On Index Coding in Noisy Broadcast Channels with Receiver Message Side Information

The index coding problem involves a sender with K messages to be transmitted across a broadcast channel, and a set of receivers each of which demands a subset of the K messages while having prior knowledge of a different subset as side information. We consider the specific instance of noisy index coding where the broadcast channel is Gaussian and every receiver demands all the messages from the source. We construct lattice index codes for this channel by encoding the K messages individually using K modulo lattice constellations and transmitting their sum modulo a shaping lattice. We introduce a design metric called side information gain that measures the advantage of a code in utilizing the side information at the receivers, and hence its quality as an index code. Based on the Chinese remainder theorem, we then construct lattice index codes for the Gaussian broadcast channel. Among all lattice index codes constructed using any densest lattice of a given dimension, our codes achieve the maximum side information gain.

An index code for a broadcast channel with receiver side information is 'locally decodable' if every receiver can decode its demand using only a subset of the codeword symbols transmitted by the sender instead of observing the entire codeword. Local decodability in index coding improves the error performance when used in wireless broadcast channels, reduces the receiver complexity and improves privacy in index coding. The 'locality' of an index code is the ratio of the number of codeword symbols used by each receiver to the number message symbols demanded by the receiver. Prior work on locality in index coding have considered only single unicast and single-uniprior problems, and the optimal trade-off between broadcast rate and locality is known only for a few cases. In this paper we identify the optimal broadcast rate (including among non-linear codes) for all three receiver unicast problems when the locality is equal to the minimum possible value, i.e., equal to one. The index code that achieves this optimal rate is based on a clique covering technique and is well known. The main contribution of this paper is in providing tight converse results by relating locality to broadcast rate, and showing that this known index coding scheme is optimal when locality is equal to one. Towards this we derive several structural properties of the side information graphs of three receiver unicast problems, and combine them with information theoretic arguments to arrive at a converse.

Index coding is an elegant and powerful idea for minimal usage of broadcast channels that investigates the fundamental limits and optimal coding schemes in information flow networks such as wireless networks, data storage networks, and content delivery networks. Minimal usage of broadcast channels is the main requirement of efficient information broadcasting in information flow networks which is the ultimate goal for successful communication. Finding the optimal index code with extreme rate for the class of index coding problems is the fundamental question in index coding. In this work, we present a new class of index coding problems with the size of maximum acyclic induced subgraph = 3. We refer to this class of index coding problems as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$class \mathbb {III}$ </tex-math></inline-formula> index coding problems. Our goal is to characterize the extreme rate (i.e., the optimal index code length for extreme rate) for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$class \mathbb {III}$ </tex-math></inline-formula> index coding problems. We show that the optimal linear index code of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$class \mathbb {III}$ </tex-math></inline-formula> index coding problem over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{2}$ </tex-math></inline-formula> achieves the extreme rate on the broadcast rate. To present this new result, we use the results of index coding network coding duality. In contrast with this new result, we also present that there exists maximum acyclic induced subgraph size 3 index coding problems with the optimal broadcast rate strictly greater than the extreme rate. To present this contrast, we use the alignment graph and the conflict hyper graph based approach.

Index coded PSK modulation over an AWGN broadcast channel, for a given index coding problem (ICP) is studied. For a chosen index code and an arbitrary mapping (of broadcast vectors to PSK signal points), a decision rule for the maximum likelihood (ML) decoder is derived. The message error performance of a receiver at high SNR is characterized by a parameter called PSK-index coding gain (PSK-ICG). The PSK-ICG of a receiver is determined by a metric called minimum inter-set distance. For a given ICP with an order of priority among the receivers, and a chosen $2^N$-PSK constellation an algorithm to find (index code, mapping) pairs, each of which gives the best performance in terms of PSK-ICG of the receivers, is proposed. No other pair of index code (of length 𝒩 with $2^N$ broadcast vectors) and mapping can give a better PSK-ICG for the highest priority receiver. Also, given that the highest priority receiver achieves its best performance, the next highest priority receiver achieves its maximum gain possible and so on in the specified order of priority.

