1. Performance analysis of LT codes with different degree distribution
Zhu Zhiliang, Liu Sha, Zhang Jiawei,
Zhao Yuli, Yu Hai
Software College, Northeastern University, Shenyang, Liaoning, China.
College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, China

2. Outline Introduction Degree distribution of LT codes
Analysis of LT codes
Average degree
Degree release probability
Average overhead factor

3. Introduction
The encoding/decoding complexity and error performance are governed by the degree distribution of LT code.
Designing a good degree distribution of encoded symbols [7]
To improve the encoding/decoding complexity and error performance
In this paper , we analysis
Ideal soliton distribution
Robust soliton distribution
Suboptimal degree distribution
Scale-free Luby distribution
Average degree
Degree release probability
Average overhead factor

4. LT process c1 a1 c2 a2 c3 a3 c4 a4 c5 a5 c6 STATE: covered = { }
processed = { }
ripple = { }
released = { }
ACTION:
Init: Release c2, c4, c6

5. LT process c1 a1 c2 a2 c3 a3 c4 a4 c5 a5 c6 STATE:
released = {c2,c4,c6}
covered = {a1,a3,a5}
processed = { }
ripple = {a1,a3,a5}
ACTION:
Process a1

6. LT process c1 a1 c2 a2 c3 a3 c4 a4 c5 a5 c6 STATE:
released = {c2,c4,c6,c1}
covered = {a1,a3,a5}
processed = {a1}
ripple = {a3,a5}
ACTION:
Process a3

7. LT process c1 a1 c2 a2 c3 a3 c4 a4 c5 a5 c6 STATE:
released = {c2,c4,c6,c1}
covered = {a1,a3,a5}
processed = {a1,a3}
ripple = {a5}
ACTION:
Process a5

8. LT process c1 a1 c2 a2 c3 a3 c4 a4 c5 a5 c6 STATE:
released = {c2,c4,c6,c1,c5}
covered = {a1,a3,a5,a4}
processed = {a1,a3,a5}
ripple = {a4}
ACTION:
Process a4

9. LT process c1 a1 c2 a2 c3 a3 c4 a4 c5 a5 c6 STATE:
released = {c2,c4,c6,c1,c5,c3}
covered = {a1,a3,a5,a4,a2}
processed = {a1,a3,a5,a4}
ripple = {a2}
ACTION:
Process a2

10. LT process c1 a1 c2 a2 c3 a3 c4 a4 c5 a5 c6 STATE:
released = {c2,c4,c6,c1,c5,c3}
covered = {a1,a3,a5,a4,a2}
processed = {a1,a3,a5,a4,a2}
ripple = { }
ACTION:
Success!

11. Ideal soliton distribution [6]
Works poor
Due to the randomness in the encoding process,
Ripple would disappear at some point, and the whole decoding process failed.
[6] M. Luby, “LT codes”, Proc. Annu. Symp. Found. Comput. Sci. (Vancouver, Canada), 2002, pp

12. Robust soliton distribution [6]
Maximum failure probability of the decoder
when encoded symbols are received
Degree distribution of Ideal Soliton Distribution
[6] M. Luby, “LT codes”, Proc. Annu. Symp. Found. Comput. Sci. (Vancouver, Canada), 2002, pp

13. Suboptimal degree distribution
Optimal degree distribution is proposed[12]
When k is large, the coefficient matrix of optimal degree distribution is too sick.
No solution.
Suboptimal degree distribution:
[12] Zhu H P, Zhang G X, Xie Z D, "Suboptimal degree distribution of LT codes". Journal of Applied Sciences-Electronics and Information Engineering. Jan 2009, Vol. 27, No. 1, pp
R is initial ripple size
E is the expected number of encoded symbols required to recovery the input symbols.

14. Scale-free Luby distribution [13]
Based on modified power-law distribution
Presenting that scale-free property have a higher chance to be decoded correctly.
A large number of nodes with low degree
A little number of nodes with high degree
P1 : the fraction of encoded symbols with degree-1
r : the characteristic exponent
A : the normalizing coefficient to ensure
[13] Yuli Zhao, Francis C. M. Lau, "Scale-free Luby transform codes", International Journal of Bifurcation and Chaos, Vol. 22, No. 4, 2012.

15. Analysis of LT codes
The encoding/decoding efficiency is evaluated by the average degree of encoded symbols.
Less average degree
Fewer times of XOR operations
Encoded symbol should be released until the decoding process finished
Degree release probability is very important
Less number of encoded symbols required to recovery the input symbols means less cost of transmitting the original data information.
The overhead should be considered
: degree distribution : average degree

16. Average degree Ideal soliton distribution
Can be calculated based on the summation formula of harmonic progression
r : Euler's constant which is similar to 0.58
Average degree of ideal soliton degree distribution is

17. Average degree Robust soliton distribution
The complexity of its average degree is

18. Average degree Suboptimal degree distribution
The complexity of its average degree is

19. Average degree Scale-free Luby distribution
Based on the properties of Scale-free
The average degree of Scale-free Luby Distribution will be small
(r-1) is the sum of a p-progression
It is obvious that the average degree of SF-LT codes is smaller
Encoding/decoding complexity of SF-LT code is much lower than the others

20. Degree release probability
[6] M. Luby, “LT codes”, Proc. Annu. Symp. Found. Comput. Sci. (Vancouver, Canada), 2002, pp
[6]
In general, r(L) should be larger than 1
At least 1 encoded symbol is released when an input symbol is processed.
𝐿 𝑘−(𝐿+1) 𝑖−2 𝑘 𝑖

21. Degree release probability Ideal soliton distribution [6]
Degree release probability Robust soliton distribution

22. Degree release probability Suboptimal degree distribution
Using limit theory, it can be expressed as , where
Suppose E encoded symbols is sufficient to recovery the k original input symbols.
At each decoding step, larger than 1 encoded symbol is released.

23. Degree release probability Scale-free Luby distribution
Initial ripple size must be bigger than Robust Soliton Distribution’s
k·P1 is bigger than 1
The complexity is

24. Degree release probability
Suboptimal degree distribution's degree release probability is bigger than the others

25. Average overhead factor
A decreasing ripple size provides a better trade-off between robustness and the overhead factor [14]
The theoretical evolution of the ripple size :
Assuming that at each decoding iteration, the input symbols can be added in to the ripple set without repetition
[14] Sorensen J. H., Popovski. P., Ostergaard J., "On LT codes with decreasing ripple size", Arxiv preprint PScache/ v1.
: the number of degree-i input symbols left
L : the size of unprocessed input symbols

26. Average overhead factor

27. Conclusion
Robust LT codes, suboptimal LT code and SF-LT code are capable to recovery the input symbols efficiently.
From the overhead factor, SF-LT codes and suboptimal LT codes need much less number of encoded symbols to recovery given number of input symbols.
The average degree of SF-LT code is smaller than the others.
SF-LT code performs much better probability of successful decoding and enhanced encoding/decoding complexity