We consider the cooperative coding of a pair of correlated memoryless source and side-information sequences wherein an encoder assisted by a side-information aided listening-helper describe the source and side-information to a receiver. We obtain a single-letter characterization of the rate region in both cases: when the decoder reconstructs just the source and when it reconstructs both the source and side-information. In the quadratic Gaussian setting we determine the optimal law characterizing the rate region using a technique introduced by Courtade in combination with a conditional version of the converse for the Gaussian CEO problem.

We consider a successive refinement source coding model in which each receiver observes its own side-information causally and is required to form its reconstruction in a causal manner. Furthermore, a one way conference link of given capacity allows Decoder 1-the helper-to send causal descriptions of what it has received so far to Decoder 2. A complete characterization of the rate distortion region is provided. The optimal helper is a scalar quantizer of its side-information which depends on the corresponding first-stage message symbol sent by the encoder. Thus, the imposition of the causality constraint on the side informations as well as on the helper's conference both renders the problem tractable and simplifies the optimal code.

A memoryless bivariate Gaussian source is transmitted to a pair of receivers over an average-power limited bandwidth-matched Gaussian broadcast channel. Based on their observations, Receiver 1 reconstructs the first source component while Receiver 2 reconstructs the second source component both seeking to minimize the expected squarederror distortions. In addition to the source transmission digital information at a specified rate should be conveyed reliably to Receiver 1-the “stronger” receiver. Given the message rate we characterize the achievable distortions region. Specifically, there is an SNR-threshold below which Dirty Paper coding of the digital information against a linear combination of the source components is optimal. The threshold is a function of the digital information rate, the source correlation and the distortion at the “stronger” receiver. Above this threshold a Dirty Paper coding extension of the Tian-Diggavi-Shamai hybrid scheme is shown to be optimal.

A number of additive noise networks are studied in the presence of a helper that observes the noise and assists the decoder by providing it with a rate-limited description of said noise. It is shown that “flash helping”-where noise descriptions are provided infrequently but with great precision-is often optimal and typically increases capacity by the maximal allowed description rate. It requires no binning. The discrete setting of the modulo-additive noise channel is also discussed.

We study data transmission and asymmetric state communication over a state-dependent two-user discrete memoryless degraded broadcast channel. In our model the encoder, to whom the state information is available either strictly-causally or causally, in addition to transmitting data reliably to each of the users must also provide the “higher quality” user with a description of the memoryless state sequence with some defined fidelity. The capacity region-i.e. the set of attainable rate-pairs given a maximal-allowed distortion when the state reconstruction fidelity is measured using a single-letter distortion function-is computed in the absence as well as in the presence of feedback from the “higher quality” output to the encoder. In the no-feedback cases we also study the constrained source-coding problem wherein the encoder is required to form a reconstruction that should predict the decoder's reconstruction within a given distortion, and establish the respective capacity regions. Finally, in the strictly-causal setting without feedback and when the encoder is required to form a reconstruction that should coincide with the decoder's reconstruction with probability close to one we characterize the Common-Reconstruction capacity region.

We consider the broadcasting of a memoryless bivariate Gaussian source over a two-user Gaussian broadcast channel where in addition to the source broadcasting a message is conveyed reliably to the “higher quality” receiver. Receiver 1 reconstructs the first source component while Receiver 2 reconstructs the second source component. Given the message rate, we characterize the set of attainable mean squared-error distortion pairs at the two receivers.

The additive noise channel is studied in the presence of a helper who observes the noise and can describe it to the receiver over a rate-limited noise-free bit-pipe. It is shown that the capacity of this network is typically the sum of the capacity of the channel in the absence of the helper and the capacity of the bit-pipe from the helper to the receiver. This holds for finite-variance stationary and ergodic noises under fairly general power-like constraints on the transmitted signal. A helper that is only cognizant of the noise is thus as helpful as an omniscient helper that is cognizant of both the noise and the transmitted message. The achievability proof is based on “flash helping” and requires no binning. Extensions to additive-noise multi-access channels are also discussed.

A Gaussian source and two data streams are to be transmitted over a Gaussian broadcast channel: the first stream, the “common stream,” is to be decoded by both receivers, and the second, the “private stream,” only by the strong receiver. Both receivers wish to estimate the source sequence, though with possibly different mean squared-errors. The quadruples of achievable rates and estimation errors are characterized, and it is shown that-once the data rates have been fixed-there is no tension between the estimation errors. Only the “equal bandwidth” case is treated, where the rate at which the source emits symbols is also the rate at which the channel is used.

