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neighbors of an encoded symbol are those symbols of the previous layer of the bipartite graph to which the encoded symbol is connected by edges. An encoded symbol is obtained by XOR-ing over all its neighbors of the previous graph layer.
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1. Find an output symbol yi that is connected to only one input symbol xj . If there is no such output symbol, the decoding algorithm halts at this point and fails to recover all the input symbols. 2. Set xj = yi . 3. Add (i.e., perform a bit-by-bit XOR) xj to all the output symbols that are connected to xj . 4. Remove all the edges connected to the input symbol xj . 5. Repeat until all {xj } are determined, or otherwise no more output symbols can be found that have exactly one neighbor. In the rst case, decoding ends successfully; in the latter, it fails. LT codes require a simple decoder, but it turns out that the degree distribution is a critical part of the design. The decoding process as described above will not even start if there is no output symbol of degree 1. This means that a good degree distribution is required for good decoding performances. Indeed, whether or not the decoding algorithm is successful depends solely on the degree distribution. The degree distribution used to produce LT codes is the Soliton Distribution or the more ef cient Robust Soliton Distribution. Further details on the degree distribution analysis and the Soliton and Robust Soliton distributions can be found in Luby [28].
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Applications. In 1998 a new company (Digital Fountain, Inc.) was launched by the inventor of LT codes (Michael Luby) [36]. The company aimed at exploiting the digital fountain s technology as a tool for downloading popular software packages, such as Adobe Acrobat; and, indeed, Adobe was an early investor, as were Cisco Systems and Sony Corporation. Obviously, Digital Fountain Inc. took a patent on LT codes, and this has been limiting utilization of LT codes in the research community as well as in other research and development centres. Digital Fountain s researchers have been working on a new generation of codes, and the group is also addressing other related research issues, such as application of the technology to video-on-demand and streaming media. Recently, the group has been focusing on congestion controls to ensure that the Digital Fountain packets behave in ways that do not cause network ow problems. Indeed, security issues are being addressed in collaboration with some universities to conceive, for example, a packet authentication system for verifying that data have come from a particular source. Launched to address the problems of streaming media and multicast, Digital Fountain like many new technology companies has found an unexpected niche. Without the requirement for TCP acknowledgments, data encoded with Luby transform codes travels faster than ordinary packets; this advantage becomes signi cant over long distances. Thus, the technology has proved particularly useful to companies that frequently transmit extremely large data les over long distances, such as movie studios and oil exploration companies. Inspired to LT codes, the work in Dimakis et al. [11] proposes using digital fountains in wireless sensor networks. The sensor network is viewed as a big DataBase with k independent data-generating nodes and n data-storage nodes. Each storage node
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is assumed to have a limited storage capacity of only one single data packet. Data packets are small and generated independently from each other. It is indeed assumed that no correlation exists between them. The k source data packets are encoded and distributed after addition of redundancy to the n storage nodes. A collector/sink node wishing to retrieve information from the sensor network needs only to query any set of storage nodes of size slightly larger than k to reconstruct the entire data sensed at the source nodes. The solution proposed is inspired by the family of protocols named Sensor Protocols for Information via Negotiation (SPIN). These protocols assume that all the nodes have enough storage space to store all the information gathered in the network. Hence the information gathered by the sensors is disseminated throughout the entire network, and a user can query any node to get the required information immediately by only communicating with one node. The solution in Dimakis et al. [11] is different in that each sensor node has not enough memory to store all the data generated in the network and thus collects a combination of the total information gathered in the network. Therefore, one single interrogation is not suf cient, but slightly more than k are required. 4.3.4 Raptor Codes Raptor codes are a recent extension of LT codes (2004) being conceived with the aim to improve the decoding probability of LT codes [29]. The decoding graph of LT codes, in fact, needs to be on the order of k log(k) in the number of edges (where k is the total number of input symbols) to make sure that all the input symbols can be recovered with high probability. Raptor codes focus on relaxing this condition such that the decoding process requires a smaller number of edges to be available. Raptor codes introduce a rst level of encoding where the input symbols are precoded and a new set of symbols comprising some redundancy is generated. The new set of symbols is then given as input to a second level of encoding that performs LT coding (see Figure 4.9). Precoding consists of a traditional erasure correcting code, say C, with a xed stretch factor. The choice of code C depends on the speci c application in mind. One possible choice is to use a Tornado code, but other choices are also possible such as
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Figure 4.9. Raptor codes: The input symbols are appended by redundant symbols (dark squares) in the case of a systematic precode. An appropriate LT code is used to generate output symbols from the precoded input symbols.
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