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

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

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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

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

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

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

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

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

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

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

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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

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

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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.

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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 P 1 : 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.

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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

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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

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

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

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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

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

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

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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.

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

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

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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 : the number of degree-i input symbols left L : the size of unprocessed input symbols [14] Sorensen J. H., Popovski. P., Ostergaard J., "On LT codes with decreasing ripple size", Arxiv preprint PScache/ v1.

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Average overhead factor

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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

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