We consider the broadcasting of a scalar memoryless Gaussian source over a two-user Gaussian broadcast channel with bandwidth expansion. In concurrent with the source description, a message that should be decoded reliably just by the “higher quality” user is sent. Conditioned on the message rate, we derive a necessary condition for the achievability of a mean-squared-error distortion pair at the two receivers, which extends the Reznic–Feder–Zamir bound. This necessary condition and a sufficient condition that we derive establish the set of achievable energy-distortion exponents for this setting.

The Rate-and-Asymmetric-State (RnAS) capacity region of a state-dependent two-user discrete memoryless degraded broadcast channel with a state-cognizant encoder is the set of rate-pairs at which data can be transmitted over the channel when the “higher quality” user-in addition to reliably decoding its message-must also reconstruct the state sequence with some defined fidelity. We show that feedback from the channel output to the encoder can increase this capacity region even for degraded broadcast channels that are memoryless with memoryless states. This capacity region is calculated here for such channels in the absence as well as in the presence of feedback when the state reconstruction fidelity is measured using a single-letter distortion function and the state sequence is revealed to the encoder in one of two different ways: strictly-causally or causally.

The rate-and-state capacity of a state-dependent channel with a state-cognizant encoder is the highest possible rate of communication over the channel when the decoder - in addition to reliably decoding the data - must also reconstruct the state sequence with some required fidelity. Feedback from the channel output to the encoder is shown to increase this capacity even for channels that are memoryless with memoryless states. This capacity is calculated here for such channels with feedback when the state reconstruction fidelity is measured using a single-letter distortion function and the state sequence is revealed to the encoder in one of two different ways: strictly-causally or causally. For the noncausal case, we provide bounds on the capacity and identify a condition under which the bounds coincide. Feedback does not increase the rate-and-state capacity when the decoder must reconstruct the state sequence perfectly or, in some settings, when the channel is Gaussian and fidelity is measured in terms of mean squared-error.

We consider the broadcasting of a single Gaussian source over a two-user Gaussian broadcast channel with bandwidth expansion. In addition to the source broadcasting the encoder sends a message reliably to the “higher quality” user. Conditioned on the message rate, we derive an outer bound on the set of attainable mean-squared error distortion pairs at the two users which extends the Reznic-Feder-Zamir bound. Based on the outer bound and an inner bound that we derive we characterize the set of achievable energy-distortion exponents for this problem.

We study the interference channel with one-sided generalized feedback and secrecy requirement. In our model Message 1 that is known just to Encoder 1 should be decoded by both receivers. Message 2-known only to Encoder 2-should be decoded by Decoder 2 and kept as secret as possible from Decoder 1. The uncertainty of Decoder 1 about Message 2 is measured by means of the equivocation rate. In addition, a noisy feedback is provided to Encoder 2. We derive an outer bound on the capacity-equivocation region for this model that is valid for the ”noisy cribbing” regime. The gap between this upper bound and the inner bound reported before is just in the Markov conditions satisfied by one of the auxiliary random variables. Furthermore, we consider a simplified causal cognitive interference model: the interference channel with a cribbing encoder. We show that when, conditioned on the input of Encoder 1, the interference channel is degraded our outer bound is tight thereby establishing the capacity-equivocation region for this case.

The Rate-and-State capacity of a state-dependent channel with a state-cognizant encoder is the highest possible rate of communication over the channel when the decoder-in addition to reliably decoding the data-must also reconstruct the state sequence with some required fidelity. Feedback from the channel output to the encoder is shown to increase this capacity even for channels that are memoryless with memoryless states. This capacity is calculated here for such channels with feedback when the state reconstruction fidelity is measured using a single-letter distortion function and the state sequence is revealed to the encoder in one of two different ways: strictly-causally or causally.

In the secure cooperative source-coding problem that we consider, Encoder 1 and Decoder 1 observe, respectively, a pair of independent identically distributed (i.i.d.) correlated sources and wish to communicate the first source to Decoder 2 subject to a distortion constraint. Encoder 1 sends a message to both the decoders over a public nonsecure channel, and then Decoder 1-the helper-sends a message just to Decoder 2 over a secure channel. An eavesdropper, which has access to memoryless side-information that is correlated with the first source, intercepts the message sent by Encoder 1 and based on both wishes to minimize its uncertainty about the first source. The secrecy measure of the system is the equivocation of the first source at the eavesdropper. We determine the rates equivocation region for this problem assuming that conditioned on the side-information of Decoder 1 the eavesdropper's side-information is independent of the first source. To that end, we derive an inner and an outer bound that depend explicitly on the proportional rate that Encoder 1 spends in the cooperation process. Subsequently, we evaluate this region for the case of a doubly symmetric binary source and Hamming distortion.

