A Novel Construction Method of Fountain Codes

Weiyang Lu ,Xuehong Lin, Jiaru Lin, Kai Niu Key Lab of Universal Wireless Communications, Ministry of Education

Beijing University of Posts and Telecommunications Beijing 100876, China

lsjlwy@bupt.edu.cn

Abstract—As the first practical fountain codes, Luby Transform (LT) codes’ performance is greatly affected by the neighbor nodes selection. For most existing LT codes, neighbor nodes are selected randomly, so the encoding process cannot be described by a determined generator matrix. Moreover, this randomness leads to low encoding efficiency when code length is short. Therefore in this paper, a novel LT codes construction method is proposed by defining a LT codes construction matrix with fixed value, which has an advantage in encoding implementation. AndKent map chaos is used for the implementation of construction matrix. Meanwhile, the neighbor nodes selection becomes randomly limited. By using the And-Or tree, an optimization model is built to solve for the optimization solution in selecting neighbor nodes. Simulation results show that the proposed construction method has a better performance in bit error rate (BER) and decoding success rate, which makes it more efficient in reducing decoding overhead.

Keywords-LT codes; construction method; And-Or tree; optimization solution

I. INTRODUCTION

Digital fountain codes are sparse matrix codes with better performance in erasure channel which are firstly proposed by M. Luby in 1998 [1]. Until 2002, Luby invented a kind of fountain codes called LT codes [2] which can be applied in practical. The symbol of LT codes is obtained by xor operation of all connected input symbols. And the number of encoded packets generated from source is potentially limitless. Raptor codes [3] enhance the performance of LT codes by precoding and cascading. They are being adopted to Multimedia Broadcast Multicast Services (MBMS) within 3GPP [4] and attract more and more attentions in many practical applications such as multicast [5], downloading in parallel [6], streaming video [7], collaborative relay networks [8], etc.

The performance of fountain codes is primarily affected by two factors: degree distribution function of the encoded symbols and the way to choose neighbor nodes. Predetermined degree distribution affects not only the encoding complexity, but also the decoding success rate. The Robust Soliton Distribution (RSD) mentioned in [2] is widely used in LT codes due to its good performance and so is in this paper. In [9], chaos is used as pseudo random number generator for LT codes. However it just utilizes the feature of chaos that is similar to ideal random sequence and achieves a better performance than using the pseudo random sequence generated by the computer.

In traditional encoding process of LT codes and Raptor codes, neighbor nodes are selected at random, so the positions of element one in the generator matrix, which represents the adjacency relations of input information, change randomly every time. The encoding process cannot be described by a generator matrix with fixed value. This makes it difficult for encoding implementation. For most existing fountain codes, only the 3GPP Raptor codes are proposed by a generator matrix. Moreover, the decoder cannot recover input symbols until a relatively high decoding overhead, which is not efficient when code length is short. In our construction method of LT codes, a construction matrix with fixed value is designed by utilizing the kinetic equation of Kent map chaos. And the neighbor nodes selection changes into a random limited. Based on the analysis of And-Or tree, an optimization model is built and the optimization solution for neighbor nodes selection is solved. Simulation results show that the proposed LT codes construction method has a lower BER and a higher decoding success rate. In general, the construction method has an advantage in encoding implementation, encoding efficiency and BER reducing.

The rest of this paper is organized as follows: Section II introduces the encoding/decoding process of LT codes. A novel construction method of LT codes based on the construction matrix with fixed value and an optimization model is proposed in Section III. Section IV describes the implementation of LT codes construction method. Simulation parameters and results are presented in Section V. Finally, we conclude our work in Section VI.

II. OVERVIEW OF LT CODES

Suppose that K input information symbols are 1 2 k, , ,s s s� ,

and N encoded symbols are 1 2, , , nt t t� . Each encoded symbol is generated independently by the following process:

Step 1: Randomly choose a degree d by the predetermined degree distribution � ;

Step 2: Choose d neighbor information symbols randomly from K input information symbols;

Step 3: Execute xor operation in d information symbols one by one to generate the encoded symbol;

Step 4: Repeat step 1, 2 and 3 until the end of encoding.

This work is supported by National Basic Research Program of China(973 Program 2009CB320401), National Key Scientific and TechnologicalProject of China (2010ZX03003-001) and Chinese Universities ScientificFund (BUPT 2012RC0101).

