Network Coding Employing Product

Coding at Relay Stations

BILAL ZAFAR

Master of Science ThesisStockholm, Sweden 2009

Network Coding Employing Product

Coding at Relay Stations

BILAL ZAFAR

Master of Science Thesis performed at

the Radio Communication Systems Group, KTH.

June 2009

Examiner: Professor Ben Slimane

KTH School of Information and Communications Technology (ICT)Radio Communication Systems (RCS)

TRITA-ICT-EX-2009:74

c© Bilal Zafar, June 2009

Tryck: Universitetsservice AB

Abstract

Network coding is a useful tool to increase the multicast capacity of networks. Thetraditional approach to network coding involving XOR operation has several limita-tions such as low robustness and can support only two users/packets at a time,per relay,in the mixing process to achieve optimal error performance.We propose the employ-ment of product coding at the relay station instead of xor andinvestigate such a systemwhere we use the relay to generate product codes by combiningpackets from differ-ent users.Our scheme uses relays to transmit only the redundancy of the product codeinstead of the whole product code.We seek to employ product coding can be able tosupport more than two users/packets per relay per slot,while maintaining a good errorperformance. Our scheme can accomodate as many users per relay as the costituentblock code allows, thus reducing the number of relays required in the network.Productcodes also offer increased robustness and flexibility as well as several other advantages,such as proper structure for burst error correction withoutextra interleaving. We com-pare the performance of such a scheme to the conventional xorscheme and see that ourscheme not only reduces the number of relays required but gives improved error perfor-mance as well as. Another encouraging result is that our scheme starts to significantlyoutperform the conventional one by introducing a gain at therelay.

iii

Acknowledgements

This work was carried out at Radio Communications Systems Laboratory (RST) in theDepartment of Electrical Engineering at the Royal Institute of Technology (KTH).

First of all I would like to thank my supervisor Prof. SlimaneBen Slimane for pro-viding me an opportunity to do the thesis in the department. His constant advise andencouragement incited me to look into new research topics. Iwould also like to thankmy opponent for providing me valuable recommendations and feedback.

Secondly I would like to thank my parents for always giving meendless love andsupport. I am also thankful to my friends at KTH and back home for their support.

v

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Report Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Network coding theory 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Symbol-level and Packet-level network codes . . . . . . . . .. . . . 72.3 Applications of network coding . . . . . . . . . . . . . . . . . . . . . 9

3 Product coding 133.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Matrix representation . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Decoding of Product codes . . . . . . . . . . . . . . . . . . . . . . . 16

4 Proposed scheme: Combination of Network and Product coding 194.1 General Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Decoding of XOR based scheme . . . . . . . . . . . . . . . . . . . . 214.4 Decoding of Product code based scheme . . . . . . . . . . . . . . . .22

5 Simulation results 27

6 Conclusions and Future Work 296.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

References 31

vii

List of Figures

2.1 Communication without relay . . . . . . . . . . . . . . . . . . . . . 62.2 Relay without using network coding . . . . . . . . . . . . . . . . . . 62.3 Relay with network coding . . . . . . . . . . . . . . . . . . . . . . . 72.4 Butterfly network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Network coding with xor . . . . . . . . . . . . . . . . . . . . . . . . 92.6 No reduction in slots without Network coding . . . . . . . . . .. . . 102.7 Reduction in slots without Network coding . . . . . . . . . . . .. . 102.8 Reduction in slots by employing both user co-operation and network

coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.9 Symbol-level codes and packet-level codes . . . . . . . . . . .. . . . 11