We consider the transmission of a memoryless bivariate Gaussian source over a two-user additive Gaussian multiple-access channel with unidirectional conferencing encoders. Here, prior to each transmission block, Encoder 1, which observes the first source component, is allowed to communicate with Encoder 2, which observes the second source component, via a unidirectional noise-free bit-pipe of given capacity. The main results of this paper are sufficient conditions and a necessary condition for the achievability of a distortion pair expressed as a function of the channel SNR and of the source correlation. The main sufficient condition is obtained by an extension of the vector-quantizer scheme suggested by Lapidoth-Tinguely, for the case without conferencing, to the case with unidirectional conference. In the high-SNR regime, and when the capacity of the conference channel is unlimited, these necessary and sufficient conditions are shown to agree. We evaluate the precise high-SNR asymptotics for a subset of distortion pairs when the capacity of the conference channel is unlimited in which case we show that a separation-based scheme attains these optimal distortion pairs. However, with symmetric average-power constraints and fixed conferencing capacity, at high-SNR, the latter separation-based scheme is shown to be suboptimal.

We consider the transmission of a bivariate Gaussian memoryless source over a two-user additive Gaussian multiple-access channel with unidirectional conferencing. Here, prior to each transmission block Encoder 1 is allowed to communicate with Encoder 2 via a unidirectional noise-free bit-pipe of given capacity. We derive a necessary condition for the achievability of a distortion pair expressed as a function of the channel SNR and of the source correlation. In the high-SNR regime, and when the capacity of the conference channel is unlimited, the necessary condition is shown to agree with a sufficient condition derived previously by the authors. We evaluate the precise high-SNR asymptotics for a subset of distortion pairs when the conferencing capacity is unlimited in which case we show that the problem is separable.

We consider a problem which is an extension of the multi-terminal source coding problem with encoder breakdown studied by Berger-Yeung. In our problem when, and only when, encoder breakdown happens the decoder gets help from a third node in the form of side-information which is degraded with respect to the un-encoded source. Inner and outer bounds on the rate-distortion region are derived. The achievable region is tight when both source components are reconstructed losslessly at the non-breakdown scenario and partially tight when the distortion measures are degraded and deterministic. Our results also explain the missing sum-rate constraint in the characterization of the rate-distortion region of the Berger-Yeung problem.

We consider the problem of transmitting a bivariate Gaussian memoryless source over a two-user additive Gaussian multiple-access channel with unidirectional conferencing encoders. Here, prior to each transmission block Encoder 1 is allowed to communicate with Encoder 2 via a unidirectional noise-free bit-pipe of given capacity. We extend the vector-quantizer scheme suggested by Lapidoth-Tinguely, for the case without conferencing, to the case with unidirectional conference and derive an achievable rate-distortion region. We compare the performance of the suggested vector-quantizer to that of the optimal scheme for lossless transmission when the capacity of the conference link is large.

We study the interference channel with one-sided generalized feedback and secrecy requirements. In our model, Message 1 that is known just to Encoder 1 should be decoded by both receivers. Message 2—known only to Encoder 2—should be decoded by Decoder 2 and kept as secret as possible from Decoder 1. The uncertainty of Decoder 1 about Message 2 is measured by means of the equivocation rate. In addition, a noisy feedback is provided to Encoder 2. We derive an achievable rate-equivocation region for this model and an outer bound for the “noisy cribbing” regime without secrecy, the gap being the Markov conditions satisfied by one of the auxiliary random variables. Furthermore, we consider a simplified causal cognitive interference model: the interference channel with a cribbing encoder. We derive an inner bound on the rate region for this model and prove that when the interference channel is degraded conditionally on the input of Encoder 1, our inner bound is tight.

