This paper proposes COPE, a new architecture for wireless mesh networks. In addition to forwarding packets, routers mix (i.e., code) packets from different sources to increase the information content of each transmission. We show that intelligently mixing packets increases network throughput. Our design is rooted in the theory of network coding. Prior work on network coding is mainly theoretical and focuses on multicast traffic. This paper aims to bridge theory with practice; it addresses the common case of unicast traffic, dynamic and potentially bursty flows, and practical issues facing the integration of network coding in the current network stack. We evaluate our design on a 20-node wireless network, and discuss the results of the first testbed deployment of wireless network coding. The results show that COPE largely increases network throughput. The gains vary from a few percent to several folds depending on the traffic pattern, congestion level, and transport protocol.

Opportunistic routing is a recent technique that achieves high throughput in the face of lossy wireless links. The current opportunistic routing protocol, ExOR, ties the MAC with routing, imposing a strict schedule on routers ’ access to the medium. Although the scheduler delivers opportunistic gains, it misses some of the inherent features of the 802.11 MAC. For example, it prevents spatial reuse and thus may underutilize the wireless medium. It also eliminates the layering abstraction, making the protocol less amenable to extensions to alternate traffic types such as multicast. This paper presents MORE, a MAC-independent opportunistic routing protocol. MORE randomly mixes packets before forwarding them. This randomness ensures that routers that hear the same transmission do not forward the same packets. Thus, MORE needs no special scheduler to coordinate routers and can run directly on top of 802.11. Experimental results from a 20-node wireless testbed show that MORE’s median unicast throughput is 22 % higher than ExOR, and the gains rise to 45 % over ExOR when there is a chance of spatial reuse. For multicast, MORE’s gains increase with the number of destinations, and are 35-200 % greater than ExOR.

Abstract—It is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite-field alphabet. It is also known that this result does not generalize to arbitrary networks. There are several examples in the literature of solvable networks with no scalar linear solution over any finite field. However, each example has a linear solution for some vector dimension greater than one. It has been conjectured that every solvable network has a linear solution over some finite-field alphabet and some vector dimension. We provide a counterexample to this conjecture. We also show that if a network has no linear solution over any finite field, then it has no linear solution over any finite commutative ring with identity. Our counterexample network has no linear solution even in the more general algebraic context of modules, which includes as special cases all finite rings and Abelian groups. Furthermore, we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field (exactly 10 % greater), so the network is not even asymptotically linearly solvable. It follows that, even for more general versions of linearity such as convolutional coding, filter-bank coding, or linear time sharing, the network has no linear solution. Index Terms—Asymptotics, flows, linear coding, network information theory, routing. I.

Some forms of ad-hoc networks need to operate in extremely performance-challenged environments where end-to-end connectivity is rare. Such environments can be found for example in very sparse mobile networks where nodes ”meet ” only occasionally and are able to exchange information, or in wireless sensor networks where nodes sleep most of the time to conserve energy. Forwarding mechanisms in such networks usually resort to some form of intelligent flooding, as for example in probabilistic routing. We propose a communication algorithm that significantly reduces the overhead of probabilistic routing algorithms, making it a suitable building block for a delay-tolerant network architecture. Our forwarding scheme is based on network coding. Nodes do not simply forward packets they overhear but may send out information that is coded over the contents of several packets they received. We show by simulation that this algorithm achieves the reliability and robustness of flooding at a small fraction of the overhead.

Traditionally, interference is considered harmful. Wireless networks strive to avoid scheduling multiple transmissions at the same time in order to prevent interference. This paper adopts the opposite approach; it encourages strategically picked senders to interfere. Instead of forwarding packets, routers forward the interfering signals. The destination leverages network-level information to cancel the interference and recover the signal destined to it. The result is analog network coding because it mixes signals not bits. So, what if wireless routers forward signals instead of packets? Theoretically, such an approach doubles the capacity of the canonical relay network. Surprisingly, it is also practical. We implement our design using software radios and show that it achieves significantly higher throughput than both traditional wireless routing and prior work on wireless network coding. 1.

We consider the problem of establishing minimum-cost multicast connections over coded packet networks, i.e., packet networks where the contents of outgoing packets are arbitrary, causal functions of the contents of received packets. We consider both wireline and wireless packet networks as well as both static multicast (where membership of the multicast group remains constant for the duration of the connection) and dynamic multicast (where membership of the multicast group changes in time, with nodes joining and leaving the group). For static multicast, we reduce the problem to a polynomial-time solvable optimization problem, ... and we present decentralized algorithms for solving it. These algorithms, when coupled with existing decentralized schemes for constructing network codes, yield a fully decentralized approach for achieving minimum-cost multicast. By contrast, establishing minimum-cost static multicast connections over routed packet networks is a very difficult problem even using centralized computation, except in the special cases of unicast and broadcast connections. For dynamic multicast, we reduce the problem to a dynamic programming problem and apply the theory of dynamic programming to suggest how it may be solved.

Abstract—Network coding substantially increases network throughput. But since it involves mixing of information inside the network, a single corrupted packet generated by a malicious node can end up contaminating all the information reaching a destination, preventing decoding. This paper introduces distributed polynomial-time rate-optimal network codes that work in the presence of Byzantine nodes. We present algorithms that target adversaries with different attacking capabilities. When the adversary can eavesdrop on all links and jam zO links, our first algorithm achieves a rate of C 0 2zO, where C is the network capacity. In contrast, when the adversary has limited eavesdropping capabilities, we provide algorithms that achieve the higher rate of C 0 zO. Our algorithms attain the optimal rate given the strength of the adversary. They are information-theoretically secure. They operate in a distributed manner, assume no knowledge of the topology, and can be designed and implemented in polynomial time. Furthermore, only the source and destination need to be modified; nonmalicious nodes inside the network are oblivious to the presence of adversaries and implement a classical distributed network code. Finally, our algorithms work over wired and wireless networks. Index Terms—Byzantine adversaries, distributed network error-correcting codes, eavesdroppers, information-theoretically optimal, list decoding, polynomial-time algorithms. I.

We consider applying network coding in settings where there is a cost associated with network use.

An outer bound on the rate region of noise-free information networks is given. This outer bound combines properties of entropy with a strong information inequality derived from the structure of the network. This blend of information theoretic and graph theoretic arguments generates many interesting results. For example, the capacity of directed cycles is characterized. Also, a gap between the sparsity of an undirected graph and its capacity is shown. Extending this result, it is shown that multicommodity flow solutions achieve the capacity in an infinite class of undirected graphs, thereby making progress on a conjecture of Li and Li. This result is in sharp contrast to the situation with directed graphs, where a family of graphs are presented in which the gap between the capacity and the rate achievable using multicommodity flows is linear in the size of the graph.

Abstract. Error correction in existing point-to-point communication networks is done on a link-by-link basis, which is referred to in this paper as classical error correction. Inspired by network coding, we introduce in this two-part paper a new paradigm called network error correction. The theory thus developed subsumes classical algebraic coding theory as a special case. In Part I, we discuss the basic concepts and prove the network generalizations of the Hamming bound and the Singleton bound in classical algebraic coding theory. By studying a few elementary examples, the relation between network error correction and classical error correction is investigated.