An improved LT encoding scheme with extended chain lengths

1. An improved LT encoding scheme with extended chain lengths Kuo-Kuang Yen, Chih-Lung Chen and Hsie-Chia Chang ISITA2012, Honolulu, Hawaii, USA, October 28-31, 2012

2. Outline • Introduction • Background • Analysis of early decoding termination • Decoding chain • Averages chain length of decoding termination of different ranges • Proposed encoding scheme • Simulation results

3. Introduction • Fountain code • Randomly generates encoding symbols without a fixed code rate • Can be designed without considering the channel condition • LT code : k input symbols encoding symbol Original data 1 2 3 4 k … XOR … a b c Degree:3 Neighbors: 1 , 3 , 4

4. [13] R. Karp, M. Luby, and A. Shokrollahi, “Finite length analysis of LT codes,” in International Symposium on Information Theory, ISIT, 2004, p. 39. [14] G. Maatouk and A. Shokrollahi, “Analysis of the second moment of the LT decoder,” in International Symposium on Information Theory, ISIT, 2009, pp. 2326–2330. Introduction • LT code • Ripple : • Received degree-1 encoding symbols • Those whose degree descends to one • The BP (Belief-Propagation) decoding terminates when these ripples are depleted. • The average number of ripples is relatively small • The decoding termination occurs more frequently [13] [14]. • For the LT codes with the robust Solitondistribution • The number of ripples is initially low. • The BP decoding terminates early • Leading to most of undecodedinput symbols. • The decoding termination within the range 0 ≤ n ≤ k/2 generally contributes to more than 90% of undecoded input symbols. n = the number of decoded input symbols

5. In this paper • Decoding chains : • Input symbols can be divided into decoding chains by the connectivity of received degree-2 encoding symbols. • The average chain length is shorter when the decoding termination occurs within the range 0 ≤ n ≤ k/2 than k/2+1 ≤ n ≤ k. • Goal : • Increasing the average chain length • Reducing the number of the decoding termination within 0 ≤ n ≤ k/2 for lower symbol loss probability Symbol loss probability : The probability that an input symbol is undecoded after the BP decoding

6. Background : LT codes and its encoding

7. Background : BP decoding

8. Background : degree distribution

9. Background : drawback of LT codes with RSD • The decoding termination within 0 ≤ n ≤ n of all undecoded input symbols : The probability of the decoding termination with n input symbols decoded

10. Background : drawback of LT codes with RSD • Percentage of undecoded input symbols resulting from the decoding termination : • The BP decoding terminates within 0 ≤ n ≤ k/2 generally contributes to more than 90% of undecoded input symbols when encoding with the robust Soliton distribution. Robust Soliton distribution ε=0.1 c=0.02 δ=0.01

11. Analysis of early decoding termination :Decoding chain • Decoding chains : • Input symbols can be divided into decoding chains by the connectivity of received degree-2 encoding symbols. • Definition : A length-L decoding chain is constituted of L different input symbols it1 , it2 , ・ ・ ・ such that φ(i1) = {c1} φ(i2) = {∅} φ(i3) = {c1} φ(i4) = {c4, c5} φ(i5) = {c5, c6} φ(i6) = {c6, c7} φ(i7) = {c4, c7}

12. Analysis of early decoding termination :Averages chain length of decoding termination • : The number of input symbols connected by received degree-2 encoding symbols . • : The probability that an input symbol belongs to a length-L decoding chain • : The average chain length The average of k,, and over two different ranges: The probability of the decoding termination with n input symbols decoded

13. Analysis of early decoding termination :Averages chain length of decoding termination • The average chain length in 0 ≤ n ≤ k/2 is much shorter than in k/2 +1 ≤ n ≤ k • We want to increase the average chain length in 0 ≤ n ≤ k/2

14. k Proposed encoding scheme k-τ τ • We separate input symbols into two groups • Neighbors of any degree-2 encoding symbol are restricted from different groups of input symbols.

15. Proposed encoding scheme • Increasing the probability of two degree- 2 encoding symbols sharing one of their neighbors • Extending the average chain length • When τ= 1 , Assuming : gdegree- 2 encoding symbols g connections to g connections to the rest k – 1 inputs k k-1 1 XOR … a b c

16. Proposed encoding scheme • The probability that one of these k− 1 input symbols is connected by the received degree-2 encoding symbols is • On average, input symbols form a single decoding chain • Each of the k input symbols has probability to be in the decoding chain The average decoding chain length The average number of redundant symbols

17. Proposed encoding scheme • For example, with k = 1000 , ε= 0.1 , Ω2= 0.5 and τ= 1 , • On average • The chain length : 179.4 • The redundant symbols :125.5. • The average chain length is extended • The remaining number of received encoding symbols after removing the redundant ones is 1100−125.5 = 974.5 • Insufficient to decode all input symbols. • As τ grows, the number of redundant symbols is reduced.

18. Simulation results • Using robust Soliton distribution of parameters c = 0.02 , δ= 0.01 ,overhead ε = 0.2 • The number τ is optimized for the symbol loss probability, • τ = 40 for k = 250 • τ = 100 for k=500 • τ = 210 for k=1000 • τ = 420 for k=2000 • Encoding symbols are transmitted without erasure.

19. Simulation results

20. Simulation results

21. Simulation results