M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi, D. A. Spielman, and V. Stemann, Practical lossresilient codes, in ##### ## ### #### ###### ### ##### ## ###### ## #####, pp. 150159, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....guarantees. The latest version of the paper and our ongoing experimental work is available at http: www.cs. bu.edu groups aces codes 1 Our techniques are based on leveraging well known all or nothing like properties of certain fast forward error correction codes (FEC) such as Tornado codes [12]. These codes can be used to facilitate the delivery of bulk content, particularly in the multicast case, as advocated using the digital fountain paradigm [2] By making minor modi cations to the codes and encrypting about 4 of all transmissions, we ensure that an adversary intercepting the ....

....an encoded le spanning n packets when n is 10,000 and m is 5,000. An alternative approach relies on the use of sparse codes based on parity check principles, which can be encoded and decoded in nearly linear time, at the expense of relaxing the optimal decoding guarantee provided in IDA [12, 2, 11]. Sparse parity check codes facilitate fast encoding and decoding and are well suited to use with large les and highly dispersed encodings. A document is divided into a collection of m xed length input symbols, or data packets, x 1 ; xm . An encoder produces a potentially unbounded ....

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M. Luby, M. Mitzenmacher, A. Shokrollahi, D. Spielman, and V. Stemann. Practical lossresilient codes. In Proceedings of the 29 th ACM Symposium on Theory of Computing, April 1997.

....into # # # encoding blocks for # ## #; receiving any # distinct encoding blocks allows the complete, efficient reconstruction of the source data. For large files, close approximations to an ideal digital fountain can be achieved with fast forward error correcting codes, such as Tornado codes [22], which we employ. Traditionally, encoded content has been most widely used in multicast applications, where transmission of a redundant symbol enables different receivers to recover from different packet losses. In our setting, encoding the content realizes a a new benefit, namely while it still ....

....Fountain approach described in the introduction is an example of a recent effort that aims to achieve the second of the above two goals in the context of Internet servers. In one instantiation of that approach, Byers, Luby, Mitzenmacher, and Rege propose the use of fast error correcting codes [22] to alleviate the need for per client state information at the source of a reliable multicast transmission [12] Another instantiation of this approach uses encoded content to facilitate stateless downloads from multiple mirror sites in parallel [11] Our work also employs this paradigm. However, ....

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M. Luby, M. Mitzenmacher, A. Shokrollahi, D. Spielman, and V. Stemann. Practical lossresilient codes. In 29th STOC, May 1997.

....in the multicast tree. Thus, FEC is a good solution for broadband communications where data is multicast over wide area networks to a large group of clients [12, 14, 20, 23, 33] For real time data services, or time sensitive and delay critical services in general, ARQ is not suitable [3, 12, 19, 22]. The feedback channel overhead and the retransmission latency are the most important limitations. For example, a retransmitted video frame may be too old for presentation when arriving at the client side. Low latency is necessary for applications supporting multimedia playouts or human ....

....supported by a single INSTANCE server with the same or better quality than in traditional server implementations. It is worthwhile to note that our prototype is based on FEC schemes that are available as public domain software. There exist several faster correcting codes, e.g. the Tornado code [12, 19], that are commercially distributed and were not available for evaluation in our prototype. Obviously, these codes promise even higher performance improvements for the integrated error management in INSTANCE. The design of the other two INSTANCE techniques, i.e. zero copy one copy memory ....

Luby, M., Mitzenmacher, M., Shokrollahi, M. A., Spielman, D. A., Stemann, V.: "Practical LossResilient Codes", Proceedings of the 29th annual ACM Symposium on Theory of Computing (STOC'97), El Paso, TX, USA, May 1997, pp. 150-159. 16

....The erasure code presented in this paper has been in use since early 1997 in a number of research papers and actual applications (e.g. 11, 17] for the development of scalable multicast communication protocols. Albeit some other erasure codes with better asymptotic performance have been proposed [8] recently, we believe that our code has many practical applications being deterministically decodable, small and simple (we have run it on machines as small as an HP100LX) and, especially, not subject to patents or other impediments to its use. 4 Conclusions We have presented a couple of tools ....

