Nour KADI, Khaldoun Al
AGHA
21st Annual IEEE International Symposium on
Personal, Indoor and Mobile Radio Communications
1

Introduction

Previous work

The drawback of LT codes

Switched Code
◦ Analysis of Switched Distribution

Simulation

Conclusions
2

A wireless ad hoc network is a decentralized
type of wireless network.
◦ wireless Sensor networks (WSN)

Each node participates in routing by
forwarding data for other nodes, and so the
determination of which nodes forward data is
made dynamically based on the network
connectivity.
3



Overcoming problems associated with
resource scarcity and unreliable channels has
been a challenge for data broadcasting
protocols in ad-hoc networks.
The network coding combined with network
broadcasting can reduce the bandwidth and
power consumption and increase the
throughput.
Rateless codes achieve a reliable broadcast
transmission.
4
b1
b2
b1
b1
b1
b2
b2
b2
b2
b1
5
(for channel coding)
6
(For network coding)
7
[4]D. S. Lun, M. M´edard, R. Koetter, and
M. Effros, “On coding for reliable
communication over packet networks,”
CoRR, vol. abs/cs/0510070, 2005.
Linear Network
Coding [4]
LT Code
Encoding &
Decoding
complexity
𝑂(𝑛3 )
Low complexity
Intermediate
nodes
Decode & re-encode
received information
Just forward
the encoded
packets
Result
Save the network
Consume lot of
resource and increase
network
the throughput
resource
8

To achieve reliable broadcasting and efficient
bandwidth utilization：
◦ Using LT-code guarantees the simple complexity of
the proposed coding scheme.
◦ Using network coding increases the throughput and
saves the network resources.

We introduce a coding scheme that efficiently
broadcasts the source packets.
9

Combines LT-code with network coding, by
enabling intermediate nodes to perform
coding.
10


The RSD is designed to decode plenty of
symbols only when it receives sufficiently
large number of encoded packets.
Hence, using LT- code in its original form
increases significantly the end-to-end packet
delay.
11

We present a degree distribution that increases the
symbol recovery probability at any time during the
decoding process, while keeping the overhead as
small as possible.
12

The sender switches from the first
distribution to the other according to the
number of encoded packets which have been
sent.
[6] S. Agarwal, A. Hagedorn, and A. Trachtenberg, “Adaptive rateless
coding under partial information,” in Information Theory and Applications
Workshop, 2008, 2008, pp. 5–11.
13
14
Definition 1. (codeword and degree):
1
2
3
4
5
Codeword with
coding candidate {1,
3, 4}, and degree = 3
Definition 2. Binary Exponential Distribution(𝐵𝐸𝐷𝑘 ):
15

Definition 3. (The Decoding Probability): Let
𝐷(𝑟|𝑑) be the probability to decode a codeword of
degree d when r − 1 of the source symbols has
been recovered.
𝐷𝑟𝑑
𝑘−𝑟+1 𝑟−1
1
𝑑−1
=
𝑘
𝑑
16

Definition 4. (The Symbol Recovery Probability): Let 𝑅𝑟 be
the probability to recover the 𝑟 𝑡ℎ source symbol. So 𝑅𝑟 =
𝑘
𝑑=1 𝑝(𝑑)𝐷(𝑟|𝑑) , where p(d) is the probability to have a
codeword of degree d.

Definition 5. Let 𝐸𝑦 be The expected number of
recovered symbols after sending y codewords. And
let the overhead
𝜃 = Y − k where 𝐸𝑌 = k.
In addition to minimize the overhead, our interest is
to maximize 𝐸𝑦 , ∀𝑦.
This could be obtained by maximizing 𝑅𝑟 , ∀r.
17

Lemma 2. To recover the first symbol at the
destination, it is more useful that the source
chooses the degree of the codeword according to
Binary Exponential Distribution 𝐵𝐸𝐷𝑘 than using
Robust Soliton Distribution (RSD).
Because 𝐸1 = 𝑝 𝑑 = 1 ∗ 𝐷 1 1 = 𝑝 𝑑 = 1 ∗ 1
(i) 𝐿𝑒𝑡 𝑝 𝑑 = 1 = 1/2
1
𝑘
(𝐵𝐸𝐷𝑘 )
𝑅
𝑘
𝑘
𝛿
(ii) 𝑝′ 𝑑 = 1 = ( + )/𝛽, where 𝑅 = 𝑐𝑙𝑛( ) 𝑘, and
𝑘
−1 𝑅
𝑅
𝑅
𝑅
𝑘
𝛽 = 𝑖=1 + 𝑅𝑙𝑛
≤ 1 + ( H( )+ 𝑙𝑛
𝑖
𝛿
𝑘
𝑅
Harmonic number of n.
𝑅
𝛿
), H(n) is the
(RSD)
We can prove that 𝑝 𝑑 = 1 > 𝑝′ 𝑑 = 1 .
18