This paper shows that the slope at each corner point of the capacity region of the general broadcast channel coincides with that of the randomized time division (hence the Marton) inner bound and the Nair-El Gamal (as well as the Korner-Marton) outer bound. We then show that the optimal superposition coding inner bound by Bandemer, El Gamal, and Kim can be simplified to the convex closure of the union of the Cover-Bergmans UX region and the Cover-van der Meulen UV region. Generalizing a result by Hajek and Pursely on the skewed binary symmetric broadcast channel, we show that for binary input broadcast channels, the UV region reduces to time division further simplifying the superposition coding inner bound. Finally we establish necessary and sufficient conditions for the optimality of the superposition inner bound for skewed binary broadcast channels.

Both index coding and non-orthogonal multiple access (NOMA) are useful techniques for a transmitter to send information to multiple cache-enabled receivers. In former works, either index coding or cache-aided NOMA is applied in the system, while the combination of index coding and cache-aided NOMA has not been fully investigated. This work is the first attempt to integrate these two techniques. A two-phase transmission algorithm is proposed to first partition receivers into index coding groups and next pair these groups up for superposition coding. Cache-aided interference cancellation (CIC) is employed at the receiver. This new method is applied to a cache-enabled fog radio access network (F-RAN). Besides, a distinct-file caching scheme with imbalanced cache size at fog access points (F-APs) is proposed. For this particular caching scheme, the two-phase algorithm can be fine-tuned in a way so that its time complexity becomes linear in the number of F-APs, which is very fast and particularly desirable from a practical viewpoint. Furthermore, simulation results show that our proposed method can significantly reduce the power consumption for transmissions over the fronthaul link of the F-RAN.

We determine the capacity region of a 3-receiver less noisy broadcast channel. The difficulty in extending the two-receiver result to three-receivers involves extending the Csiszar-sum lemma to three or more sequences, a standard difficulty in this area. In this work we bypass the difficulty by using a new information inequality, for less noisy receivers, that is employed in the converse. We also generalize our result to obtain the capacity region for a class of less noisy receivers.

A secret key agreement framework involving three users is considered in which each of the users 1 and 2 intends to share a secret key with user 3 and users 1 and 2 are eavesdroppers with respect to each other. There is a generalized discrete memoryless multiple access channel (GDMMAC) from users 1 and 2 to user 3 where the three users receive outputs from the channel. Furthermore, there is a feedback channel from user 3 to users 1 and 2 through which user 3 sends information extracted from the received output from the GDMMAC to increase the key rates. We consider both noiseless and noisy feedback. In the case of noiseless feedback, a public channel of unlimited capacity from user 3 to users 1 and 2 is used only once. In the case of noisy feedback, a noisy broadcast channel (BC) from user 3 to users 1 and 2 can be repeatedly used, like GDMMAC. In both setups, inner bounds of the secret key capacity region are derived. The secret key capacity region is derived in some special cases where the channel inputs and outputs form Markov chains in certain orders. For illustration, the corresponding results are also derived and discussed for Gaussian channels. The cases with noiseless feedback, noisy feedback, and no feedback at all are compared with each other.

The capacity region of general discrete-memoryless broadcast channels (DMBCs) with two receivers is an open problem of considerable research interest. The optimality of superposition coding in three specific cases of the DMBC is considered. For a DMBC with binary input, symmetric output and output cardinality at most 3, superposition coding is shown to be optimal. For equal-capacity DMBCs with any input cardinality, superposition coding is shown to be suboptimal if each channel has a capacity-achieving input distribution that is not capacity-achieving for the other channel. For an equal-capacity DMBC with binary input, superposition coding is shown to be optimal if and only if the two channels are more-capable comparable even without utput symmetry. These results improve upon the previously known conditions for optimality of superposition coding in DMBCs.