M.Luby, M.Mitzenmacher, A.Shokrollahi, D.Spielman, and V.Stemann. "Practical lossresilient codes", Proc. of the Twenty-Ninth Annual ACM Symp. on Theory of Computing, El Paso, Texas, 4-6 May 1997.

....where t is the number of erasures. Note also that the assumption of deterministic encoding decoding is essential since it is possible to construct probabilistic encoding decoding methods that ensure recovery of binary sequences with at most t erasures using t(1 ffl) check bits; see Luby at al. [9]. A number of recursive constructions of low complexity codes for different transmission models is known in the literature; see Ahlswede et al. 1] Alon and Luby [2] Dumer [6] Spielman [13] The idea of constructing low complexity codes for all models is essentially the same. Namely, first the ....

M. G. Luby, M. Mitzenmacher, M. Amin Shokrollahi, D. A. Spielman, and V. Stemann, "Practical lossresilient codes," Proc. 29th ACM Annual Sympos. on the Theory of Computing (STOC'97), ACM (1997), pp. 150--159.

....32. 64) for this parameter. Coupled with packet sizes of about 1 KB, it follows that our assumption of l k does not hold for medium large sized files. Two approaches are possible to deal with this problem. The first one relies on the use of better codes, such as the Tornado codes presented in [10]. These codes have better asymptotic performance than our Vandermonde code, so they can be used even for large values of k, as shown in [4] Tornado codes are non deterministic, meaning that the number of packets necessary for a successful decoding varies, and the decision to stop receiving can ....

M.Luby, M.Mitzenmacher, A.Shokrollahi, "Practical LossResilient Codes", in Proc. of the 29 th ACM Symp. on Theory of Computing, 1997.

....can produce n 0 packets. The encoding costs are then bounded by the value of k 0 ; the receiver efficiency decreases slightly, in that the condition of having a minimum number of packets must hold on each group of n 0 packets rather than on the whole file. As an alternative, a recent paper [9] presents a different encoding algorithm, based on probabilistic techniques, which is very efficient for large values of k, requiring constant time per packet produced. 6 Conclusions and future work We have presented a congestion control algorithm for multicast data transfer on the MBone, ....

M.Luby, M.Mitzenmacher, A.Shokrollahi, D.Spielman, and V.Stemann. "Practical lossresilient codes", Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, page 150, El Paso, Texas, 4-6 May 1997.

....Moreover, 3] showed that many concepts that were first developed for the erasure channel carry over to the case of other more complicated channels. For this reason, we will start in this paper a systematic study of capacity achieving sequences of low density codes over the erasure channel. In [4] the authors introduced a simple algorithm for correcting erasures in a low density paritycheck code. To describe the result, we need some piece of notation. We visualize low density Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974, USA, email: poswald research.bell labs.com y ....

....is connected to a check node of degree i. Let i and ae i denote the fraction of edges of left degree and right degree i, respectively. A degree distribution for the graph is then the pair ( ae) where = x) P i i x i Gamma1 and ae = ae(x) P i ae i x i Gamma1 . The main result of [4] states that their simple recovery algorithm is successful on a random graph with degree distribution ( ae) and initial erasure probability ffi if ffi(1 Gamma ae(1 Gamma x) x (1) on the interval (0; ffi) The capacity of the erasure channel with probability ffi is 1 Gamma ffi. On the ....

[Article contains additional citation context not shown here]

M. Luby, M. Mitzenmacher, A. Shokrollahi, D. Spielman, and V. Stemann, "Practical lossresilient codes," in Proceedings of the 29th annual ACM Symposium on Theory of Computing, pp. 150--159, 1997.

....Moreover, 3] showed that many concepts that were first developed for the erasure channel carry over to the case of other more complicated channels. For this reason, we will start in this paper a systematic study of capacity achieving sequences of low density codes over the erasure channel. In [4] the authors introduced a simple algorithm for correcting erasures in a low density paritycheck code. To describe the result, we need some piece of notation. We visualize low density parity check codes as bipartite graphs between a set of left nodes called variable nodes and a set of right nodes ....

