ON THE PRACTICAL PERFORMANCE OF RATELESS CODES

∗

Miroslav Kneˇzevi´c, Vesselin Velichkov, Bart Preneel and Ingrid Verbauwhede

K.U. Leuven, ESAT/COSIC and IBBT

Kasteelpark Arenberg 10, B-3001 Leuven-Heverlee, Belgium

Keywords:

Coding, error-correction codes, rateless codes, online codes.

Abstract:

We propose a new parameter for optimizing the performance of rateless codes by minimizing latency and data

overhead. We call it Latency-Overhead Product (LOP). It is based on ideas used in the analysis of digital

circuits. We demonstrate the effectiveness of the LOP parameter for a speciﬁc class of rateless codes called

Online codes (OC). We give results from experiments in ideal channel and simulated wireless channel with

losses. In the second part of our experiments we analyze the relationship between the message block size

and the performance of rateless codes. With the results from these experiments we extend the results from

the performance of Luby Transform (LT) codes published in (Vukobratovic and Despotovic, 2005) by adding

ﬁgures for Online codes.

1 INTRODUCTION

Rateless codes (RC) (Luby et al., 2001; Luby, 2002;

Maymounkov, 2002; Shokrollahi, 2006; MacKay,

2005) are a class of error correction codes, used for

encodingdata transmitted over channels with variable

packet error rate. An example of such a channel is

a wireless link. Due to the fading effect present in

wireless channels, the capacity of the latter is not con-

stant. Non-constant channel capacity results in vary-

ing packet error rate. Although rateless codes are well

suited for channels with varying error rate, their use

often causes data overhead and delays. Minimizing

both data overhead and delay is critical for achiev-

ing optimal performance of rateless codes. In the pre-

sented research, we address the mentioned problem

for the case of rateless codes. We support our con-

clusions with experimental results for a speciﬁc class

of rateless codes called Online codes (Maymounkov,

2002).

∗

This work is funded partially by IBBT, Katholieke Uni-

versiteit Leuven (OT/06/40) and FWO projects (G.0300.07

and G.0450.04). This work was supported in part by

the IAP Programme P6/26 BCRYPT of the Belgian State

(Belgian Science Policy), by the EU IST FP6 projects

(ECRYPT) and by the IBBT-QoE project of the IBBT.

2 OUR CONTRIBUTION

We propose a new parameter - Latency-Overhead

Product (LOP) for evaluating and optimizing the per-

formance of rateless codes. We use the proposed pa-

rameter to experimentally ﬁnd the optimal number of

the blocks into which a message should be divided

prior to being encoded with rateless codes. We evalu-

ate the optimal block size in terms of latency and data

overhead.

3 LATENCY-OVERHEAD

PRODUCT

The proposed metric - Latency-Overhead Product

(LOP) is inspired by ideas used in the analysis of dig-

ital circuits. In digital circuits theory there exists the

so called parameter Area-Delay Product (ADP). ADP

is widely used as a measure for evaluating the optimal

performanceof the integrated circuit in terms of mini-

mum delay and consumed area. Analogously to ADP,

we propose the Latency-OverheadProduct (LOP) as a

measure for evaluating the optimal number of blocks

into which a message should be broken prior to be-

ing encoded with rateless codes. LOP evaluates op-

timality in terms of minimum latency and minimum

overhead.

173

Kneževi

c M., Velichkov V., Preneel B. and Verbauwhede I. (2008).

ON THE PRACTICAL PERFORMANCE OF RATELESS CODES.

In Proceedings of the International Conference on Wireless Information Networks and Systems, pages 173-176

DOI: 10.5220/0002021801730176

 SciTePress

4 LOP AND RATELESS CODES

Let us consider an encoder for a ﬁle of size n blocks.

Let δ be the average overhead and t

total decoding

time of the algorithm. We deﬁne Latency-Overhead

Product as:

LOP = LOP(n) = δ

(1)

where α and β are weight parameters of the LOP

function. By ﬁxing the weights α and β to speciﬁc

values we are choosing whether the LOP minimiza-

tion will be done with a stronger preference for small

overhead (α > β) or small delay (α < β). For exam-

ple if we are more interested in having small overhead

rather than small delay (i.e. we want the delay also to

be small, but this is not as important as having a small

overhead) then we can indicate this by giving a larger

weight value for α and smaller weight value for β e.g.

