We consider the problem of two transmitters wishing to exchange information through a relay in the middle. The channels between the transmitters and the relay are assumed to be synchronized, average power constrained additive white Gaussian noise channels with a real input with signal-to-noise ratio (SNR) of snr. An upper bound on the capacity is 1 2 log(1 + snr) bits per transmitter per use of the medium-access phase and broadcast phase of the bi-directional relay channel. We show that using lattice codes and lattice decoding, we can obtain a rate of 1 2 log( 1 2 + snr) bits per transmitter, which is essentially optimal at high SNR. The main idea is to decode the sum of the codewords modulo a lattice at the relay followed by a broadcast phase which performs Slepian-Wolf coding with structured codes. For asymptotically low SNR, joint decoding of the two transmissions at the relay (MAC channel) is shown to be optimal. We also show that if the two transmitters use identical lattices with minimum angle decoding, we can achieve the same rate of 1 2 log( 1 2 + snr). The proposed scheme can be thought of as a joint physical layer, network layer code which outperforms other recently proposed analog network coding schemes.

The index coding problem has recently attracted a significant attention from the research community due to its theoretical significance and applications in wireless ad-hoc networks. An instance of the index coding problem includes a sender that holds a set of information messages X = {x1, . . . , x k } and a set of receivers R. Each receiver ρ = (x, H) ∈ R needs to obtain a message x ∈ X and has prior side information comprising a subset H of X. The sender uses a noiseless communication channel to broadcast encoding of messages in X to all clients. The objective is to find an encoding scheme that minimizes the number of transmissions required to satisfy the receivers' demands with zero error.In this paper, we analyze the relation between the index coding problem, the more general network coding problem and the problem of finding a linear representation of a matroid. In particular, we show that any instance of the network coding and matroid representation problems can be efficiently reduced to an instance of the index coding problem. Our reduction implies that many important properties of the network coding and matroid representation problems carry over to the index coding problem. Specifically, we show that vector linear codes outperform scalar linear codes and that vector linear codes are insufficient for achieving the optimum number of transmissions.

Abstract-We consider the problem of data exchange by a group of closely-located wireless nodes. In this problem each node holds a set of packets and needs to obtain all the packets held by other nodes. Each of the nodes can broadcast the packets in its possession (or a combination thereof) via a noiseless broadcast channel of capacity one packet per channel use. The goal is to minimize the total number of transmissions needed to satisfy the demands of all the nodes, assuming that they can cooperate with each other and are fully aware of the packet sets available to other nodes. This problem arises in several practical settings, such as peer-to-peer systems and wireless data broadcast. In this paper, we establish upper and lower bounds on the optimal number of transmissions and present an efficient algorithm with provable performance guarantees. The effectiveness of our algorithms is established through numerical simulations.

We consider scenarios where wireless clients are missing some packets, but they collectively know every packet. The clients collaborate to exchange missing packets over an error-free broadcast channel with capacity of one packet per channel use. First, we present an algorithm that allows each client to obtain missing packets, with minimum number of transmissions. The algorithm employs random linear coding over a sufficiently large field. Next, we show that the field size can be reduced while maintaining the same number of transmissions. Finally, we establish lower and upper bounds on the minimum number of transmissions that are easily computable and often tight as demonstrated by numerical simulations.

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Abstract-In this paper, we focus on the fundamental problem of finding the optimal encoding for the broadcasted packets that minimizes the overall number of transmissions. We show that this problem is NP-complete over GF (2) and establish several fundamental properties of the optimal solution. We also propose a simple heuristic solution for the problem based on graph coloring and present some empirical results for random settings.

We consider the problem of securing a multicast network against a wiretapper that can intercept the packets on a limited number of arbitrary network edges of its choice. We assume that the network employs the network coding technique to simultaneously deliver the packets available at the source to all the receivers. We show that this problem can be looked at as a network generalization of the wiretap channel of type II introduced in a seminal paper by Ozarow and Wyner. In particular, we show that the transmitted information can be secured by using the Ozarow-Wyner approach of coset coding at the source on top of the existing network code. This way, we quickly and transparently recover some of the results available in the literature on secure network coding for wiretap networks. Moreover, we derive new bounds on the required alphabet size that are independent of the network size and devise an algorithm for the construction of secure network codes. We also look at the dual problem and analyze the amount of information that can be gained by the wiretapper as a function of the number of wiretapped edges.

Abstract-In the multicast network coding problem, a source needs to deliver packets to a set of terminals over an underlying communication network . The nodes of the multicast network can be broadly categorized into two groups. The first group incudes encoding nodes, i.e., nodes that generate new packets by combining data received from two or more incoming links. The second group includes forwarding nodes that can only duplicate and forward the incoming packets. Encoding nodes are, in general, more expensive due to the need to equip them with encoding capabilities. In addition, encoding nodes incur delay and increase the overall complexity of the network. Accordingly, in this paper, we study the design of multicast coding networks with a limited number of encoding nodes. We prove that in a directed acyclic coding network, the number of encoding nodes required to achieve the capacity of the network is bounded by 3 2 . Namely, we present (efficiently constructible) network codes that achieve capacity in which the total number of encoding nodes is independent of the size of the network and is bounded by 3 2 . We show that the number of encoding nodes may depend both on and by presenting acyclic coding networks that require ( 2 ) encoding nodes. In the general case of coding networks with cycles, we show that the number of encoding nodes is limited by the size of the minimum feedback link set, i.e., the minimum number of links that must be removed from the network in order to eliminate cycles. We prove that the number of encoding nodes is bounded by (2 + 1) 3 2 , where is the minimum size of a feedback link set. Finally, we observe that determining or even crudely approximating the minimum number of required encoding nodes is an -hard problem.