....it is connected to a check node of degree i. Let i and ae i denote the fraction of edges of left degree and right degree i, respectively. A degree distribution for the graph is then the pair ( ae) where = x) P i i x i Gamma1 and ae = ae(x) P i ae i x i Gamma1 . The main result of [4] states that their simple recovery algorithm is successful on a random graph with degree distribution ( ae) and initial erasure probability ffi if ffi(1 Gamma ae(1 Gamma x) x (1) on the interval (0; ffi) In this case, we say that the pair ( ae) affords ffi. The capacity of the erasure ....

[Article contains additional citation context not shown here]

M. Luby, M. Mitzenmacher, A. Shokrollahi, D. Spielman, and V. Stemann, "Practical lossresilient codes," in Proceedings of the 29th annual ACM Symposium on Theory of Computing, pp. 150--159, 1997.

....paritycheck codes. He suggests a natural decoding algorithm for these codes, and proves a good bound on the fraction of errors that can be corrected. As the codes that Gallager builds are derived from regular graphs, we refer to them as regular codes. Following the general approach introduced in [7] for the design and analysis of erasure codes, we consider error correcting codes based on random irregular bipartite graphs, which we call irregular codes. We introduce tools based on linear programming for designing linear time irregular codes with better error correcting capabilities than ....

....directly apply to such graphs. Instead, he constructs explicit graphs of large girth to which his analysis does apply. The main contribution of this paper is the design and analysis of low density parity check codes based on irregular graphs. This work follows the general approach introduced in [7] for the design and analysis of erasure codes. There it is shown that using irregular graphs yields codes with much better performance than regular graphs. In accordance with [7] we consider errorcorrecting codes based on random irregular bipartite graphs, which we call irregular codes. We ....

[Article contains additional citation context not shown here]

M. Luby, M. Mitzenmacher, M. A. Shokrollahi, D. A. Spielman, and V. Stemann, "Practical LossResilient Codes", Proc. 29 th Symp. on Theory of Computing, 1997, pp. 150--159.

....a new fast algorithm for the setting of of Gaussian noise, and leave several open questions regarding the construction of these codes and the associated decoding algorithms. 1 Introduction Simple linear time erasure codes with nearly optimal correction properties were introduced and analyzed in [2, 3]. The analysis of these codes are based on the analysis of a simple stochastic process on an irregular random bipartite graph. In [4] a similar analysis was used to develop and prove bounds on the behavior of irregular low density parity check codes, extending the previous work of Gallager [1] on ....

....and prove bounds on the behavior of irregular low density parity check codes, extending the previous work of Gallager [1] on regular low density parity check codes. In this work, we consider low density parity check codes that handle both errors and erasures, building on the previous work of [2, 3, 4]. Our work generalizes and unifies the previous analyses in a natural way. As a result of this unification, we find simple codes for both erasures and errors with associated linear time encoding and decoding algorithms and provable performance bounds. For convenience, we call these codes LDEE ....

[Article contains additional citation context not shown here]

M. Luby, M. Mitzenmacher, M. A. Shokrollahi, D. A. Spielman, and V. Stemann, "Practical LossResilient Codes", Proc. 29 t h Symp. on Theory of Computing, 1997, pp. 150--159.

....rate 1 4 codes on 16,000 bits over a binary symmetric channel, previous low density parity check codes can correct up to approximately 16 errors, while our codes can correct over 17 . Our improved performance comes from using codes based on irregular random bipartite graphs, based on the work of [7]. Previously studied low density parity check codes have been derived from regular bipartite graphs. We report experimental results for our irregular codes on both binary symmetric channels and Gaussian channels. In some cases our results come very close to reported results for turbo codes, ....

.... under belief propagation decoding, has been the subject of much recent experimentation and analysis [2, 11, 12, 16] The interest in these codes stems from their near Shannon limit performance, their simple descriptions and implementations, and their amenability to rigorous theoretical analysis [6, 7, 8, 9, 11, 16, 17]. Moreover, there appears to be a connection between these codes and turbo codes, introduced by [1] In particular, the turbo code decoding algorithm can be understood as a belief propagation based algorithm [10, 5] and hence any understanding of belief propagation on low density parity check ....

[Article contains additional citation context not shown here]

M. Luby, M. Mitzenmacher, M. A. Shokrollahi, D. A. Spielman, and V. Stemann, "Practical LossResilient Codes", Proc. 29 t h Symp. on Theory of Computing, 1997, pp. 150--159.