___________________________________ 978-1-4673-2101-3/12/$31.00 ©2012 IEEE

The value of each encoded symbol, together with its degree and neighbors, are then put into a packet and sent to the receiver for the decoding process. The encoded symbols are transmitted under erasure channel continuously. The transmitted bits are either received correctly or completely erased. When the receiver has received more than K encoded symbols, the decoding process using belief propagation (BP) decoding algorithm would begin as follows:

Step 1: Search encoded symbol nt with degree one. If there is no such encoded symbol, the decoding procedure stops, and tries to receive at least one more encoded symbol;

Step 2: Set ks equal to nt , and then execute xor operation

for ks and the encoded symbols connected to it with their degrees decrease by one;

Step 3: Repeat step 1 and 2 until the entire ks are decoded, or the encoded symbol with degree one cannot be found.

RSD degree distribution � �u d mentioned above can be defined as follows [2]:

� � � � � �� � � �

� �

� �

� � � �

1

1 1

1 2,1

. . 1, , 1

ln /=

0

k

i

d du d

d d

dK

dd K

d d

R Ks t ddK RR R Kd d

K Rothers

� �

� �

�

��

� � � ��� � � � ����� � � �� �� � � � �� �� �

��

�

�

�

(1)

where � �ln /R c k K� � is the average number of encoded symbols with degree one. c is a suitable constant greater than zero and � is the allowable failure probability.

III. LT CODES CONSTRUCTION METHOD

From the encoding process of traditional LT codes, we know that all the degree and neighbor nodes of each encoded symbol are selected randomly. So the adjacency relations of input information change randomly every time, which means the positions of element one in the generator matrix cannot be predicted. This makes it difficult for encoding implementation and optimization. For most existing fountain codes, only the 3GPP Raptor codes are proposed by a generator matrix. Moreover, the decoder cannot recover input symbols until it has received more than K encoded symbols. When the code length is short, most information cannot be recovered. This makes the encoding efficiency low and BER high at the beginning.

A novel construction method of LT codes is proposed in this section. The construction method has an advantage in encoding implementation, encoding efficiency and BER reducing. Rather than selecting neighbor nodes randomly, the encoder will choose them in a random limited way. The encoding process can be described by a construction matrix whose element value is fixed.

A. LT codes construction matrix The construction matrix with fixed value is implemented by

the chaotic sequence. Chaos is a kind of complex and random phenomenon in deterministic nonlinear dynamical systems. The features of chaos such as sensitivity to initial conditions and control parameters, inherent randomness and ergodicity make it similar to ideal random sequence. Once the initial value is given, the rest sequence can be deduced by the kinetic equation. That is to say, the value of the chaos pseudo random numbers is predetermined by its initial value. Kent map [10], which is one kind of well-studied chaotic sequence, is used as pseudo random number generator to implement the LT codes construction method whose construction matrix has fixed value. The kinetic equation of Kent map can be defined as follows:

1

/ 0(1- ) / (1- ) 1

n nn

n n

x m x mx

x m m x

� �� � �� (2)

where 0< <1m , hence if 0 [0,1]x � , for all 1n � , [0,1]nx � .

The selection of neighbor nodes is described by a LT codes

construction matrix LTC with fixed value as shown in Fig 1.

The construction matrix LTC is filled with one or zero. The

columns of LTC represent the input information symbols, and

element one in each line of LTC represent the neighbor nodes used to generate each encoded symbol. The construction matrix

LTC is composed of n sparse matrixes iG and one sparse

matrix LTG , where 1, 2...i n . iG has the size of (1+ / 2 )K S K� , where K can be divided by 2S . A detailed

description of iG is shown in the figure below. Elements in the

area beyond two fold lines are zeros in iG . While LTG is the generator matrix of the norm LT codes with the size of ( (1+ / 2 )+1)N n K S K� � . The total number of element one in

each line of iG is equal to the degree of corresponding encoded

symbol, so is in LTG . The number and the position of element

one in each line of iG and LTG can be deduced with the help of the kinetic equation of Kent map. Hence the LT codes

construction matrix LTC with fixed value can be predetermined

with three parameters: the initial value 1(1)x of one Kent map chaos used to generate degree of each encoded symbols, the

initial value 2 (1)x of the other Kent map chaos used to generate adjacency relations and the number of current

encoded symbol _y num .The detail implementation of this process is shown in Section IV.