We derive the capacity region of the two-user dirty-paper Gaussian multiaccess channel (MAC) with conferencing encoders. In this MAC, prior to each transmission block, the transmitters can hold a conference in which they can communicate with each other over error-free bit pipes of given capacities. The received signal suffers not only from additive Gaussian noise but also from additive interference, which is known noncausally to the transmitters but not to the receiver. The additive interference is modeled as Gaussian or uniform over a sphere. We show that the interference can be perfectly mitigated, i.e., that the capacity region without interference can also be achieved in its presence. This holds irrespective of whether the transmitters learn the interference before or after the conference. It follows as a corollary that also for the MAC with degraded message sets, the interference can be perfectly mitigated if it is known noncausally to the transmitters. To derive our results, we generalize Costa's single-user writing-on-dirty-paper achievability result to channels with dependent interference and not-necessarily Gaussian noise.

The two-user discrete memoryless state-dependent interference channel models a scenario in which two encoders transmit a pair of independent messages to two receivers where the signal intended for one receiver causes interference at the other receiver. Message 1 sent by Encoder 1 should be decoded by both receivers while Message 2 sent by Encoder 2 should be decoded by Decoder 2. The channel law is governed by an i.i.d. state process and the state sequence is available non-causally to both Encoder 2 and Decoder 2. It is further assumed that Encoder 2 cribs causally from Encoder 1. We derive an achievable rate-region for this model and show that it is tight when the output at Decoder 1 is degraded w.r.t. the output at Decoder 2 conditionally on the state and the input of Encoder 1.

The (non-causal) cognitive interference channel, studied recently by Liang et. al., is a model for a classical two-user discrete memoryless interference channel, over which two transmitters send a pair of independent messages. It is assumed that the first message is shared by both encoders, whereas the second message is known only to Encoder 2-the cognitive transmitter. Receiver 2 needs to decode both messages, and Receiver 1 should decode only the first message while Message 2 should be kept as secret as possible from Receiver 1. The level of secrecy is measured by the equivocation rate. For this model the capacity-equivocation region has been derived by Liang et. al. In this work we dispense of the assumption that Message 1 is shared a-priori by both encoders. Instead, we study the case in which Encoder 2 observes causally a feedback output of the channel and derive an achievable rate-equivocation region for this model. For a simplified model in which Encoder 2 cribs causally from Encoder 1 we establish the capacity-equivocation region for a degraded interference channel.

We study the transmission of a memoryless bivariate Gaussian source over an average-power-constrained one-to-two Gaussian broadcast channel. The transmitter observes the source and describes it to the two receivers by means of an average-power-constrained signal. Each receiver observes the transmitted signal corrupted by a different additive white Gaussian noise and wishes to estimate the source component intended for it: Receiver 1 wishes to estimate the first source component and Receiver 2 wishes to estimate the second. Our interest is in the pairs of expected squared-error distortions that are simultaneously achievable at the two receivers. We prove that an uncoded transmission scheme that sends a linear combination of the source components achieves the optimal power-versus-distortion trade-off whenever the signal-to-noise ratio is below a certain threshold. The threshold is a function of the source correlation and the distortion at the receiver with the weaker noise.

A side-information scalable model in which each receiver observes its own side-information, while the side-information pair is stochastically degraded with respect to the source, is considered. It is further assumed that a one way conference link with given capacity exists between the decoder observing the higher quality side-information and the one observing the degraded side-information. Our contribution is in deriving an outer bound on the rate-distortion region for this model. Specifically, we propose a modification to the technique presented by Tian-Diggavi which accounts for the presence of the conference channel.

We derive two inner bounds on the rate region of the partially cooperative relay broadcast channel. The first inner bound is based on the Cover and El Gamal generalized strategy for the original relay channel, while the second inner bound is based on the strategy proposed recently by Chong, Motani, and Garg, for the relay channel, which combines backward decoding and simultaneous decoding. Both rate regions subsume the inner bound reported by Liang and Kramer which is based just on the decode-and-forward strategy for the common message. A broadcast channel with cooperating decoders is a broadcast channel wherein the receivers, once observing their outputs, may exchange messages via a pair of channels with specified capacities. An achievable rate region for this model is derived based on the first coding strategy for the partially cooperative relay broadcast channel. This region subsumes the Dabora and Servetto rate region. A converse result is proved for this model based on the converse result of Nair and El Gamal for the broadcast channel and Willems's converse proof for the multiple-access channel with cooperating encoders.

An achievable rate for the discrete memoryless relay channel with receiver-transmitter feedback is proposed based on block-Markov superposition encoding. The achievable rate can also be extended to Gaussian channels. A second achievable rate for the Gaussian relay channel based on a Schalkwijk-Kailath type scheme is presented. For some channels both achievable rates strictly improve upon all previously known achievable rates. For the discrete memoryless relay channel also a converse result is provided.

