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Degree Distribution of XORed Fountain codes 1 Lucie Nodin, Anya Apavatjrut, Claire Goursaud, Jean-Marie Gorce

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Planning 2 Part I : Overview Wireless sensor network Fountain codes Network coding Part II : Contribution Theoretical analysis of the degree distribution of the XORed Fountain code Theoretical approach to preserve the degree distribution Application to LT and Raptor Codes Conclusion

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Part I: Overview 3 An approach to network coding of fountain code in a wireless sensor network

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Wireless Sensor Network 4

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Fountain Codes 5

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6 Principle of Fountain Codes : LT Encoding Process Randomly choose the degree d of the packet from the Robust Soliton Distribution Uniformly select d distinct fragments among K and apply a bitwise sum (XOR) between these d fragments. f2f2 f2f2 f1f1 f1f1 f2f2 f2f2 f2f2 f2f2 f1f1 f1f1 f1f1 f1f1 f2f2 f2f2 f3f3 f3f3 f2f2 f2f2 f1f1 f1f1 f2f2 f2f2 f3f3 f3f3 f3f3 f3f3 f2f2 f2f2 f3f3 f3f3 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 K = block length d = degree of the packet Source packet

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Fountain Codes 7 Principle of Fountain Codes : LT Decoding Process: Belief Propagation Find the encoded packet that have degree one. Degree one packet is considered as a decoded fragment of information. If none exists the decoding process halts at this step. Remove the combination of this decoded fragments from other un- decoded packets. Repeat these steps iteratively until all the packets are decoded successfully or until the decoding process halts due to the lack of degree one packet.

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Fountain Codes 8 Principle of Fountain Codes : LT Decoding Process: Belief Propagation good degree distribution : large-> encoded packets cover all initial fragments small-> ensure decoding capability f2f2 f2f2 f1f1 f1f1 f2f2 f2f2 f3f3 f3f3 f1f1 f1f1 f2f2 f2f2 p1p1 f1f1 f1f1 p2p2 f2f2 f2f2 p3p3 f2f2 f2f2 f3f3 f3f3 p4p4 f1f1 f1f1 f3f3 f3f3 p5p5 f2f2 f2f2 p6p6 f2f2 f2f2 f3f3 f3f3 p7p7 f2f2 f2f2 f3f3 f3f3 K = block length d = degree of the packet

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Fountain Codes 9 Degree Distribution Robust Soliton Distribution is the optimal distribution for the BP decoding [Luby2002] Ideal Soliton Distribution Robust Soliton Distribution where, and

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Fountain Code 10 Degree Distribution Degree distribution of Raptor code – precode+weakened LT code [Shokrollahi2006] where and is the overhead which allows to recover the initial data

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Network Coding 11 R R Packet 1 Packet 2 Packet XORed

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Part II: Contribution 12 Related work Decode and Reencode Successive encoding by relay nodes [Gummadi et al.2008] XORing algorithms are implemented at the relay nodes in order to preserve the target degree distribution [Apavatjrut et al.2010, Champel2009] In this work… Whereas the previous works focus on algorithm implementation, this work focuses on theoretical analysis.

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Part II: Contribution 13 Theoretical analysis of the degree distribution of the XORed Fountain code Theoretical approach to preserve the degree distribution Application to LT and Raptor Codes

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XORing Fountain Codes 14 Insight of XORing packets encoded with fountain codes Packet Header f2f2 f2f2 f1f1 f1f1 f2f2 f2f2 f2f2 f2f2 f1f1 f1f1 f1f1 f1f1 f2f2 f2f2 f3f3 f3f3 f2f2 f2f2 f1f1 f1f1 f2f2 f2f2 f3f3 f3f3 f3f3 f3f3 f2f2 f2f2 f3f3 f3f3 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 110 100 010 Ex. K=3

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XORing Fountain Codes 15 Insight of XORing packets encoded with fountain codes example 0101 1000 1101 0101 1100 1001 d R = degree of the resulting packet after a XOR operation d 1 = degree of the first packet d 2 = degree of the second packet o = number of degree overlap between the two packets