3.1 Product code with parameters [nm, k1 k2, d1 d2] . . . . . . . . . . . . 153.2 (144, 64) product code . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1 Multi-user communication using relay . . . . . . . . . . . . . . .. . 204.2 Product coding for multiple users for the proposed scheme . . . . . . 214.3 (n,k)∗(l, N) product code . . . . . . . . . . . . . . . . . . . . . . . . 224.4 Network coding for multiple users with XOR . . . . . . . . . . . .. 234.5 four-user communication using relay(s) . . . . . . . . . . . . .. . . 234.6 Product coding for four users for the proposed scheme . . .. . . . . 244.7 Network coding for four users using XOR . . . . . . . . . . . . . . .244.8 Product coding for four users employing proposed scheme. . . . . . . 254.9 Product coding for the case of Hamming (15,11)∗(7,4) product code . 254.10 Truth table of XOR based network coding . . . . . . . . . . . . . .. 264.11 Truth table of Product coding based network coding for Hamming(7,4)

code on the column. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1 BER:Proposed Scheme vs XOR based network coding . . . . . . .. 285.2 Effect of relay gain on BER:Proposed Scheme vs XOR based network

coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

ix

List of Abbreviations

ARQ Automatic Repeat and RequestAWGN Additive White Gaussian NoiseBER Bit Error RateBPSK Binary Phase Shift KeyingCRC Cyclic Redundancy CheckDMC Discrete Memoryless channelsFEC Forward Error CorrectionGMD Generalized Minimum DistanceMAP Maximum AposterioriMIMO Multiple Input Multiple OutputML Maximum LikelihoodRS Reed-SolomonSNR Signal to Noise Ratio

xi

Chapter 1

Introduction

1.1 Background

Since its conception in [1] network coding has generated a lot of interest and has ledto new research and development envisaging a revolution in the field of informationflow. There have been a few modifications suggested over the years in the conventionalscheme employing xor at the relay. We propose the employmentof product coding atthe relay station instead of xor and investigate such a system where we use the relayto generate extra redundancy using product coding. Finallywe will compare the per-formance of such a scheme to the conventional xor scheme. Thescope of this thesisinvolves information flow through communication networks,meaning that the sourcedata goes through at least two hops (channels) before arriving at the destination node,and the intermediate relays don’t just retransmit the data,they mix the informationfrom different users or channels or time intervals.

The traditional approach to network coding which involves XOR-ing has several limi-tations. This scheme limits the number of users that can be combined at relays to twoonly for the case of optimal error performance, because of the nature of the XOR op-eration. Not only does our scheme allow us to combine more than two users at a relay,it can be used to combine a large number of users thus reducingthe number of relaysrequired.

1.2 Previous Work

Network coding was proposed in [1] to solve the problem of network multicast capac-ity, it is shown that rate from source to sinks can approach the minimum of min-cutcapacities of individual sinks if we employ network coding,thus generalizing the min-cut theory for single sink networks [2] to multicast networks. [2] shows that interme-diate node encoding is not required to achieve min-cut capacity in single source singlesink networks i.e. the minimum of all single cut-sets is the maximum achievable flow.There was a lot of interest in the field after [1], resulting intremendous amount of re-search work. In [3], along with giving a clear construction approach for linear networkcodes, it is shown that linear network coding can be used to achieve min-cut capacity,and that each sink can receive information at its own min-cutcapacity. A simple andeffective framework using transfer matrices for network coding is provided by [4], thus

1

2 CHAPTER 1. INTRODUCTION

in-case of multicast, if a source sink pair has a non-singular transfer matrix, it can havea permissible network code. In [5] random network coding is proposed in order to in-crease robustness and to provide a framework for distributed network coding withoutthe requirement of a central control.

Network coding can also be employed for benefits other than multicast capacity. [6]shows how to employ network coding for recovering from non-ergodic link failures.[7] discusses the use of network coding for secure information transfer by proposing ascheme for wiretapped networks. Path diversity and distributed randomness of networkcoding can also be used to detect byzantine changes to data [8], since an acceptablenew packet cannot be formed without knowing the other nodes’packets. [9] describesa cross-layer approach to network coding thus reducing physical transmission energyrequirement for Ad Hoc networks [10]. [11] studies distributed network coding in anetwork with multiple sources transmitting to multiple sinks and shows that source andnetwork coding cannot , in general, be separated without losses, except for special twosource two sink networks. [12] studies coding gain in case ofnetwork coding for com-bination networks.