We consider a one-to-two Gaussian broadcasting problem where the transmitter observes a memoryless bi-variate Gaussian source and each receiver wishes to estimate one of the source components. The transmitter describes the source pair by means of an average-power-constrained signal and each receiver observes this signal corrupted by a different additive white Gaussian noise. From its respective observation, Receiver 1 wishes to estimate the first source component and Receiver 2 wishes to estimate the second. We seek to characterize the pairs of expected squared-error distortions that are simultaneously achievable at the two receivers. Our result is that below a certain SNR-threshold an ldquouncoded schemerdquo that sends a linear combination of the source components is optimal. We present a lower bound on this threshold in terms of the source correlation and the distortion at the receiver with weaker channel noise.

We derive the capacity region of the Gaussian version of Willemspsilas two-user MAC with conferencing encoders. This setting differs from the classical MAC in that, prior to each transmission block, the two transmitters can communicate with each other over noise-free bit-pipes of given capacities. The derivation requires a new technique for proving the optimality of Gaussian input distributions in certain mutual information maximizations under a Markov constraint. We also consider a Costa-type extension of the Gaussian MAC with conferencing encoders. In this extension, the channel can be described as a two-user MAC with Gaussian noise and Gaussian interference where the interference is known non-causally to the encoders but not to the decoder. We show that as in Costa's setting the interference sequence can be perfectly canceled, i.e., that the capacity region without interference can be achieved.

A successive refinement model in which each receiver observes its own side-information, while the side-information pair is stochastically degraded with respect to the source, is considered. It is further assumed that a one way conference link with given capacity exists between the decoder observing the degraded side-information and the one observing the higher quality side-information. Inner and outer bounds on the rate-distortion region are derived. Our bounds are partially tight in the sense that the characterization of the primary encoder rates is conclusive, the remaining gap being in the characterization of the conference encoder rate.

A partially cooperative relay broadcast channel is a network with one source node and two destination nodes wherein destination 1 acts as a relay to assist destination 2. We derive two inner bounds on the rate region of this model. The first inner bound is based on the Cover-El Gamal generalized strategy for the original relay channel, while the second inner bound is based on the strategy proposed by Chong-Motani-Garg, for the relay channel, which combines backward decoding and simultaneous decoding. Both rate regions subsume the inner bound reported by Liang and Kramer which is based just on the decode-and-forward strategy for the common message.

We propose an encoding scheme for the Gaussian relay channel with receiver-transmitter feedback based on the Schalkwijk-Kailath coding strategy for the memoryless Gaussian single-user channel with feedback. The scheme has the advantage over previous schemes for the relay channel of being of very low complexity and, for certain channel parameters, achieving much higher rates.

We propose to send a Gaussian source over an average-power limited additive white Gaussian noise channel by transmitting a linear combination of the source sequence and the result of its quantization using a high dimensional Gaussian vector quantizer. We show that, irrespective of the rate of the vector quantizer (assumed to be fixed and smaller than the channel's capacity), this transmission scheme is asymptotically optimal (as the quantizer's dimension tends to infinity) under the mean squared-error fidelity criterion. This generalizes the classical result of Goblick about the optimality of scaled uncoded transmission, which corresponds to choosing the rate of the vector quantizer as zero, and the classical source-channel separation approach, which corresponds to choosing the rate of the vector quantizer arbitrarily close to the capacity of the channel

A coding technique is proposed for the discrete memoryless relay channel with noiseless feedback from relay to transmitter. The achievability result we obtain is analogous to a result proposed by Cover and El Gamal that combines cooperation and transmission of compressed data

A converse result is proved for the discrete memoryless relay channel in the presence of relay-transmitter causal feedback. The result subsumes the Cover-El Gamal max-flow min-cut upper bound on the capacity of the one-way relay channel, and it is used to establish the partial feedback capacity of the semideterministic relay channel

Asymptotic nonexistence bounds for unequal error protecting codes with two protection levels are considered. We show that the improved estimates on the possible distance distributions for codes may sometimes yield sharper upper bounds than the previously known ones on the higher significance protection level of both nonlinear and linear codes having two protection levels.

Achievable rates over the discrete memoryless relay channel with partial feedback configurations are proposed. Specifically, we consider partial feedback from the receiver to the sender as well as partial feedback from the relay to the sender. These achievable rates are calculated for the general Gaussian and the$Z$relay channels and are shown to improve on the known one-way achievable rates.

