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 LT code : k input symbols encoding symbol
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. Introduction LT code The average number of ripples is relatively small
[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.
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 Soliton distribution
The number of ripples is initially low.
The BP decoding terminates early
Leading to most of undecoded input 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. We want to increase the average chain length in 0 ≤ n ≤ k/2
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. Proposed encoding schemek τ
k-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 :
g degree- 2 encoding symbols
g connections to
g connections to the rest k – 1 inputs k 1
k-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