( 1)2KnS

12KS

1 S / 2K K/ 2S K

1 2 _( (1), (1),y )LT numG x x

1G

nG

2S

N

......

......

0

0

1 S / 2K K/ 2S K2S

12KS

1 2 _( (1), (1),y )i numG x x

Figure 1. LT codes construction matrix LTC

Based on this LTC , the selection of neighbor nodes is limited between a / 2K length area that changes every time with the encoding process at first. The 2S information nodes in

the middle area between two fold lines of iG are selected at most / 2K S times more than the information nodes in the

boundary. After (1 / 2 )n K S encoded symbols, K input

information symbols are selected randomly. The (1 / 2 )n K S

is the conversion threshold when most 2S information nodes in the middle have been correctly recovered. The parameters S and n are the optimization variables needed to be solved for.

B. Analysis of LT codes based on And-Or tree We would analyze the optimization solution with the help

of And-Or tree [11]. An And-Or tree is defined as follows: let lT be a tree of depth 2l . Each leaf node is labeled with either

zero or one. Among them, nodes at depth 0,2, 4, , 2 2l �� are labeled as an “OR” node whose value are equal to the “OR” of its children, and nodes at depth 1,3,5, , 2 1l �� are labeled as an “AND” node whose value are equal to the “AND” of its children.

Assume that any “OR” node has i children with a

probability of i� , where 1ii

� � .Similarly, assume that any

“AND” node has i children with a probability of i� , where 1ii

� � . The root node is estimated to be zero with

probability ly , and “OR” nodes of depth 2 is evaluated to zero

with a probability of 1ly � . Luby gave the following theory:

The root node of And-Or tree lT is estimated to be zero with

probability 1( )l ly f y � , where 1ly � is the probability with

which 1lT � ’s root node is evaluated to zero. Here we have:

� � � �� �1 1f x x� � � � (3)

where � �= iii

x x� �� and � �= iii

x x� �� .

As shown in [7], ly is the probability that an information symbol has not been recovered after l BP iteration decoding. It reflects the decoding error rate of BP decoding algorithm when the input symbols are selected at random.

C. Optimization model

Since the 2S information nodes in the proposed LT codes construction method are selected at most / 2K S times more than others, once the information symbol is recovered during

/ 2K S times selection, the probability of decoding success rate rP tends to be one and performance gains tend to be saturated.

This probability rP can be described as /21- K S

ly , where ly is a function of degree distribution � , the average degree of

encoded symbol � and the decoding overhead � . And rP isexpected to be higher with a low decoding overhead � .Meanwhile the proportion of middle information nodes in the K input information symbols is 2 /S K , this proportion is expected to be higher.

Therefore the optimization model can be described as: /21- K S

r lP y (4)

(1 ) / (1 / 2 )r lossK p K S n� (5)

� � � �� �

/ 2

0,(2 / ) 1

1

' '

(1 )( 1)

'

lim (1- ) [0.9,1)

( )

1 1

. . ( ) ( ) / (1)( )

(1)

r

r

K SlS K

l l

x

y

y f y

f x x

s t x xx e

�

� �

� �

�

�

�

� �

�

�

�

��

� ��� � � � �� ��

(6)

Solving the problem above, parameter S and decoding

overhead r� are obtained. So the encoded symbols received by

the decoder are (1+ )rK � . Suppose that the packet loss rate of

erasure channel is lossp , so the information symbols transmitted

by the encoder are (1+ )/r lossK p� . Meanwhile considering the

structure of LTC , the threshold (1+ )/r lossK p� should be divided by 1+ / 2K S . So parameter n can be expressed as

(( (1+ )/ )/(1+ / 2 ))r lossn floor K p K S�� , where ( )floor � means the biggest integer that satisfies the conditions.

IV. IMPLEMENTATION OF LT CODES CONSTRUCTION METHOD

In this section, an algorithm is proposed to implement the LT codes construction method, which is described in the following in details:

Step 1: Degree selection

Suppose that the number of input symbols is K , according

to the RSD degree distribution � �u d , the interval of (0,1) is divided into K non-overlapping sub-intervals corresponding to a different degree value d . Then generate a K length sequence of Kent map. The location of the Kent map random number in the interval determines the degree d of each encoded symbol.

As long as the initial value 1(1)x of Kent map sequence is known, the degree of K input symbols is predetermined. The

degree is the number of element one in each line of LTC

Step 2: Optimization solution

Based on And-Or tree, the optimization model can be build.