α = 2, β = 1. The resultant value for the LOP function

is LOP = δ

In general, the overhead δ and latency t

are func-

tions of several parameters. In realistic environments

they depend on the state of the channel. For a very

general case we can derive a theoretical condition that

needs to be fulﬁlled in order to obtain optimal value

for the block size n for a given rateless code:

∂LOP

∂n

= αδ

α−1

∂δ

∂n

+ βt

β−1

∂t

∂n

= 0 . (2)

In other words, given the parameters α and β, using

LOP we can ﬁnd an optimal n such that Eq. (3) holds.

∂δ

∂n

= −

∂t

∂n

. (3)

Here δ and t

depend on the type of rateless code that

is used and on the state of the channel. If Eq. (3) has

a real solution we can achieve an optimal value for n.

We show the effectiveness of the LOP in prac-

tice for a speciﬁc class of rateless codes called Online

codes. Before presentingour experimentalresults, we

give a short description of Online codes.

5 ONLINE CODES

Online codes (OC) are a type of rateless codes suit-

able for channels with an unknown or variable loss

rate. OC were introduced by Petar Maymounkov

in (Maymounkov, 2002). They are efﬁcient error-

correction codes, requiring O(1) time to generate

each encoding block and O(n) time to decode an orig-

inal message of length n. Detailed treatment of On-

line codes is given in (Maymounkov,2002)and (May-

mounkov and Mazieres, 2003).

Table 1: Calculating the Optimal Block Size using LOP pa-

rameter - Ideal Channel.

Message size Overhead Latency LOP

n [%] [s] [ms]

667 31.23 0.22 21.46

833 28.36 0.23 18.50

1,333 21.38 0.26 11.88

2,000 16.93 0.30 8.60

4,000 8.21 0.55 3.71

10,000 5.32 1.26 3.57

12,000 5.00 1.82 4.59

20,000 4.76 3.92 8.88

40,000 4.05 12.52 20.54

In our experiments we use LOP to measure the

optimal number of message blocks when message is

encoded with Online codes in two cases: ideal chan-

nel and simulated wireless channel with losses. We

simulate a channel with losses with a Gilbert-Elliot

model. Next we describe our experimental settings

and provide the obtained results.

6 EXPERIMENTS AND RESULTS

In order to show the usefulness of the pro-

posed parameter LOP we have implemented Online

codes (Maymounkov, 2002).

. With Online codes we

conduct two sets of experiments: (1) ﬁnding an op-

timal block size with LOP; (2) measuring the gen-

eral performance of Online codes in respect to packet

size vs. total overhead. The ﬁrst experiment is con-

ducted for two cases: (a) ideal channel (b) channel

with losses, simulated according to a Gilbert-Elliot

model.

6.1 Optimal Block Size with Lop

In this set of experimentswe use LOP in order to eval-

uate the optimal blocksize of a message encoded with

OC. We run 10, 000 simulations per measurement.

One simulation consists of encoding and decoding of

a test ﬁle, using Online codes. In all simulations we

use a test ﬁle of size 1 MB. During experiments we

change the size of the blocks into which the test ﬁle is

broken, resp. we change the number of blocks com-

posing the ﬁle. In each simulation we measure the

two parameters of the LOP function: (1) Overhead δ

- the difference between the number of original mes-

sage blocks n and the total number of sent message

The source code for our Online codes imple-

mentation is licensed under GPL and is available at

http://sourceforge.net/projects/onlinecodes

WINSYS 2008 - International Conference on Wireless Information Networks and Systems

174

Table 2: Calculating the Optimal Block Size using LOP pa-

rameter - Wireless Channel.