The earlier papers [1], [3] assumed error free channels for network coded networks,but recently network coding for networks and channels with transmission errors hasreceived a lot of attention from researchers around the world. In [13], [14] and [15]network error correction is proposed for network coding. Itis shown that network errorcorrection is a generalization of point to point error correction and various performancebounds can be formed. The separation of network and channel codes, without loss ofoptimality, is achieved in [16] and [17], provided the channel code codeword length isinfinitely long and channels are statistically independentdiscrete memoryless channels(DMCs), thus reducing the problem to error free channel network coding as in [1], [3].

[18] studies the properties of error correction codes for networks in the presence ofvarious types of errors and describes their minimum rank as compared to minimumdistance for ordinary error correction codes. [19], [20] show that channel and networkcoding separation leads to loss of optimality thus decodingerrors due to channel noise,while using channel codes, are not avoidable. [21] shows an analogous result for Arefnetworks [22] (networks consisting of deterministic relaycoding and no interference).

Joint network-channel codes in the form of parallel concatenated turbo codes are pre-sented in [23]. Network coding for distributed antenna systems is proposed in [24]and significant improvement in diversity performance is achieved at lower cost. Packetlevel network coding is presented in [25] for erasure channels.

[26] gives a list of publications related to network coding.

Product coding was proposed by Elias in [27] to achieve long powerful codes operatingclose to shannon’s capacity [28] while being relatively simple to decode. Concatenatedcodes were later developed from product codes in [29] [30] [31] [32] [33]. Turbo de-coding of product codes was first investigated in [34]. Iterative decoding of productcoding was developed in [35].

Combination of network coding and product coding has been proposed in the past.Works such as [36] investigate this concept in order to improve error performance.

1.3. PROBLEM STATEMENT 3

However [36] is fundamentally different from our research as that uses distributedproduct coding which involves product coding at the start atone place instead of ourscheme in which we use product coding at all relays. In contrast to [36] we only focuson the comparison between our scheme and conventional network coding. Our work isalso novel in the sense that we only use relays to transmit theredundancy of productcodes and not the whole product codes which results in increased throughput.

1.3 Problem Statement

Network coding introduced in [1], can be used to significantly increase the capacityof communication systems. While product codes can be used to increase robustness ofcommunication systems. They are also extremely flexible in terms of rate, performanceand . We wish to combine these properties in one system by using product codes insteadof the XOR operation used in conventional network coding. XORing leads to manydisadvantages of its own so replacing it with a coding schemesuch as product codingcan lead to increased gain. The basic aim of this thesis is to:

• Replace Xor with a better/ smarter coding scheme.

• Device a way to combine Network coding and Product coding in one communi-cation system to reduce the number of relays required.

• Ascertain the gain/power margin when we replace Xor with product coding.

1.4 Report Outline

• Chapter 2 explains the main concept behind network coding and enlistsits var-ious advantages such as increased capacity, reduction in slots, increase in diver-sity order etc. It also explains the difference between symbol-level and packet-level network codes. The chapter finishes with a list of applications for networkcoding.

• Chapter 3 explains the concept of product coding, how they are formed,theirparticular advantages and their decoding.

• Chapter 4 explains how we propose to combine network coding with productcoding and the expected advantages. The system model is alsopresented.

• Chapter 5gives the results of our simulation along with a discussion about them.We laso list some proposals for future work.

Chapter 2

Network coding theory

2.1 Introduction

To solve the problem of multicast capacity and to attain maximum throughput in net-works, a method called network coding was developed in [1]. The underlying principleof this method is to allow the mixing of data at the relay stations and implement a de-coding scheme at the receiver which enables it to extract thedata useful to it.

Linear network coding is the form of network coding in which the data at relay sta-tions is combined linearly. If we have S source nodes and D destination nodes and Rrelay nodes, then each node generates a new packet by linearly combining the receivedpackets. A new message Xi is related to the received message Mk by

Xi =

S∑

k=1

g(k)i Mk

Where g(k)i are coefficients from the Galoi field GF(2s ). Each of the D destination

nodes tries to solve the linear problem X=GM, for which at least S packets must bereceived.