And the optimal parameter S and decoding overhead r� areobtained. Considering the packet loss rate of erasure channel and division constraints, then choose a proper optimal parameter n satisfying the conditions.

Step 3: Neighbor nodes selection

After optimization parameters S and n are obtained, neighbor nodes are selected according to the LT codes construction matrix. For each encoded symbol, a K length sequence of Kent map is generated. If the number of encoded

symbols is less than (1 / 2 )n K S , select d largest values

among position ( -1) 1j S� to ( -1) /2j S K� of the Kent map

sequence, where [1,1+ / 2 ]j K S� . The initial value of j is one, and j adds one during each iteration. When j is equal to 1+ / 2K S , j returns to one. With the change of j , different position of the Kent map sequence is selected recurrently. Else

if the number of encoded symbols is larger than (1 / 2 )n K S ,select d largest values from the K length Kent map sequence. Since Kent map sequence can be described by its kinetic equation, the corresponding positions of d largest values in the

sequence are predetermined as long as the initial value 2 (1)x

and the number of current encoded symbol _y num is known. Then the d input symbols in the input vector with the same positions will be selected as neighbors for the encoded symbol.

This is the position of element one in each line of LTC . Hence LTC with fixed value is implemented by the chaotic sequence.

Step 4: Xor operation execution

Execute xor operation for d neighbor symbols to generate the encoded symbol. The degree, the number of the encoded

symbol and the initial value of the Kent map sequence are added to the packet head for decoding. Due to the feature of chaos, the receiver can recover the adjacency relation of the encoded symbols correctly, thus reduces packets head overhead.

Step 5: Repeat step 3 and 4 until the of encoding process.

Analyzing the encoding process of the proposed LT codes construction method, we can know that some middle input symbols have more selection times than other nodes, so they can participate more in the encoding process. Once the decoder has received some encoded symbols, that information could be recovered earlier. Beyond a threshold, most of those nodes would have been correctly recovered. The selection of neighbor nodes returns to a random at this time. Those already recovered nodes would help other nodes in the decoding process, which reduces the whole BER. The parameters S and n are optimized by the optimization model to maximize the performance gain.

(1 / 2 )n K S

[( -1) 1,( -1) /2]j S j S K� �

Figure 2. The algorithm flow chart for LT codes construction method

The implementation of LT codes construction method is simplified by an algorithm flow chart shown in Fig 2.

V. SIMULATION RESULTS AND ANALYSIS

To show superiority of the proposed LT codes construction method, simulation results are presented in this section. Kent map chaos is used as pseudo random number generator for both

original LT codes and the proposed construction method. The detailed simulation parameters are listed in Table 1.

The calculation of optimization parameter S and decoding

overhead r� based on And-Or tree is shown in Fig 3. Different values of parameter S are selected according to the division constraints. As shown in the figure, considering S is equal to 250, the decoding success rate of middle nodes tends to be one when decoding overhead � is close to 0.2. This means that nearly all those information nodes can be correctly recovered. Similarly, when S is equal to 100, the decoding overhead �level off to 0.15. However if the decoding overhead � is expected to reduce to 0.1, the parameter S needs to be decreased to 50, which makes the proportion of middle information nodes in the input information symbols quit small.

So let parameter =100S and the decoding overhead r� is near to 0.15. In order to change the situation before the performance

gain is saturated, set conversion decoding overhead =0.1r� . At

this point, >0.9rP , which satisfies the objective function.

TABLE I. SIMULATION PARAMETERS

Parameter name Values

LT Distribution parameters

� 0.5 c 0.05

Total input symbols K 1000 Symbol length 5

Packet loss rate lossp 0.5

Decoding overhead � 0.0:0.05:0.25 Kent map parameter m 0.7

After optimization parameter S and decoding overhead r�are determined, the information symbols transmitted by the

encoder are (1+ )/r lossK p� . lossp is supposed to be 0.5 in the simulation. Considering the precision of random number generation in computer and packets loss error of the erase

channel, let =0.6lossp in the calculation, and choose

parameter =300n . Hence after (1 / 2 )=1800n K S encoded symbols, neighbor nodes are selected randomly among all the input information symbols.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Decoding Overhead