Message size Overhead Latency LOP

n [%] [s] [ms]

667 49.55 1.53 375.65

833 42.10 1.50 256.86

1,333 30.37 0.99 91.31

2,000 22.85 0.91 47.51

4,000 10.61 1.17 13.17

10,000 6.23 2.11 8.19

12,000 5.88 2.47 8.54

20,000 5.32 5.63 15.95

40,000 4.79 14.55 33.38

blocks necessary for successful decoding and (2) La-

tency t

- the total time measured from start of the

encoding until the end of decoding.

In Tables 1 and 2 are shown results from calculat-

ing LOP for the cases of ideal channel and channel

with losses respectively. In each simulation we calcu-

late the value of the LOP function, using the measured

values for δ and t

. For our test settings, we give pref-

erence to minimizing the packet overhead over min-

imizing the computational costs. That results in the

values α = 2, β = 1 for the weight parameters of the

LOP function.

From Tables 1 and 2 we can see that for ideal

channel the minimum value of the LOP parameter

is 3.57 ms and for channel with losses this value is

equal to 8.19 ms. Both values are obtained for the

case when the original message is broken down to

10, 000 blocks. The encoding and decoding of this

message, using Online codes results in 1.26 s latency

and 5.32 % overhead for ideal and 2.11 s latency and

6.23 % overhead for channel with losses respectively.

These values are optimalunderthe setup conditionsof

the experiment (globalparameters ofthe Online codes

algorithm and LOP constants - α and β).

The data from Tables 1 and 2 is used to con-

struct the graphs in Figs. 1 and 2 respectively. The

two graphs represent the values of Latency-Overhead

Product as a function of the message size n.

6.2 Packet Size Vs. Total Overhead

In this set of experiments we investigate the relation-

ship between the packet size and the total overhead

when transmitting a test ﬁle encoded with Online

codes. We transmit a ﬁle of size 1M Byte using blocks

of different size varying from 25 Bytes to 500 Bytes.

This causes the number of blocks in which the origi-

nal ﬁle is broken to vary from 40, 000 to 2, 000 blocks

respectively. The test ﬁle is transmitted over the sim-

Figure 1: LOP as a function of the message size - Ideal

Channel.

Figure 2: LOP as a function of the message size - Wireless

Channel.

ulated wireless channel. The channel is simulated ac-

cording to the Gilbert-Elliot model where packet error

rate is packet length dependent.

The results from this test are presented in four

graphs in Fig. 3. We have one graph for each block

size. Each graph represents a histogram showing

the total number of experiments (ordinate) where the

overhead has a speciﬁc value (abscissa). White bars

represent the overhead on the transmitting side and

are more interesting for us. They show the total over-

head of the transmitting ﬁle. The black bars give

just an overhead at the receiving side. Observing the

Fig. 3 we can notice that with increasing the packet

size black and white bars are getting more separated.

This behavior is natural as increasing the packet size

causes larger packet error rate.

To summarize the results from our two sets of

experiments: using LOP, given the weight parame-

ters α and β, we experimentally obtain the value of

n = 10,000 blocks as an optimal message size for

both ideal channel and channel with losses. Observ-

ing the simulation results for the relationship between

packet size vs. total overhead it is straightforward

to conclude that the use of smaller packets results in

smaller total overhead and hence - in better perfor-

mance. On the other hand, by decreasing the packet

size we increase the number of packets and as a result

decoding becomes computationally more expensive.

ON THE PRACTICAL PERFORMANCE OF RATELESS CODES

175

Figure 3: Packet size vs. total overhead for four different packet sizes - channel with losses.

7 CONCLUSIONS

In this paper we have proposed a new parameter -

Latency-Overhead Product(LOP) - for ﬁnding the op-

timal block size for a message encoded with rateless

codes. We implemented Online codes and showed

how LOP can successfully be used to determine the

optimal block size in terms of best compromise be-

tween latency and data overhead. Experimenting with

the behavior of Online codes in simulated wireless

environment, where the packet error rate is packet-

length dependant, we showed that the use of smaller

block sizes results in a smaller total overhead. Al-

though we experimentally proved the effectiveness of

LOP for evaluating and optimizing the performance

of rateless codes, this parameter can be further inves-

tigated from theoretical point of view. This is a possi-

ble direction for the future work.

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