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**Degree Distribution of XORed Fountain codes**

Theoretical derivation and Analysis Lucie Nodin, Anya Apavatjrut, Claire Goursaud, Jean-Marie Gorce

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**Planning Part I : Overview Part II : Contribution Conclusion**

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 An approach to network coding of fountain code in a wireless sensor network Wireless Sensor Network Fountain codes Network Coding

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**Wireless Sensor Network**

Overview: A set of independent sensor nodes spatially distributed in a large area Limitation: Battery life, limited computational capability, limited resource Requirement: Energy awareness, Robustness, Reliability

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**Fountain Codes Characteristics: erasure block code**

Benefits: rateless, universal, limited feedback channel is required Limitation: overhead, additional computational complexity, redundancy Choices of fountain codes: LT, Raptor (due to its low decoding complexity of the Belief Propagation algorithm)

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**Fountain Codes 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. Source packet f1 f2 f3 K = block length d = degree of the packet f1 f1 f2 f2 f1 f2 f2 f2 f2 f3 f3 f3 p1 p2 p3 p4 p5 p6 p7

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**Fountain Codes 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 Principle of Fountain Codes : LT**

Decoding Process: Belief Propagation good degree distribution : large-> encoded packets cover all initial fragments small-> ensure decoding capability f1 f2 p1 f2 f1 p2 f2 p3 f2 f3 p4 f3 f1 f3 p5 f2 p6 f2 f3 p7 f2 K = block length d = degree of the packet f1 f2 f3

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**Fountain Codes 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 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 Network Coding ?**

Overview: processing of information at intermediate nodes Benefits: redundancy optimization, packet diversity Question : How to properly apply XOR operations among the encoded packets at relay nodes R? ? Packet 1 R Packet XORed Packet 2

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**Part II: Contribution Related work In this 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 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 Insight of XORing packets encoded with fountain codes Packet Header f1 f2 f3 Ex. K=3 f1 f1 f2 f2 f1 f2 f2 f2 f2 f3 f3 f3 p1 p2 p3 p4 p5 p6 p7 1 1 1

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XORing Fountain Codes Insight of XORing packets encoded with fountain codes example no overlap with overlap 1 1 1 1 1 1 1 1 1 1 1 1 dR = degree of the resulting packet after a XOR operation d1 = degree of the first packet d2 = degree of the second packet o = number of degree overlap between the two packets

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**XORing Fountain Codes Overlap probability**

Assuming that d1≤d2 , the probability that o fragments overlap when XORing two packets with degree d1 and d2 can be expressed as f1 f2 f3 f4 f5 f6 f7 f k-1 fk K = block length d1 = degree of the first packet d2 = degree of the second packet O = number of degree overlap between the two packets

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XORing Fountain Codes 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 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 pn is the degree distribution of the packet p1 once n XORs is done. The degree distribution are initialized as:

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

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

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**Preserving the Degree Distribution: Theoretical Approach**

Question How to select d1 and d2 in order to obtain the target degree dR Solution Find joint probability of picking (d1,d2) complex with 2xK unknown variables Fixing degree d1 and find probability of picking d K unknown variables Pchoice = probability of picking d2

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**Preserving the Degree Distribution: Theoretical Approach**

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**

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**

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

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**Application to LT and Raptor Codes**

By solving the system of equations for Raptor code : Pchoice can be determined as: 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**

Validation of the obtained results with simulations Examples for LT codes : d1=1 Resulting degree distribution from one XOR between LT encoded packets when d1=1 and d2 is chosen according to Pchoice distribution

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**Application to LT and Raptor Codes**

Validation of the obtained results with simulations Examples for LT codes : d1=2 Resulting degree distribution from one XOR between LT encoded packets when d1=2 and d2 is chosen according to Pchoice distribution

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**Application to LT and Raptor Codes**

Validation of the obtained results with simulations Examples for LT codes : d1=98 Resulting degree distribution from one XOR between LT encoded packets when d1=98 and d2 is chosen according to Pchoice distribution

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**Application to LT and Raptor Codes**

Validation of the obtained results with simulations Examples for LT codes : d1=99 Resulting degree distribution from one XOR between LT encoded packets when d1=99 and d2 is chosen according to Pchoice distribution

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Conclusion 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 [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 –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.,

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