Let us see how network coding can be deployed in case of wireless networks, and itsues.Consider the case of two users connected to a base station,all channels are rayleighfading channels because of their wireless nature

Figure 2.1 shows the case where we don’t employ relays at all,here we only needtwo waveforms and the capacity for such a system is given by

C =1

2[log2(1 + γ1) + log2(1 + γ2)]

Figure 2.2 shows the case where we employ a relay without using network coding,i.e. we use the relay only for decoding and forwarding the data, here we need fourwaveforms and the capacity is given by

C =1

4[log2(1 + γ1 + γr) + log2(1 + γ2 + γr)]

5

6 CHAPTER 2. NETWORK CODING THEORY

Figure 2.1: Communication without relay

Figure 2.2: Relay without using network coding

Figure 2.3 shows the case where we employ a relay with networkcoding by Xoringthe data of the two users at the relay station, we need three waveforms for this and thecapacity is given by

C =1

3[log2(1 + max[γ1, γ2]) + log2(1 + min[γ1, γ2] + γr)]

Let’s take a look at how we can improve the network throughputusing networkcoding. Consider the butterfly network of figure 2.4, here we don’t employ networkcoding at the relay stations, instead we just use them in a simple store and forwardscheme. Clearly this scheme can only achieve a multicast rate of 1.5 bits per timeslot(E can transmit either s1 or s2 in one timeslot).

Instead if we employ network coding and xor the data of both users at the relay, asshown in figure 2.5, we can achieve a multicast rate of 2 bits per timeslot.

Network coding can also be used to provide diversity in wireless networks, Figure

2.2. SYMBOL -LEVEL AND PACKET-LEVEL NETWORK CODES 7

Figure 2.3: Relay with network coding

2.6 and 2.7 shows how we can employ network coding to reduce the number of slots ina network, while keeping the same diversity order. As we can see in figure 2.6, withoutnetwork coding we will have to use either two relays for simulataneous transmissionof the packets from the relays to the Base station or use one relay but two time slotsto send the two packets seperately, while in network coding we have to send just oneXORed packet so we need just one relay and one time slot.

We can reduce the number of slots even further by employing both user co-operationand network coding, as shown in figure 2.8.

It should be noted however that the errror performance of XOR-based network codingdecreases significantly when we try to combine more than two users at a time at a relaydue to the nature of the XOR operation and the fact that the redundancy generated bythe relay in this case is not a powerful error correction mechanism.

2.2 Symbol-level and Packet-level network codes

We can classify network coding into two types: symbol level and packet level codes.They are suitable for differing situations and are processed differently.

If we have S input channels and D output channels at a relay. A new message onthe ith (i=1,...,D) channel Xi is related to the received message Mk by

Xi =S

∑

k=1

gi,kMk

Where gi,k ∈ (0 ,..., N-1) is a scalar and denotes the encoding variable with N as thealphabet size of the network code. Mk , (k=1,...,S) is the input from the kth channel.

8 CHAPTER 2. NETWORK CODING THEORY

Figure 2.4: Butterfly network

For packet-level codes, Xis and Mks are vectors in N, i.e Xi =[ x1,x2 ,..., xL] andMk=[ m1, m2,...,mL ] with xa and mb (a,b=1,...,L) having a range of 0,...,N-1. HereL is the length of the vector. For Packet-level codes side information, usually of fixedlength, is present in the headers including Cyclic Redundancy Check (CRC) and in-formation regarding the linear coding scheme. This property is especially useful forrandom codes [5] where central control is not present. For large enough L, side infor-mation is transmitted at negligible expense and perfect error detection is assumed(i.ewhole packets are discarded upon detection of an error). Thus the packet-level codescan be modeled using erasure channels.

For symbol-level codes, Xi s and Mks are scalars having a range of 0,...,N-1. Sideinformation, such as CRC, is not possible or unsuitable due to inefficiency, leading tolack of error detection. Thus the channels are modeled as discrete memoryless chan-nels (DMCs). These codes find implementation in networks such as circuit switchedAd Hoc wireless networks [10]. These codes require a synchronization mechanism inaddition to channel model, and a central control is also required as sink has knowledgeof the incoming source symbols structure.

Symbol-level codes and packet-level codes, though quite different, are interconnected.Perfect synchronization and error detection can both be present together in some net-work codes and altogether absent in some. These can be regarded as packet level codeswith symbol level properties. Thus these two types are not distinct. Figure 2.9 showstheir relationship in which we see that there exist symol-level codes that have error de-tection and are also self-contained while packet level codes can contain synchronizationas well.

