1. Degree Distribution of XORed Fountain codes
Theoretical derivation and Analysis
Lucie Nodin, Anya Apavatjrut, Claire Goursaud, Jean-Marie Gorce

2. 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

3. Part I: Overview
An approach to network coding of fountain code in a wireless sensor network
Wireless Sensor Network
Fountain codes
Network Coding

4. 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

5. 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)

6. 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

7. 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.

8. 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

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

10. 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

11. 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

12. 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.

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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:

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

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

21. 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

22. 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

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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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.

31. 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.,

32. Thank you