Dec

odin

g S

ucce

ss R

ate

of M

iddl

e N

odes

S=50S=100S=125S=250

Figure 3. The calculation of optimization parameters

0 0.05 0.1 0.15 0.2 0.25

10-2

10-1

100

Decoding Overhead

BE

R

Original LT codesOptimization LT codes Proposed method without optimization

Figure 4. The comparison of BER with different decoding overhead

0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Decoding Overhead

Def

ined

Dec

odin

g S

ucce

ss R

ate

Original LT codesOptimization LT codesProposed method without optimization

Figure 5. The comparison of defined decoding success rate with different decoding overhead

The performance comparison of the proposed LT codes construction method and the original LT codes in BER with different decoding overhead is shown in Fig 4. The figure shows that the BER is deduced with the increase of decoding overhead. Compared with the original LT codes, the proposed optimization LT codes construction method has a lower BER, approximately reducing decoding overhead by 0.05. The proposed method without optimization chooses the parameters S and n freely, so it has a lower performance gain in BER. This proves that the proposed LT codes construction method indeed has superiority in reducing BER.

Meanwhile, in some application, not all input information symbols are required to be recovered correctly. We defined a decoding success rate when more than ninety percent of the input information symbols are correct decoded. The comparison of the proposed LT codes construction method and the original LT codes in decoding success rate with different decoding overhead is shown in Fig 5. As the figure shows, the decoding success rate is higher with the increase of decoding overhead. The proposed optimization LT codes construction method has 53% higher decoding success rate at most than the original LT codes, and the proposed method without optimization can improve the decoding success rate to 37% at most, thus improves the encoding efficiency.

VI. CONCLUSIONS

A novel construction method of LT codes is proposed in the paper by designing a construction matrix whose elements can be deduced by the kinetic equation of Kent map chaos. The construction matrix makes it much more convenient in encoding implementation. Meanwhile the selection of neighbor nodes becomes random limited, which enhances the encoding efficiency when the code length is short. Objective model is

built to obtain the optimization solution in neighbor nodes selection. Seen from the simulation results, the proposed construction method reduces BER and improves encoding success rate, which is efficient in reducing decoding overhead.

REFERENCES

[1] J.W. Byers, M. Luby, M. Mitzenmacher, A. Rege, “A Digital Fountain Approach to Reliable Distribution of Bulk Data,” Proceedings of ACM SIGCOMM’ 98, pp 56–67, Vancouver, September 1998.

[2] M. Luby, “LT codes,” in Proc. 43rd Ann. IEEE Symp. Found. Comp. Sci., 2002.

[3] A. Shokrollahi, “Raptor codes,” IEEE Trans. Inf. Theory, vol. 52, pp. 2551–2567, June. 2003.

[4] 3GPP, “3GPP TS 26.346 V7.0.0, Technical Specification Group Services and System Aspects; Multimedia Broadcast/Multicast Service; Protocols and Codes”, September. 2007.

[5] Shih-Kai Lee, Yen-Ching Liu, Hsin-Liang Chiu, Yung-Chih Tsai, “Fountain Codes With PAPR Constraint for Multicast Communications,” IEEE Transactions on Broadcasting, pp: 319-325, 2011.

[6] Michael Luby, Tiago Gasiba, Thomas Stockhammer, Mark Watson,“Reliable Multimedia Download Delivery in Cellular Broadcast Networks,” IEEE Transactions on Broadcasting, pp:235-246, 2007.

[7] Hyung Rai Oh, Hwangjun Song, “Mesh-Pull-Based P2P Video Streaming System Using Fountain Codes,” ICCCN 2011, pp: 1-6, 2011.

[8] Andreas F. Molisch, Neelesh B. Mehta and Jonathan S. Yedidia, “Performance of Fountain Codes in Collaborative Relay Networks”, IEEE Trans. Wireless Commun., Vol. 6, No. 11, pp. 4108-4118, November. 2007.

[9] Qian Zhou, Zengqiang Chen, “Application of Chaos in Digital Fountain Codes,” ICYCS 2008, pp: 2786–2791, 2008.

[10] H.Sakai, H.Takumaru, “Autocorrelation of a certain chaos”, IEEE Transaction on Acoustics, Speech and Signal Processing, 28(5) pp: 588-589, October, 1980.

[11] M. Luby, Mitzenmacher and A. Shokrollahi, “Analysis of random processes via and-or tree evaluation,” the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 364–373, 1998.