2.3. APPLICATIONS OF NETWORK CODING 9

Figure 2.5: Network coding with xor

2.3 Applications of network coding

Above mentioned benefits of network coding enable the following uses of networkcoding

• Alternative to forward error correction (FEC) and automatic repeat request (ARQ)

• Robust and resilient to network spying attacks

• Digital file distribution

• P2P file sharing

• Throughput increase

• Bidirectional low energy transmission

10 CHAPTER 2. NETWORK CODING THEORY

Figure 2.6: No reduction in slots without Network coding

Figure 2.7: Reduction in slots without Network coding

Figure 2.8: Reduction in slots by employing both user co-operation and network coding

2.3. APPLICATIONS OF NETWORK CODING 11

Figure 2.9: Symbol-level codes and packet-level codes

Chapter 3

Product coding

3.1 Introduction

Forward error correction (FEC) codes can be most suitably used to increase spectrumefficiency in wireless communications. However FEC codes with increased codinggains and decreased overhead are limited by Shannon’s channel capacity theorem [28],which states that BER performance of a code is bounded. In general it is more diffi-cult to decode powerful codes since they have high decoding complexity. It is thereforevery important to form codes that approach theoretical maximum while being relativelyless complex. Using a long and powerful code is a requirementto utilize the maximumcapacity of a channel.

Product coding can be used to form long and powerful codes using simple constituentcodes. They were proposed by Elias in [27]. They are part of a class of codes that offersperformance close to the Shannon bound. They are also extremely flexible in terms ofrate, performance and complexity. Product codes have had animportant role in ob-taining many theoretical results in coding theory. For example in [27], Elias presentsproduct codes that exhibit a non-vanishing code rate and a non-vanishing code frac-tional minimum distance (ratio between minimum distance and length of the code),asymptotically. These codes were the first ever to exhibit asymptotic properties. Prod-uct coding concept was later developed into concatenated coding in [29] [30] [31] [32][33].

Product coding is also ideally suited to wireless communications as wireless commu-nication channels suffer from multi-path fading and noise leading to burst errors in thetransmitted information. In general we have to use interleaving to correct burst errors,as interleaving converts burst errors into random errors that can then be corrected us-ing channel codes. However, interleaving method is limitedby the maximum delaythe system can accommodate. Product coding, though, has a proper structure for bursterror correction and thus does not require extra interleaving.

Product codes have high error correcting capabilities eventhough their minimum dis-tances are much smaller than those of optimal codes of comparable length. One reasonis that product codes are extremely efficient in correcting burst errors as any error pat-tern that is restricted to a number of rows that is less than half the minimum distance

13

14 CHAPTER 3. PRODUCT CODING

of the column code can be corrected, similarly a burst error restricted to a number ofcolumns less than half the minimum distance of the row code can be corrected. Also,for random errors, if the number of errors in each row is less than half the minimumdistance of the error correction code used in that row, then these errors can be corrected,similarly, if the number of errors in each column is less thanhalf the minimum distanceof the error correction code used in that column, then these errors can be corrected. AMaximum likelihood decoder is also capable of correcting these error patterns.

Another advantageous feature of a product code is that its covering radius (maximumhamming weight of a correctable error pattern from the all zero codeword) is usually,much greater than half of its minimum distance, leading to the useful property of pos-sible correctable error patterns exceeding half the minimum distance of the code. Thuserror patterns on the row that exceed half the minimum distance of the row code anderror patterns on the column that exceed half the minimum distance of the columncode might still be correctable using a maximum likelihood or a near maximum likeli-hood decoder (a decoder that comes close to achieving maximum-likelihood decodingwithout requiring nearly as much computation per decoded asthe Viterbi-algorithmdecoder.).

Product codes can be simply represented as a set of matrices in which each row andeach column are code words in different constituent codes. The concept behind prod-uct codes is simple in which we construct long block codes by using two or even moreshorter codes. Consider two block codes C1 and C2 with rates R1 and R2 respectively,length n and m respectively, dimensions k1 and k2 respectively, and minimum ham-ming distance d1 and d2 respectively. Thus the parameter of C1 and C2 are [n, k1, d1]and [m, k2, d2]. A product code P is obtained by

• By placing k1 × k2 bits in an array with k1 rows and k2 columns.

