓ, i) incurs a unit delay on (i, j). In the classical model, every edge incurs a unit delay on every adjacent edge and the cut-set bound is a well-known outer bound on the capacity region. In this paper, we investigate the multimessage multicast network (MMN) consisting of independent channels, where each channel is associated with a set of edges and each edge may incur zero delay on some other edges. Our result reveals that the capacity region of the MMN with independent channels and zero-delay edges lies within the classical cut-set bound despite a violation of the unit-delay assumption.

We prove the strong converse for the N -source Gaussian multiple access channel. In particular, we show that any rate tuple that can be supported by a sequence of codes with asymptotic average error probability <;1 must lie in the Cover-Wyner capacity region. Our proof consists of the following. First, we perform an expurgation step to convert any given sequence of codes with asymptotic average error probability <;1 to codes with asymptotic maximal error probability <;1. Second, we quantize the input alphabets with an appropriately chosen resolution. Upon quantization, we apply the wringing technique (by Ahlswede) on the quantized inputs to obtain further subcodes from the subcodes obtained in the expurgation step, so that the resultant correlations among the symbols transmitted by the different sources vanish as the blocklength grows. Finally, we derive upper bounds on achievable sum-rates of the subcodes in terms of the type-II error of a binary hypothesis test. These upper bounds are then simplified through judicious choices of auxiliary output distributions. Our strong converse result carries over to the Gaussian interference channel under strong interference as long as the sum of the two asymptotic average error probabilities <;1.

This paper establishes that the strong converse holds for some classes of discrete memoryless multimessage multicast networks (DM-MMNs) whose corresponding cut-set bounds are tight, i.e., coincide with the corresponding sets of achievable rate tuples. Our strong converse result implies that for any DM-MMN of these classes, the average error probabilities of any sequence of codes operated at a rate tuple belonging to the exterior of the cut-set bound must tend to one (and are not simply bounded away from zero) as the block length grows. Examples in the classes of DM-MMNs include wireless erasure networks, DM-MMNs consisting of independent discrete memoryless channels (DMCs) as well as single-destination DM-MMNs consisting of independent DMCs with destination feedback. Our elementary proof technique leverages properties of the Rényi divergence.

We consider a communication network consisting of nodes and directed edges that connect the nodes. Each edge receives a symbol from a node and outputs a symbol to a node in each time slot. An edge (ℓ, i) terminating at node i is said to incur zero delay on an edge (i, j) originating from node i if the following holds in each time slot: Node i receives the symbol output from (ℓ, i) before encoding the symbol to be transmitted on (i, j). Otherwise, (ℓ, i) is said to incur a delay on (i, j). In the classical model, every edge incurs a unit delay on every other edge and the cut-set bound is a well-known outer bound on the capacity region. However, if an edge is allowed to incur zero delay on another edge, then there exists a two-node network whose capacity region exceeds the classical cut-set bound. In this paper, we investigate the multimessage multicast network (MMN) consisting of independent channels and study the capacity region under an edge-delay model where an edge may incur zero delay on some other edges. Our edge-delay model subsumes the classical model. Our result reveals that the capacity region of the MMN with independent channels and zero-delay edges lies within the classical cut-set bound despite a violation of the classical unit-delay assumption.

This paper investigates the asymptotic expansion for the size of block codes defined for the additive white Gaussian noise (AWGN) channel with feedback under the following setting: A peak power constraint is imposed on every transmitted codeword (i.e., maximum per-codeword power constraint), and the average error probability of decoding the transmitted message is non-vanishing as the blocklength increases. It is well-known that the presence of feedback does not increase the first-order asymptotics (i.e., capacity) in the asymptotic expansion for the AWGN channel. The main contribution of this paper is proving an upper bound on the asymptotic expansion for the AWGN channel with feedback. Combined with existing achievability results for the AWGN channel, our result implies that the presence of feedback does not improve the second- and third-order asymptotics.

This paper establishes that the strong converse holds for some classes of discrete memoryless multimessage multicast networks (DM-MMNs) whose corresponding cut-set bounds are tight, i.e., coincide with the set of achievable rate tuples. The strong converse for these classes of DM-MMNs implies that all sequences of codes with rate tuples belonging to the exterior of the cut-set bound have average error probabilities that tend to one. Examples in the classes of DM-MMNs include wireless erasure networks, DM-MMNs consisting of independent discrete memoryless channels (DMCs), and single-destination DM-MMNs consisting of independent DMCs with destination feedback. Our proof technique leverages the properties of the Rényi divergence.

In a network, a node is said to incur a delay if its encoding of each transmitted symbol involves only its received symbols obtained before the time slot in which the transmitted symbol is sent (hence the transmitted symbol sent in a time slot cannot depend on the received symbol obtained in the same time slot). A node is said to incur no delay if its received symbol obtained in a time slot is available for encoding its transmitted symbol sent in the same time slot. Under the classical model, every node in a discrete memoryless network (DMN) incurs a unit delay, and the capacity region of the DMN satisfies the well-known cut-set outer bound. In this paper, we propose a generalized model for the DMN where some nodes may incur no delay. Under our generalized model, we obtain a new cut-set outer bound, which is proved to be tight for some two-node DMN and is shown to subsume an existing cut-set bound for the causal relay network. In addition, we establish under the generalized model another cut-set outer bound on the positive-delay region-the set of achievable rate tuples under the constraint that every node incurs a delay. We use the cut-set bound on the positive-delay region to show that for some two-node DMN under the generalized model, the positive-delay region is strictly smaller than the capacity region.