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XORing Fountain Codes 16 Overlap probability Assuming that d 1 ≤d 2, the probability that o fragments overlap when XORing two packets with degree d 1 and d 2 can be expressed as K = block length d1 = degree of the first packet d2 = degree of the second packet O = number of degree overlap between the two packets f1f1 f1f1 f2f2 f2f2 f3f3 f3f3 f4f4 f4f4 f5f5 f5f5 f6f6 f6f6 f7f7 f7f7 f k-1 f k-1 fkfk fkfk

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XORing Fountain Codes 17 Degree probability for a packet resulting from one XOR Probability of getting resulting packet with degree by applying the total law of probabilities

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XORing Fountain Codes 18 Degree probability for a packet resulting from several XORs By XORing N+1 packets together, N XORs successive are done on two packets at each steps: Where p n is the degree distribution of the packet p 1 once n XORs is done. The degree distribution are initialized as:

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XORing Fountain Codes 19 Degree probability for a packet resulting from several XORs P(d) Degree (d)

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Degree probability for a packet resulting from several XORs When, Soliton Distribution Gaussian Distribution XORing Fountain Codes 20 Randomly applying XOR operations -> decoding inefficiency

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Preserving the Degree Distribution: Theoretical Approach 21 Question How to select d 1 and d 2 in order to obtain the target degree d R Solution Find joint probability of picking (d1,d2) complex with 2xK unknown variables Fixing degree d 1 and find probability of picking d2 K unknown variables Pchoice = probability of picking d2

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Preserving the Degree Distribution: Theoretical Approach 22 Matrix representation Such that represents the targeted resulting degree distribution represents a matrix of overlaps’ probabilities with coefficient represents the degree probability distribution of how to choose the second packets in order to obtain a specific

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Preserving the Degree Distribution: Theoretical Approach 23 How to determined ? Too difficult to be determined by matrix inversion Estimation with the least square method and

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Application to LT and Raptor Codes 24 By solving the system of equations for LT code : P choice can be determined as: Degree Distribution Pchoice of the degrees to choose to recover Robust Soliton distribution for packets resulting from one XOR Irregularity of P choice

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By solving the system of equations for Raptor code : P choice can be determined as: Application to LT and Raptor Codes 25 Degree Distribution Pchoice of the degrees to choose to recover weaken Robust Soliton distribution for packets resulting from one XOR

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Application to LT and Raptor Codes 26 Validation of the obtained results with simulations Examples for LT codes : d 1 =1 Resulting degree distribution from one XOR between LT encoded packets when d1=1 and d2 is chosen according to P choice distribution

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Application to LT and Raptor Codes 27 Validation of the obtained results with simulations Examples for LT codes : d 1 =2 Resulting degree distribution from one XOR between LT encoded packets when d1=2 and d2 is chosen according to P choice distribution

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Application to LT and Raptor Codes 28 Validation of the obtained results with simulations Examples for LT codes : d 1 =98 Resulting degree distribution from one XOR between LT encoded packets when d1=98 and d2 is chosen according to P choice distribution

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Application to LT and Raptor Codes 29 Validation of the obtained results with simulations Examples for LT codes : d 1 =99 Resulting degree distribution from one XOR between LT encoded packets when d1=99 and d2 is chosen according to P choice distribution

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Conclusion 30 Theoretical Analysis of the degree distribution of XORed fountain codes as well as a technique to preserve the degree has been proposed. The theoretical derivation in this work can be used as a way to recover a given degree distribution after XOR operations. This can later be applied to all the network coding-like application with fountain codes. Our theoretical and simulation results highlight that, under a certain conditions of packet selection, the target degree is reachable without the need to decode the packet entirely at the relay.

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References 31 [Luby2002] M. Luby, “LT codes,” The 43rd Annual IEEE Symposium on Foundations of Computer Science, Proceedings., pp. 271 – 280, 2002. [Shokrollahi2006] A. Shokrollahi, “Raptor codes,” IEEE Transactions on Information Theory, vol. 52, no. 6, pp. 2551 –2567, june 2006. [Gummadi et al.2008] R. Gummadi and R. Sreenivas, “Relaying a fountain code across multiple nodes,” in IEEE Information Theory Workshop, 2008, pp. 149–153. [Apavatjrut et al.2010] A. Apavatjrut, “Towards increasing diversity for the relaying of LT fountain codes in wireless sensor network”, to be published in IEEE Communications Letters. [Champel2009] M.-L. Champel, “LT network codes,” INRIA, Tech. Rep., 2009.

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