# Performance analysis of LT codes with different degree distribution

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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

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

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

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

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

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

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

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

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

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!

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

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

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.

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.

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

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

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

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

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

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

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

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.

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

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