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Company LOGO F OUNTAIN C ODES, LT C ODES AND R APTOR C ODES Susmita Adhikari Eduard Mustafin Gökhan Gül

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O UTLINE 2. Fountain Codes 4. LT Codes 5. Raptor Codes 7. Conclusion 1. Motivation 3. Degree Distribution 2Kiel, February 2008

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Binary Erasure Channel M OTIVATION p 0 1 0 1 p 1-p e Capacity = (1 - p) 3Kiel, February 2008

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Automatic Repeat Request (ARQ) Wasteful usage of bandwidth, network overloads and intolerable delays. Forward Error Correcting (FEC) codes Reed-Solomon codes, LDPC codes, Tornado codes. Rate should be determined in compliance with the erasure probability p. High computational cost of overall encoding and decoding. M OTIVATION 4Kiel, February 2008

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FOUNTAIN CODES Rateless: Number of code symbols, which can be generated from input info symbols is potentially unlimited. Capacity achieving: Decoder can recover info symbols from any set of code symbols, which is only slightly longer than input message length. Universal: Fountain Codes are near optimal for any BEC. 5Kiel, February 2008

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F OUNTAIN C ODES Encoding: Example: 6Kiel, February 2008 For info symbols s 1, s 2 …s K, code symbols t 1, t 2 … and random generator matrix G,

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FOUNTAIN CODES Decoding: For info symbols s 1, s 2 …s K, code symbols t 1, t 2 … and random generator matrix G, Example: Easier to calculate using Tanner Graph 7Kiel, February 2008

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F OUNTAIN C ODES... ?? ? 1 [ [ ] 1 1 1 0 ]... 0 01 8Kiel, February 2008

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Encoding complexity of and decoding complexity of, which makes them impractical. F OUNTAIN C ODES... 9Kiel, February 2008

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D EGREE D ISTRIBUTION Degree is the number of edges connecting an encoded symbol to the input symbols. Degree distribution is the probability density function of all degrees used for encoding. Average degree of the encoded symbols should be at least for a realiable decoding. Degree distribution affects the encoding and decoding costs. Kiel, February 200810

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LT C ODES First practical implementation of Fountain Codes, proposed by Michael Luby. LT codes are rateless and universal. Computational complexity of both the encoding and the decoding process increase logarithmically with the increase of the data length. Therefore, compared to Fountain Codes, LT codes provide a considerably reduced computational cost while achieving the capacity. 11Kiel, February 2008

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LT C ODES - “Encoding” Encoding Algorithm Divide the message M into equi-length parts of k bits resulting in K number of symbols. M k 001100010110 12Kiel, February 2008

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LT C ODES - “Encoding” d=3 Encoding Algorithm Randomly choose the degree d of the encoding symbol from a degree distribution. 13Kiel, February 2008

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LT C ODES - “Encoding” Encoding Algorithm Choose randomly distinct input symbols as the edges of the encoding symbol in the tanner graph. 14Kiel, February 2008 d=3 001100010110 1 1 2 2 3 3 4 4 5 5 6 6 1 1 4 4 6 6

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LT C ODES - “Encoding” Encoding Algorithm Determine the encoding symbol as bitwise modulo 2 sum of the edge symbols. 15Kiel, February 2008 d=3 001100010110 1 1 2 2 3 3 4 4 5 5 6 6 11

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LT C ODES - “Encoding” d d v v 2 2 (1,3) 2 2 (1,2) 2 2 (5,6) 2 2 (4,6) 1 1 (2) 1 1 (5) 3 3 (1,4,6) 1 1 (3) 1 1 1 1 (4) 253461 16Kiel, February 2008

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LT C ODES - “Decoding” Decoding Algorithm Find a check node with degree one and assign its value to the corresponding input symbol. If there is no such a node halt the decoding and report the failure of decoding. Add this value to all check nodes connected to this input symbol. Remove all edges from the graph, which are connected to the related input symbol. Repeat the first three steps until all input symbols are recovered. 17Kiel, February 2008

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d d v v 2 2 (1,3) 2 2 (1,2) 2 2 (4,6) 1 1 (5) 3 3 (1,4,6) 1 1 (3) LT C ODES - “Decoding”.... Decoding Failure! 18Kiel, February 2008

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d d v v 2 2 (1,3) 2 2 (1,2) 2 2 (4,6) 1 1 (5) 3 3 (1,4,6) 1 1 (3) 1 1 (4)... LT C ODES - “Decoding” Decoding Successful! 19Kiel, February 2008

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R APTOR C ODES An extension of LT codes, introduced by Shokrollahi. Core idea - “To relax the condition of recovering all input symbols and to require only a constant fraction of input symbols be recoverable.” Idea achieved by concatenation of an LT code and a precode. LT code recovers a large proportion of input symbols. Precode recovers the fraction unrecovered by LT code. Encoding and decoding complexity increases linearly with K. 20Kiel, February 2008

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R APTOR C ODES “Encoding” 21Kiel, February 2008

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22 Received Symbols Erased Symbol Recovered Message Symbols Decoding Successful! Unrecovered Symbols R APTOR C ODES “Decoding”

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Raptor Codes can efficiently be used over noisy channels with the same encoding scheme that we have previously described and BP algorithm using soft inputs. AWGN Channel with E s /N 0 =-2.83 dB and K=9500 info- bits, N av =20737 code-bits, Cap=0.5bit/symbol, R av =0.458bit/symbol. Tends to approach the capacity with the increase of message length on both AWGN and the fading channel with Rayleight distribution. 23Kiel, February 2008 RAPTOR CODES on Noisy Channels

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Capacity achieving Raptor Codes haven’t been proven yet for other symetric channels. However, it is proven that Raptor Codes are not universal for all rates for symetric channels other than BEC. Generalized Raptor Codes outperform ordinary Raptor Codes using rate-compatible distribution arrangement on BSM and AWGN channels. RAPTOR CODES on Noisy Channels 24Kiel, February 2008

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C ONCLUSION Fountain Codes Advantage: Rateless, universal and capacity achieving. Disadvantage: Higher encoding and decoding complexity. LT Codes Advantage: Lower complexity than Fountain Codes. Disadvantage: Complexity increases logarithmically with the message length. Raptor Codes Advantage: The lowest complexity achievable Can be applied to arbitrary channels efficiently. 25Kiel, February 2008

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