....using a particular message passing decoder. Since for every given pair of degree sequences ( ae) we can determine the resulting threshold value ffi , it is natural to search for those pairs which maximize this threshold. 3 This was done, with great success, in the case of erasure channels [5, 6, 7]. In the case of most other channels of interest the situation is much more complicated and new methods must be brought to bear on the optimization problem. Fig. 2 compares the performance of the (3; 6) regular LDPCC (which is the best regular such code) with the performance of the best irregular ....

....oe, as well as the raw input bit error probability P b . The empirical evidence presented in Fig. 2 together with the results presented in Section 3 beg the question of whether LDPCCs can achieve capacity on a given binary input memoryless channel. The only result in this direction is that of [5] which gives an explicit sequence of degree distributions such that, in the limit, the codes induced by these degree distributions achieve capacity on an erasure channel. The following Theorem, due to Gallager, imposes, at least for the BSC, a necessary condition in order for LDPCCs to achieve ....

[Article contains additional citation context not shown here]

M. Luby, M. Mitzenmacher, A. Shokrollahi, D. Spielman, and V. Stemann, "Practical lossresilient codes," in Proceedings of the 29th annual ACM Symposium on Theory of Computing, pp. 150--159, 1997.

....of encoding and decoding operations. As detailed in subsequent sections, the encoding and decoding processing times for standard Reed Solomon erasure codes are prohibitive even for moderate values of k and n. The alternative we propose is to avoid this cost by using the much faster Tornado codes [11]. As always, there is a tradeoff associated with using one code in place of another. The main drawback of using Tornado codes is that the decoder requires slightly more than k of the transmitted packets to reconstruct the source data. This tradeoff is the main focus of our comparative simulation ....

....of redundant packets for a total of n = k encoding packets all of a fixed length P . 5.1 Theory In this section, we describe in some detail the construction of a specific Tornado code and explain some of the general principles behind Tornado codes. This construction is based on ideas from [11, 12]. We begin by providing intuition behind Reed Solomon codes. We think of the ith source data packet as containing the value of a variable x i , and we think of the jth redundant packet as containing the value of a variable y j that is a linear combination of the x i variables over an appropriate ....

[Article contains additional citation context not shown here]

M. Luby, M. Mitzenmacher, A. Shokrollahi, D. Spielman, V. Stemann, "Practical LossResilient Codes." In Proceedings of the 29 th ACM Symposium on Theory of Computing, 1997.

....neighboring packets to the left, as depicted on the right side of Figure 1. The structure of the bipartite random graphs must be specially chosen to guarantee both rapid encoding and decoding and the erasure property described below. A detailed, technical description of these codes is provided in [8] and [9] Tornado codes have the following erasure property: to reconstruct the source data, it suffices to recover slightly more than k of the n packets stored in the graph. We say the reception overhead is ffl if (1 ffl)k encoding packets are required to reconstruct the source data. 2 The ....

....The erasure codes listed in Tables 2 and 3 as Vandermonde [16] and Cauchy [2] are standard implementations of Reed Solomon erasure codes, based on Vandermonde matrices and Cauchy matrices, respectively. Both Tornado A and Tornado B codes were designed using some of the principles described in [8] and [9] The implementations were not carefully optimized, so their running times could be improved by constant factors. All experiments were benchmarked on a Sun 167 MHz UltraSPARC 1 with 64 megabytes of RAM running Solaris 2.5.1. All runs are with packet length P = 1KB. For all runs, a file ....

M. Luby, M. Mitzenmacher, A. Shokrollahi, D. Spielman, V. Stemann, "Practical LossResilient Codes." In Proceedings of the 29 th ACM Symposium on Theory of Computing, 1997.

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M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi, D. A. Spielman, and V. Stemann, Practical lossresilient codes, in ##### ## ### #### ###### ### ##### ## ###### ## #####, pp. 150159, 1997.

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M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi, D. A. Spielman, and V. Stemann, Practical lossresilient codes, in ##### ## ### #### ###### ### ##### ## ###### ## #####, pp. 150159, 1997.

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M.Luby, M.Mitzenmacher, A.Shokrollahi, D.Spielman, and V.Stemann. "Practical lossresilient codes", Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, page 150, El Paso, Texas, 4-6 May 1997.

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