Citation Context

...s mathematiques algorithmiques École Polytechnique Fédérale de Lausanne 1015 Lausanne, Switzerland amin.shokrollahi@epfl.ch September 19, 2005 Abstract 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. W...

Citation Context

...than to their dimension. Therefore, for small rates the encoding and decoding algorithms for these codes are slow. This turns out to be quite limiting in many applications, such as those described in =-=[5]-=-, since the codes used there are of extremely low rate. This suggests that the encoding/decoding times of traditional coding technologies may not be adequate for the design of scalable data transmissi...

Citation Context

...re faster algorithms based on fast polynomial arithmetic, but these algorithms are often too complicated in practice.) However, quadratic running times are still too large for many applications. 2sIn =-=[3]-=- the authors construct codes with linear time encoding and decoding algorithms that can come arbitrarily close to the capacity of the BEC. These codes, called Tornado codes, are very similar to Gallag...

Citation Context

...h parameters (n,ΩD(x)) are sufficient to recover at least (1 − δ)n input symbols via BP decoding, where δ = (ε/4)/(1 + ε). Proof. We use the analysis of the decoding process as described in [3] or in =-=[12]-=-. Consider a set of n(1 + ε/2)+1 output symbols and set up the graph associated to these output symbols. This graph is a random graph with edge degree distributions ι(x) and ω(x) corresponding to the ...

Citation Context

.... One possible choice would be a Tornado code [3], though other choices are also possible (for example an LT-code with the appropriate number of output symbols, or an irregular Repeat-Accumulate Code =-=[11]-=-). 15s6 Raptor Codes with Good Asymptotic Performance In the last section we encountered two types of Raptor Codes. For one of them, the LT-codes, the overhead and the space were close to 1, while the...

Citation Context

...de the sum of its neighbors among the message nodes is zero. (See Figure 3.) BP decoding of LDPC codes over an erasure channel is very similar to the BP decoding of LT-Codes [3]. It has been shown in =-=[18]-=- that this decoding algorithm is successful if and only if the graph induced by the erased message positions does not contain a stopping set. A stopping set is a set of message nodes such that their i...

Citation Context

...ue of the input symbol. Let pi be the probability that an edge in the decoding graph carries a value 1 from an output symbol at step i of the decoding process. Then, a standard tree analysis argument =-=[16]-=- shows the recursion pi+1 = ω(1 − ι(1 − pi)), where ω and ι are the output and the input edge degree distributions, respectively. Note that this recursion is only valid if we can assume that the messa...

Citation Context

...+ ε) = (1 − R)/2 with O(n log(1/ε)) arithmetic operations. Examples of such codes are Tornado codes [3], right-regular codes [13], and certain types of repeat-accumulate codes. The reader can consult =-=[14]-=- for other types of such capacity-achieving examples. We remark, however, that it is not necessary for Cn to be capacity-achieving, since we only require that the decoder be able to decode up to (1 − ...

Citation Context

...ction 8 we develop a new set of ideas and design efficient systematic versions of Raptor Codes. Raptor Codes were discovered in the late 2000, and patented in late 2001 [8]. Independently, Maymounkov =-=[9]-=- later discovered the idea of pre-coding to obtain linear time codes. His results are similar to parts of Section 6. Raptor codes have been highly optimized and are being used in commercial systems Di...

Citation Context

...he BP decoder can decode Cn on a BEC with erasure probability δ = (ε/4)/(1 + ε) = (1 − R)/2 with O(n log(1/ε)) arithmetic operations. Examples of such codes are Tornado codes [3], right-regular codes =-=[13]-=-, and certain types of repeat-accumulate codes. The reader can consult [14] for other types of such capacity-achieving examples. We remark, however, that it is not necessary for Cn to be capacity-achi...

Citation Context

...es are also possible (for example an LT-code with the appropriate number of output symbols, or an irregular Repeat–Accumulate code [12], or an irregular Repeat–Accumulate code with bounded complexity =-=[13]-=-). VI. RAPTOR CODES WITH GOOD ASYMPTOTIC PERFORMANCE In the preceding section, we encountered two types of Raptor codes. For one of them, the LT-codes, the overhead was close to , and the space was cl...

Citation Context

...e with the original elements. Index Terms—Binary erasure channel (BEC), graphical codes, LT-codes, networking. I. INTRODUCTION THE binary erasure channel (BEC) of communication wasintroduced by Elias =-=[1]-=- in 1955, but it was regarded as a rather theoretical channel model until the large-scale deployment of the Internet about 40 years later. On the Internet, data is transmitted in the form of packets. ...

Citation Context

...zero to degree one, and has infinitely many terms. The distribution of the size of the output ripple at each point in time is more difficult to calculate and we refer the reader to the upcoming paper =-=[10]-=- for details. The reader is referred to Luby’s paper for a description of LT-Codes with a distribution Ω(x) with Ω ′ (1) = O(log(k)) and for which the BP decoder is a reliable decoder of overhead k(1 ...

Citation Context

...mbols from any set of output symbols whose size is close to optimal with high probability. We call such Fountain Codes universal. The first class of such universal Fountain Codes was invented by Luby =-=[6, 7, 1]-=-. The codes in this class are called LT-Codes. The distribution used for generating the output symbols lies at the heart of LT-Codes. Every time an output symbol is generated in an LT-Code, a weight d...

Citation Context

... received output symbols. In Section 8 we develop a new set of ideas and design efficient systematic versions of Raptor Codes. Raptor Codes were discovered in the late 2000, and patented in late 2001 =-=[8]-=-. Independently, Maymounkov [9] later discovered the idea of pre-coding to obtain linear time codes. His results are similar to parts of Section 6. Raptor codes have been highly optimized and are bein...

Citation Context

...h BP decoding. This analysis relies on the exact calculation of the error probability of the LT-decoder, derived in [10], combined with the calculation of the error probability for certain LDPC codes =-=[15]-=-. 19s7.1 Design of the Output Degree Distribution Following [1], we call an input symbol released at time T if at least one neighbor of that input symbol becomes of reduced degree one after T input sy...

Citation Context

...to design the output degree distribution in such a way as to ensure that a large fraction of the k input symbols are recovered. To solve this design problem, we use an idea communicated to us by Luby =-=[17]-=-: We try to keep the expected ripple size larger than or equal to c � (1 − x)k, for some positive constant c. The rationale behind this choice is that if deletion and insertion of elements into the in...