An inner bound on the deterministic-code capacity region of the two-user discrete memoryless arbitrarily varying general broadcast channel (AVGBC) was characterized by Jahn, assuming that the common message capacity is nonzero; however, he did not indicate how one could decide whether the latter capacity is positive. Csiszaacuter and Narayan's result for the single-user arbitrarily varying channel (AVC) establishes the missing part in Jahn's characterization. Nevertheless, being based on Ahlswede's elimination technique, Jahn's characterization is not applicable for symmetrizable channels under state constraint. Here, the various notions of symmetrizability for the two-user broadcast AVC are defined. Sufficient non-symmetrizability condition that renders the common message capacity of the AVGBC positive is identified using an approach different from Jahn's. The decoding rules we use establish an achievable region under state and input constraints for the family of degraded message sets codes over the AVGBC

The reliability exponents of two discrete-time single-server timing channel models, considered by Bedekar and Azizog∼lu, are determined for rate zero as well as for all rates between the corresponding critical rate and channel capacity. In both models, for rates between zero and the critical rate, we provide random-coding lower bound and straight-line combined with sphere-packing upper bound on the reliability exponent.

Asymptotic nonexistence bounds for unequal error protecting codes with two protection levels are considered. We show that the improved estimates, reported by Litsyn on the possible distance distributions for codes, may sometimes yield sharper upper bounds than the previously known ones, for both nonlinear and linear codes having two protection levels

An achievable region for the two-user discrete memoryless multiple-access channel (DMMAC) with noiseless feedback is proposed. The proposed region includes the Cover-Leung region, with the inclusion being, for some channels, strict. This inner bound is demonstrated for the ideal two-user Poisson multiple-access channel with noiseless feedback, in which case it is shown to improve on the Cover-Leung rate-sum.

The Poisson broadcast channel with degraded message sets models a bandwidth unlimited optical broadcast network over which a sender communicates a common message to a pair of terminals and a private message to just one of them. The capacity region for this communication model is determined by the Ko/spl uml/rner-Marton result, which in the Poisson regime is shown to coincide with the capacity region of the degraded broadcast channel. We present explicit code construction and practical decoding rules which attain the corresponding capacity region, while attempting to maximize the attainable error exponents at the two distinct terminals. This is accomplished by computing the error exponents attainable by the suggested decoding rules, using the technique developed by Wyner and further extended by Bross and Shamai.

The reliability exponent E(R) of the Bedekar-Azizoglu discrete-time geometric-server timing channel with service rate /spl mu/ packets per slot is determined for all rates R between the critical rate R/sub c/ and the channel capacity. For rates between 0 and R/sub c/ , we provide a random-coding lower bound E/sub r/(R) and a sphere-packing upper bound E/sub sp/(R) on E(R).

The single-user and two-user (multiple-access) Poisson arbitrarily varying channel (AVC) with input and state (peak and average power) constraints, but unlimited in bandwidth, are considered. For both cases, the deterministic and random code capacity with the average probability of error criterion is obtained. In the single-user case, A.D. Wyner's decoder (see ibid., vol.34, p.1449-61, 1988) attains the deterministic-code capacity whereas for the two-user case, a "nearest neighbor" decoder that belongs to the class of /spl beta/-decoders introduced by I. Csiszar and J. Korner (see Proc. 3rd Czechoslovak-Soviet-Hungarian Seminar on Information Theory, p.25-44, 1980) is shown to attain the deterministic-code capacity region as claimed.

The error exponent of the two-user Poisson multiple-access channel under peak and average power constraints, but unlimited in bandwidth, is considered. First, a random coding lower bound on the error exponent is obtained, and an extension of Wyner's (1988) single-user codes is shown to be exponentially optimum for this case as well. Second, the sphere-packing bounding technique suggested by Burnashev and Kutoyants (see Probl. Inform. Transm., vol.35, no.2, p.3-22, 1999) is generalized to the case at hand and an upper bound on the error exponent, which coincides with the lower bound, is derived. Thus, this channel joins its single-user partner as one of very few for which the reliability function is known.

The binary input N-user adder channel models a communication media accessed simultaneously by N users. Each user transmits a binary codeword of length n chosen from its codebook and the channel output consists of a componentwise arithmetic sum of the binary digits. Van Tilborg (1978) gave an upper bound on the size of a uniquely decodable code for the two-user case. His work is generalized here to the N-user case. The results give interesting information on the existence and properties of such codes.