ON THE INTERMEDIATE
SYMBOL RECOVERY RATE OF
RATELESS CODES
Ali Talari, and Nazanin Rahnavard
IEEE TRANSACTIONS ON COMMUNICATIONS,VOL. 60, NO. 5, MAY 2012
1
OUTLINE
Introduction
Related work
Rateless Code Design with High ISRR
RCSS: Rateless Code Symbol Sorting
Conclusion
2
INTRODUCTION
 Design new rateless codes with close to optimal
intermediate symbol recovery rates (ISRR) employing
genetic algorithms.
 Next, propose an algorithm to further improve the
ISRR of the designed code, assuming an estimate of
the channel erasure rate is available.
3
INTRODUCTION
Encoder S
Source symbol
x={𝑥1 , 𝑥2 ,.. 𝑥𝑘 }
Ω 𝑥 =
𝑘
Ω𝑖 𝑥 𝑖
Channel
encoded
symbol 𝑐𝑖
Limiteless
𝜀
𝑧 ≈ 0 𝑎𝑡 0 ≤ 𝛾 ≤ 1
Decoder D
Iteratively
decoding
𝑖=1
z
The fraction of decoded input symbol. z ∈ [0,1]
γ
γ : intermediate range.
γ𝑠𝑢𝑐𝑐
𝜀
γ𝑠𝑢𝑐𝑐 is coding overhead.
Channel erasure rate
4
RELATED WORK
The intermediate range of rateless codes can be
divided into three regions.[4]
1
2
 𝑧 ∈ [0, ],
1 2
2 3
𝑧 ∈ [ , ],
2
3
𝑧 ∈ ( , 1)
γ ∈ [0,0.693], γ ∈ [0.693, 0.824],
γ ∈ [0.824, 1]
 designs optimal degree distributions, only optimal in one
intermediate region.
 k is infinite.
[4] S. Sanghavi, “Intermediate performance of
rateless codes,” in Proc. 2007 IEEE Inf. Theory
Workshop, pp. 478–482.
5
RELATED WORK
[7] A. Beimel, S. Dolev, and N. Singer, “RT
oblivious erasure correcting,” IEEE/ACM Trans.
Netw., vol. 15, no. 6, pp. 1321–1332, 2007.
Employ feedbacks from D to keep S aware of z.[5][7]
Transmit output symbols in the order of their
ascending degree.[6]
[5] A. Kamra,V. Misra, J. Feldman, and D. Rubenstein, “Growth codes:
maximizing sensor network data persistence,” in Proc. 2006 Conf.
Applications, Technologies, Architectures, Protocols Computer
Commun., vol. 36, no. 4, pp. 255–266.
[6] S. Kim and S. Lee, “Improved intermediate
performance of rateless codes,” in Proc. 2009 Int. Conf.
Advanced Commun. Technol., ICACT, vol. 3, pp. 1682–
6
RATELESS CODE DESIGN WITH HIGH ISRR
Design degree distributions for rateless coding
with various k’s employing multi-objective genetic
algorithms.
 A. Tune the degree distribution Ω(.) considering all three
intermediate regions of 0 ≤ γ ≤ 1.
7
A. DECISION VARIABLES AND OBJECTIVE
FUNCTIONS
Maximize 𝑧0.5,Ω(.) , 𝑧0.75,Ω(.) , and 𝑧1,Ω(.)
 conflicting objective functions, so
 employ multi-objective optimization methods to design desired
distributions.
A high ISRR have Ω(.)’s with much smaller maximum
degree.
 consider degree distributions with maximum degree of 50.
 decision variables {Ω1 , Ω2 , . . . , Ω50 }, where
50
𝑖=1 Ω𝑖
= 1.
8
A. DECISION VARIABLES AND OBJECTIVE
FUNCTIONS (FOR ASYMPTOTIC CASE)
 𝑣𝑙 : the probability that an input symbol is not
recovered after l decoding iterations, where 𝑣0 = 1.

𝛽 𝑦 =
Ω′(𝑦)
Ω′(1)γ(𝑦−1)
,
and
𝛿
𝑦
=
𝑒
.
Ω′(1)
 {𝑣𝑙 }𝑙 is convergent .
𝑉γ,Ω(.) : the fixed point that is the final error rate of
a rateless decoding with Ω(.) and γ.
 hence, 𝑧γ,Ω(.) = 1 − 𝑉γ,Ω(.)
9
A. DECISION VARIABLES AND OBJECTIVE
FUNCTIONS (FOR FINITE K)
Find z for finite k we employ Monte-Carlo method by
averaging z for a large enough number of decoding
simulation experiments for kϵ{102 , 103 , 104 }.
𝑧γ,Ω(.) , in this case, are found by numerical simulations.
10
B. OPTIMIZED RATELESS CODES FOR HIGH
ISRR
Employ NSGA-II multi-objective optimization
algorithm[15] to find the distributions that have
optimal z at three selected γ′ 𝑠.
The results are available online at
 http://cwnlab.ece.okstate.edu/research
 four databases of degree distributions optimized for
k ϵ {102 , 103 , 104 , ∞}.
[15] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan,
“A fast and elitist multiobjective genetic algorithm:
NSGA-II,” IEEE Trans. Evol. Comput., vol. 6, no. 2, pp.
182–197, Apr. 2002.
11
C. PERFORMANCE EVALUATION OF THE
DESIGNED CODES
 A weighted function
 From [4],
 𝑍0.5 = 0.3934,
 𝑍0.75 = 0.5828,
 𝑍1 = 1
𝑍𝑟 is the upper bound on z.