This paper studies binary hypothesis testing with a single sensor that communicates with two decision centers over a memoryless broadcast channel. The main focus lies on the tradeoff between the two type-II error exponents achievable at the two decision centers. In our proposed scheme, we can partially mitigate this tradeoff when the transmitter has a probability larger than 1/2 to distinguish the alternate hypotheses at the decision centers, i.e., the hypotheses under which the decision centers wish to maximize their error exponents. In the cases where these hypotheses cannot be distinguished at the transmitter (because both decision centers have the same alternative hypothesis or because the transmitter’s observations have the same marginal distribution under both hypotheses), our scheme shows an important tradeoff between the two exponents. The results in this paper thus reinforce the previous conclusions drawn for a setup where communication is over a common noiseless link. Compared to such a noiseless scenario, here, however, we observe that even when the transmitter can distinguish the two hypotheses, a small exponent tradeoff can persist, simply because the noise in the channel prevents the transmitter to perfectly describe its guess of the hypothesis to the two decision centers.

5G downlink communications are expected to suffer more from the intermittent link connectivity, and studying the erasure broadcast channel (BC) provides the fundamental understanding of their performance. Recently, the capacity region of two-user erasure BC with single-user delayed channel state information (CSI) was characterized, where one receiver does not provide its channel state to the transmitter and the other receiver, while the other receiver feeds back its state globally with unit delay. Surprisingly, the capacity region with single-user delayed CSI matches that of the erasure BC with global delayed CSI of both users. When two links have same erasure probabilities, this result is valid even when a common message intended for both users is required besides two private messages. The capacity in this case is easily achieved by adding a phase multicasting the common message after the transmission of private messages. However, when erasure probabilities are unequal, this scheme is clearly not capacity-achieving since the rate of the last phase will be limited by the weaker user having larger erasure probability. We propose a scheme which transmits the common message with rate higher than the link capacity of the weaker user in the last phase but still can ensure the decodability. The key is to simultaneously create equations of the common message at the weaker receiver and re-transmit private bits before the last phase. The proposed scheme achieves the converse under unequal link erasure probabilities, and the capacity with common and private messages is fully characterized under single-user delayed CSI.

In this paper, we study duality relations for the fading broadcast channel (BC) under additive white Gaussian noise and a strict ordering of the fading distributions without channel state information at the transmitter and the natural corresponding dual fading multiple-access channel (MAC). We show that if the fading distribution for the weaker user is non-deterministic, the achievable rate region using superposition coding and successive decoding and Gaussian signaling, which is conjectured to be the capacity region, is different from the dual MAC region. Duality holds only in the case of one-sided fading, where the fading distribution for the weaker user is deterministic. Despite this lack of duality, we propose to use the dual MAC in order to approximatively solve a non-convex problem for the BC. Specifically, we consider the problem of weighted sum-rate optimization for the BC and give, under some assumptions, upper bounds on the error incurred using this procedure.

Coding with a single fixed rate may not achieve the maximum throughput of a compound channel if the transmitter only has access to channel state statistics. Superposition coding for broadcast channels can be employed to increase the throughput. We study the optimal rate allocation for finite level superposition coding to maximize the throughput. Results show 2-level superposition coding is adequate to achieve most of the throughput gain in a quasi-static Rayleigh fading channel.

We characterize the capacity of broadcast channel in MU-MIMO system with fixed relays. By analyzing the similarities between the model of MU-MIMO with relays and the circuit with multiple capacitors, we give a method to estimate the end to multi-end broadcast capacity. In our method, we first partition the relay system into simple point to point components and calculate the capacity of each point to point link; then, we get the sum capacity of the broadcast channel, by calculating the point to point capacities in "parallel" or "cascade" way. Due to the difficulty of finding close-form solution to the non-convex problem, we estimate the upper and lower bounds of the broadcast capacity.

In this paper, we consider the problem of on-demand source coding for collaborative data dissemination in vehicular ad-hoc networks (VANETs). Specifically, we address the problem of index code construction for a big-data network. In this setup, users request a certain subset of data from a central server, while possessing some side information. The central server broadcasts a compressed message to meet all the users' demands. We consider a practical and generalized scenario where the side information and requested data packets of a user may be coded. Constructing index codes for such a setup is known to be an NP-hard problem. For this scenario, we propose a deep learning based approach to construct index codes through low-rank matrix completion. The proposed deep neural network performs matrix factorization to obtain a low-rank index code. We show that the proposed deep learning based approach to construct index codes is not only computationally feasible for networks with large data packets, but also provides better performance compared to the alternating minimization based approach for index code construction.