We introduce LT codes, the first rateless erasure codes that are very efficient as the data length grows.

Abstract—Overlay networks have emerged as a powerful and highly flexible method for delivering content. We study how to optimize throughput of large transfers across richly connected, adaptive overlay networks, focusing on the potential of collaborative transfers between peers to supplement ongoing downloads. First, we make the case for an erasure-resilient encoding of the content. Using the digital fountain encoding approach, end hosts can efficiently reconstruct the original content of size from a subset of any symbols drawn from a large universe of encoding symbols. Such an approach affords reliability and a substantial degree of application-level flexibility, as it seamlessly accommodates connection migration and parallel transfers while providing resilience to packet loss. However, since the sets of encoding symbols acquired by peers during downloads may overlap substantially, care must be taken to enable them to collaborate effectively. Our main contribution is a collection of useful algorithmic tools for efficient summarization and approximate reconciliation of sets of symbols between pairs of collaborating peers, all of which keep message complexity and computation to a minimum. Through simulations and experiments on a prototype implementation, we demonstrate the performance benefits of our informed content-delivery mechanisms and how they complement existing overlay network architectures. Index Terms—Bloom filter, content delivery, digital fountain, erasure code, min-wise sketch, overlay, peer-to-peer, reconciliation. I.

LT-Codes are a new class of codes introduced in [1] for the purpose of scalable and fault-tolerant distribution of data over computer networks. In this paper we introduce Raptor Codes, an extension of LT-Codes with linear time encoding and decoding. We will exhibit a class of universal Raptor codes: for a given integer k, and any real ε> 0, Raptor codes in this class produce a potentially infinite stream of symbols such that any subset of symbols of size k(1 + ε) is sufficient to recover the original k symbols with high probability. Each output symbol is generated using O(log(1/ε)) operations, and the original symbols are recovered from the collected ones with O(k log(1/ε)) operations. We will also introduce novel techniques for the analysis of the error probability of the decoder for finite length Raptor codes. Moreover, we will introduce and analyze systematic versions of Raptor codes, i.e., versions in which the first output elements of the coding system coincide with the original k elements. 1

Abstract—The proliferation of applications that must reliably distribute large, rich content to a vast number of autonomous receivers motivates the design of new multicast and broadcast protocols. We describe an ideal, fully scalable protocol for these applications that we call a digital fountain. A digital fountain allows any number of heterogeneous receivers to acquire content with optimal efficiency at times of their choosing. Moreover, no feedback channels are needed to ensure reliable delivery, even in the face of high loss rates. We develop a protocol that closely approximates a digital fountain using two new classes of erasure codes that for large block sizes are orders of magnitude faster than standard erasure codes. We provide performance measurements that demonstrate the feasibility of our approach and discuss the design, implementation, and performance of an experimental system. Index Terms—Content delivery, erasure codes, forward error correction, reliable multicast, scalability. 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.

We present a protocol for anonymous communication over the Internet. Our protocol, called P 5 (Peer-to-Peer Personal Privacy Protocol) provides sender-, receiver-, and sender-receiver anonymity. P 5 is designed to be implemented over the current Internet protocols, and does not require any special infrastructure support. A novel feature of P 5 is that it allows individual participants to trade-off degree of anonymity for communication efficiency, and hence can be used to scalably implement large anonymous groups. We present a description of P 5, an analysis of its anonymity and communication efficiency, and evaluate its performance using detailed packet-level simulations.

Abstract—Extrinsic information transfer (EXIT) charts are a tool for predicting the convergence behavior of iterative processors for a variety of communication problems. A model is introduced that applies to decoding problems, including the iterative decoding of parallel concatenated (turbo) codes, serially concatenated codes, low-density parity-check (LDPC) codes, and repeat–accumulate (RA) codes. EXIT functions are defined using the model, and several properties of such functions are proved for erasure channels. One property expresses the area under an EXIT function in terms of a conditional entropy. A useful consequence of this result is that the design of capacity-approaching codes reduces to a curve-fitting problem for all the aforementioned codes. A second property relates the EXIT function of a code to its Helleseth–Kløve–Levenshtein information functions, and thereby to the support weights of its subcodes. The relation is via a refinement of information functions called split information functions, and via a refinement of support weights called split support weights. Split information functions are used to prove a third property that relates the EXIT function of a linear code to the EXIT function of its dual. Index Terms—Concatenated codes, duality, error-correction coding, iterative decoding, mutual information.

We survey constructions and applications of digital fountains, an abstraction of erasure coding for network communication. Digital fountains effectively change the standard paradigm where a user receives an ordered stream of packets to one where a user must simply receive enough packets in order to obtain the desired data. Obviating the need for ordered data simplifies data delivery, especially when the data is large or is to be distributed to a large number of users. We also examine barriers to the adoption of digital fountains and discuss whether they can be overcome.

Abstract—Stopping sets determine the performance of low-density parity-check (LDPC) codes under iterative decoding over erasure channels. We derive several results on the asymptotic behavior of stopping sets in Tanner-graph ensembles, including the following. An expression for the normalized average stopping set distribution, yielding, in particular, a critical fraction of the block length above which codes have exponentially many stopping sets of that size. A relation between the degree distribution and the likely size of the smallest nonempty stopping set, showing that for a I

We show how asymptotic estimates of powers of polynomials with non-negative coefficients can be used in the analysis of low-density parity-check (LDPC) codes. In particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular LDPC code ensembles. We then consider the binary erasure channel (BEC). Using these estimates we derive lower bounds on the error exponent, under iterative decoding, of LDPC codes used over the BEC. Both regular and irregular code structures are considered. These bounds are compared to the corresponding bounds when optimal (maximum likelihood) decoding is applied.