𝑊𝑟 is a tunable weight.
 We can find Ω(.) of interest by setting the appropriate
weights and selecting the Ω(.) that minimizes F(Ω(.)).
12
C. PERFORMANCE EVALUATION OF THE
DESIGNED CODES
Summary:
1. Slightly differ from the
distributions proposed in [4].
2. The maximum degree is 19
from database observation.
3. As k decreases, large degrees
are also eliminated.
13
C. PERFORMANCE EVALUATION OF THE
DESIGNED CODES
14
C. PERFORMANCE EVALUATION OF THE
DESIGNED CODES
15
RCSS: RATELESS CODED SYMBOL SORTING
The channel erasure rate ε may be available at S.
ε may be exploited as a side information to further
improve the ISRR of rateless codes.
16
A. RCSS: RATELESS SYMBOL SORTING
ALGORITHM
Main idea:
 choose an output symbol 𝑐𝑖 to be transmitted with
𝑃𝑑𝑒𝑐 𝑖 that is the highest probability among the
remaining output symbols.
𝑃𝑑𝑒𝑐 𝑖 =Pr(𝑐𝑖 can recover an input symbol at D).
The decoder generate no feedback and RCSS can
only employ the information available at source.
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A. RCSS: RATELESS SYMBOL SORTING
ALGORITHM
source S
 generates 𝑚 =
𝐾γ𝑠𝑢𝑐𝑐
1−
ε output symbols, with erasure rate ε.
 maintains a probability vector 𝜌 = [𝜌 1 , 𝜌 2 , ⋯ 𝜌(𝑘)],
𝜌 𝑗 = Pr(𝑥𝑗 still not recovered at D).
N(𝑐𝑖 ) is a set containing index of input symbols that
are neighboring to 𝑐𝑖 .
𝑃𝑑𝑒𝑐 (𝑐𝑖 ) = Pr(𝑐𝑖 can recover an input symbol at D.)
 Assume 𝑃𝑑𝑒𝑐 (𝑐𝑖 ) = 0, if 𝑐𝑖 has been previously
transmitted.
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A. RCSS: RATELESS SYMBOL SORTING
ALGORITHM
An output symbol 𝑐𝑖 can recover an
input symbol 𝑥𝑗 iff all 𝑥𝑤 , 𝑤 ∈
N(𝑐𝑖 ) − 𝑗 have already been recovered.
Initial()
Input
symbol
Output
symbol
𝜌 1 = 1 a1
𝜌 2 = 1 a2
𝜌 3 = 1 a3
𝜌 4 = 1 a4
𝜌 5 = 1 a5
Action:
c1 N(1)={1,3} 𝑃𝑑𝑒𝑐 (1)
=0
c2 N(2)={3} 𝑃𝑑𝑒𝑐 (2)
c3 N(3)={2,4} 𝑃𝑑𝑒𝑐 (3)
= 1- ε
c4 N(4)={1} 𝑃𝑑𝑒𝑐 (4)
c5 N(5)={4,5} 𝑃𝑑𝑒𝑐 (5)
=0
= 1- ε
=0
c6 N(6)={5}
= 1- ε
𝑃𝑑𝑒𝑐 (6)
transmit 𝑐2 and update 𝜌 3 .
19
A. RCSS: RATELESS SYMBOL SORTING
ALGORITHM
Input
symbol
𝜌 1 = 1 a1
𝜌 2 = 1 a2
𝜌 3 = ε1 a3
𝜌 4 = 1 a4
𝜌 5 = 1 a5
Output
symbol
c1 N(1)={1,3} 𝑃𝑑𝑒𝑐 (1)
c2 N(2)={}
𝑃𝑑𝑒𝑐 (2)
c3 N(3)={2,4} 𝑃𝑑𝑒𝑐 (3)
c4 N(4)={1} 𝑃𝑑𝑒𝑐 (4)
c5 N(5)={4,5} 𝑃𝑑𝑒𝑐 (5)
c6 N(6)={5}
𝑃𝑑𝑒𝑐 (6)
20
A. RCSS: RATELESS SYMBOL SORTING
ALGORITHM
21
B. RCSS LOWER AND UPPER
PERFORMANCE BOUNDS
Lemma 1: The performance of RCSS is upper
bounded by z = γ for ε → 0.
Lemma 2: The performance of RCSS is lower
bounded by the performance of [6] (where
symbols are only sorted based on their degree) for
ε → 1.
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C. COMPLEXITY AND DELAY INCURRED
BY RCSS
RCSS result in delays in transmission, since
 all output sorted before transmission.
Eliminate the delay increases the memory
requirements
 Save an off-line version of 𝜋𝑜𝑓𝑓−𝑙𝑖𝑛𝑒 , 𝑁𝑜𝑓𝑓−𝑙𝑖𝑛𝑒 (𝑐𝑖 ).
Overall complexity
 𝑂(𝑘)
𝑂(𝑘 2 )
23
D. PERFORMANCE EVALUATION OF RCSS
24
E. EMPLOYING RCSS WITH CAPACITYACHIEVING CODES
k =103
LT code parameters
c = 0.05
𝛿 = 0.01
25
F. RCSS FOR VARYING 𝜀
26
CONCLUSION
Design degree distributions that have optimal
performance at all three selected points employing
multi-objective genetic algorithms.
Proposed RCSS that exploits erasure rate ε and
rearranges the transmission order of output
symbols to further improve the ISRR of rateless
codes.
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