We consider the two-hop interference channel (IC), which consists of two source-destination pairs communicating with each other via two relays. We analyze the degrees of freedom (DoF) of this network when the relays are restricted to perform linear schemes, and the channel gains are constant (i.e., slow fading). We show that, somewhat surprisingly, by using vector-linear strategies at the relays, it is possible to achieve 4/3 sum-DoF when the channel gains are real. The key achievability idea is to alternate relaying coefficients across time, to create different end-to-end interference structures (or topologies) at different times. Although each of these topologies has only 1 sum-DoF, we manage to achieve 4/3 by coding across them. Furthermore, we develop a novel outer bound that matches our achievability, hence characterizing the sum-DoF of two-hop ICs with linear schemes. We also generalize the result to the multi-antenna setting, where each node has M antennas, and the relays are restricted to one-shot linear schemes. We further extend the result to the case of complex channel gains, by intuitively viewing each complex node with M antennas as a real node with 2M antennas, and characterize the sum-DoF to be 2M - 1/3.

We consider the two-hop interference channel (IC) with constant real channel coefficients, which consists of two source-destination pairs, separated by two relays. We analyze the achievable degrees of freedom (DoF) of such network when relays are restricted to perform scalar amplify-forward (AF) operations, with possibly time-varying coefficients. We show that, somewhat surprisingly, by providing the flexibility of choosing time-varying AF coefficients at the relays, it is possible to achieve 4/3 sum-DoF. We also develop a novel outer bound that matches our achievability, hence characterizing the sum-DoF of two-hop interference channels with time-varying AF relaying strategies.

In a network, a node is said to incur a delay if its encoding of each transmitted symbol involves only its received symbols obtained before the time slot in which the transmitted symbol is sent (hence the transmitted symbol sent in a time slot cannot depend on the received symbol obtained in the same time slot). A node is said to incur no delay if its received symbol obtained in a time slot is available for encoding its transmitted symbol sent in the same time slot. In this paper, we investigate the generalized discrete memoryless network (DMN) where some nodes may incur no delay. We call the capacity region of the generalized DMN under the constraint that every node incurs a delay the positive-delay region. We establish the cut-set outer bound on the positive-delay region. In addition, we use our cut-set outer bound to show that in some two-node generalized DMN, the positive-delay region is strictly smaller than the capacity region.

In a network, a node is said to incur a delay if its encoding of each transmitted symbol involves only its received symbols obtained before the time slot in which the transmitted symbol is sent (hence the transmitted symbol sent in a time slot cannot depend on the received symbol obtained in the same time slot). A node is said to incur no delay if its received symbol obtained in a time slot is available for encoding its transmitted symbol sent in the same time slot. In the classical discrete memoryless network (DMN), every node incurs a delay. A well-known result for the classical DMN is the cut-set outer bound. In this paper, we generalize the model of the DMN in such a way that some nodes may incur no delay, and we obtain the cut-set outer bound for the generalized DMN.

We consider a two-way relay channel (TWRC) in which two terminals exchange messages with the help of a relay between them. The two terminals transmit messages to the relay through the Multiple Access Channel (MAC) and the relay transmits messages to the two terminals through the Broadcast Channel (BC). We assume that the MAC and the BC do not interfere with each other, and each terminal receives signals only from the relay but not the other terminal. All the nodes are assumed to be full-duplex, which means that they can transmit and receive information at the same time. A transmission scheme for the Gaussian TWRC is said to be analog-relaying if the relay does not need any codebook for encoding. The simplest analog-relaying scheme is amplify-and-forward (AF), under which the relay amplifies the received codeword and forwards the resultant codeword to the two terminals. In this paper, we propose a new analog-relaying scheme called amplify-and-modulo (AM) based on lattice operations. AM is a slight modification of AF. Under AM, the relay first amplifies the received codeword followed by reducing the power of the amplified codeword using the modulo-lattice operation, and then forwards the resultant codeword to the two terminals. After receiving the codeword transmitted by the relay, each terminal subtracts its own information before decoding. We prove an achievable rate region for AM, and obtain a necessary and sufficient condition under which AM outperforms AF. In addition, we show by graph that AM can achieve a strictly higher equal-rate than AF and another existing analog-relaying scheme together under some scenario.

We study the three-node point-to-point relay network which consists of two terminal nodes and one relay node between them and investigate practical network coding schemes for the network. The rate region achievable by the practical network coding schemes is obtained, and a procedure for constructing the linear network codes that achieve the rate pairs in the achievable rate region is presented. In addition, we show that the use of network coding rather than routing alone enlarges the achievable rate region, in particular increases the maximum equal-rate throughput.

We consider a two-way relay channel (TRC) in which two terminals exchange messages with the help of a relay between them. The two terminals transmit messages to the relay through the Multiple Access Channel (MAC) and the relay transmits messages to the two terminals through the Broadcast Channel (BC). We assume that the MAC and the BC do not interfere with each other, and each terminal receives signals only from the relay but not the other terminal. All the nodes are assumed to be full-duplex, which means that they can transmit and receive information at the same time. The TRC is said to be without feedback if each terminal node cannot use its previously received information for encoding its message. Otherwise, the TRC is said to be with feedback. We obtain an outer bound on the capacity region of the discrete memoryless TRC without feedback and prove that the outer bound is tighter than the cut-set outer bound. In addition, we show that using feedback can enlarge the capacity region of some discrete memoryless TRC.

We introduce variable-rate linear network coding for single-source finite acyclic network. In this problem, the source of a network transmits messages at different rates in different time sessions and every nonsource node in the network decodes the messages if possible. We propose two efficient algorithms for implementing variable-rate linear network coding under different circumstances.