Lemma 3. To recover the last symbol, it is more
useful to use Soliton Distribution.
𝑑
𝑘
In this case r=k, from proposition 1, 𝐷(𝑘|𝑑) = .
𝑅𝑘 =
1
𝑘
𝑅′ 𝑘 =
𝑑
𝑘
𝑑=1 2𝑑
1 1
𝑘 𝑘
+
1
𝑘
=
2
𝑘
1−
1
𝑘
𝑑=2 𝑑−1
1 𝑘+1
2
=
1 1
𝑘 𝑘
−
𝑘+1
2𝑘
(BED)
+ 𝐻(𝑘 − 1) (Ideal Soliton distribution)
And we can prove that 𝑅′ 𝑘 > 𝑅𝑘 .
19

Lemma 4. RSD outperforms 𝐵𝐸𝐷𝑘 only after recovering
70% of the source packets at the destination.
𝑅′ 𝑟
𝑘−𝑟+1 𝑅+1
=
+
𝛽
𝐾2
𝑘
𝑅−1
𝑑=2
𝑅
𝑅𝑙𝑛
1
𝑅
𝑅2
𝛿
+
∙ Γ𝑑 +
+
𝑑−1 𝑘
𝑘 𝑘−𝑅
𝑘
𝑑 ∙ Γ𝑘
𝑅
Using a dichotomic technique, we find that when r − 1
is inferior to 0.70*k then 𝑅′′ 𝑟 is superior to 𝑅′ 𝑟 and
this is reversed when r − 1 becomes superior to
0.70*k.
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
Definition 6. Let’s consider an incremental decoder S.
If the decoder S receives r codewords, then it decodes
them in an ascending order according to their degree d.
First step, it decodes the packets with d = 1. Then,
using the packets which are recovered from the first
step, it decodes the packets with d = 2 . For a step i, it
decodes the packets with d = i by using the packets
which are recovered from steps 1,2,. . . ,i-1. If a packet
of degree d couldn’t be decoded at step i = d, then it
will be ignored in the next steps.
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
Proposition 2. The expected number of recovered
symbols after sending k codewords according to
𝐵𝐸𝐷𝑘 is at least 0.70*k.
Prove this proposition for the incremental decoder.
Assume that S receives Y codewords, then
𝑌
2
Step 1：𝐸𝑌 [1] = 𝑌 × 𝜑 𝑑 = 1 × 1 =
Step 2： 𝐸𝑌 𝑖 = 𝑌 × 𝜑 𝑑 = 𝑖 × 𝐷((
𝑌
= 𝑖
2
× 𝐷((
𝑖−1
𝑗=1 𝐸𝑌
𝑖−1
𝑗=1 𝐸𝑌
𝑗 +1)|𝑖)
𝑗 +1)|𝑖)
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Then, the total expected number of
recovered symbols is
𝑘
𝐸𝑌 =
𝐸𝑌 𝑖 =
𝑖=1
𝑌
2
𝑘
+
𝑌 × 𝜑 𝑑 = 𝑖 × 𝐷((
𝑖−1
𝑗=1 𝐸𝑌
𝑗 +1)|𝑖)
𝑖=2
Using a dichotomic technique, we find that
when Y=k then 𝐸𝑌 =0.70*k.
And this result is valid for a non-restricted
decoder.
23

To adapt LT code to the case where some
input symbols are already known at the
destination.
◦ Generate high degree with higher probability
k : total number of input symbols
l : the number of input symbols already known at the destination.
24



Using Opnet simulator
The simulation continues until k source packets
generated by the source node are delivered to all
other nodes.
At each time slot, an intermediate node 𝑁𝑖 can
transmit one packet, which is received by its
neighbors with a probability (1-𝛽).

Using distributed TDMA scheduling

Running the simulation 30 times

The parameters of RSD
◦ C=0.2, 𝛿 = 0.1
25
[4] P. Pakzad, C. Fragouli, and A. Shokrollahi,
“Coding schemes for line networks,” CoRR, vol.
abs/cs/0508124, 2005.

Comparison with 2 other LT-code based schemes.
◦ Original LT code
◦ Forward & Re-code[4]
(for line network)
 Adapt LT-code with RSD to network coding

Comparison with a random linear network coding
schemes.
◦ Probabilistic NC[8]
 Random linear network coding with a probabilistic approach
[8] C. Fragouli, J. Widmer, and J.-Y. L. Boudec, “Efficient
broadcasting using network coding,” IEEE/ACM Trans. Netw,
26
vol. 16, no. 2, pp. 450–463, 2008.
loss rate 𝛽 = 10%
30 static node in line network
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𝛽 = 10%
30 static node in line network
𝛽 = 30%, k=10
The deliver ratio is the number of
delivered packets at the destination
over the total number of the source
packets.
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50 nodes placed randomly on
a 100 × 100𝑚2 area.
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


Considered the broadcast traffic in ad-hoc
wireless networks, We have presented a novel
rateless code which outperforms LT code as it
allows intermediate nodes to process the
received packets.
The proposed distribution increases the
decoding probability of any received symbol.
The simulation shows that our scheme
reduces the number of transmissions and
increases the packet delivery ratio.
30