• Coding the k2 rows by using code C1 forming a matrix of k2 rows and n columns.

• Coding the columns using code C2.

The resulting code P, illustrated in figure 3.1 will have the parameters [nm, k1 k2,d1 d2] with a rate of R1R2.

Let’s take the example of a (12, 8) hamming code and use it to build a two dimen-sional product code. This ode takes 8 information bits and appends 4 redundancy/paritybits to it to form a 12bit codeword,I I I I I I I I R R R RIf we use the same code for both rows and columns, we will get a (144, 64) productcode shown in figure 3.2

Where Rr denotes a redundancy/parity bit for the row code and Rc denotes that forthe column code. Rrc denotes the redundancy calculated for the new redundancy bits.

3.2 Matrix representation

C1 and C2 with rates R1 and R2 respectively, length n and m respectively, dimensionsk1 and k2 respectively, and minimum hamming distance d1 and d2 respectively. Thus

3.2. MATRIX REPRESENTATION 15

Figure 3.1: Product code with parameters [nm, k1 k2, d1 d2]

the parameter of C1 and C2 are [n, k1, d1] and [m, k2, d2]. A product code P is obtainedby

• By placing k1×k2 bits in an array with k1 rows and k2 columns

• Coding the k2 rows by using code C1 forming a matrix of k2 rows and n columns

• Coding the columns using code C2

The resulting code P, illustrated in figure 8 will have the parameters with a rate ofR1R2.

Another way of describing product codes which is more general is by using matri-ces. For the codes C1 and C2, described above, the product code P is an [nm, k1k2,d1d2] code whose codewords p can be represented by all nm matricesin which the rowrows belong to C1 and the columns belong to C2 , regardless of whether the constituentcodes are linear or non-linear.

If G1 is the generator marix of C1 and G2 is the generator marix of C2 , then thegenerator matrix for P can be generated by taking the kronecker product of these two.

Gp = G2χG1

Whereχ denotes a kronecker product.Thus P can be written as

P = C1χC2

A codeword p can then be found by

p = GT2 uG1

where u is a k1k2 binary matrix and GT2 is the transpose of G2 . p will be a n×m binarymatrix. The minimum distance of P is much greater than that ofC1 and C2 , howeverthe fractional distance is much smaller than C1 and C2 ,let f1 and f2 be the fractional

16 CHAPTER 3. PRODUCT CODING

Figure 3.2: (144, 64) product code

minimum distances of the codes C1 and C2 , and fp be that of the resulting productcode then we see that ,

f1=̃d1

m

f2=̃d2

n

fp=̃d1d2

mn

Clearly we have,

fp = f1f2 < f1, f2

The smaller fractional minimum distance should make the product codes less ef-ficient in correcting errors according to classical coding theory but we now know thatthe error correcting capabilities of a code also depend on its covering radius, which isquite large for product codes and thus leads to high error correction performance.

3.3 Decoding of Product codes

More powerful codes are, in general more difficult to decode,since the decoding com-plexity increases very fast as the minimum distance of a codeincreases. This does notapply to product codes and codes related to them, since decoding them involves suc-cessively decoding their constituent codes and thus the complexity is more dependenton the complexity of the much smaller constituent codes rather than the minimum dis-tance of the whole product code.

There have been many methods developed to decode product codes such as General-ized minimum distance decoding [29], Maximum likelihood decoding, Turbo decoding[34] [35], as well as the simplest method described by Elias himself in [27].This thesis,

3.3. DECODING OFPRODUCT CODES 17

however, doesn’t focus on the efficient decoding of product codes, but on its benefitswith respect to network coding. However we will briefly look at the other methods ofdecoding product codes too.

• Elias’s method:In this method, the rows are decoded using the decoder for code C1, which de-codes up to half of d1, the columns are decoded using a decoder for C2, whichdecodes up to half of d2, thus this decoder can correct up to [(d1d2 )/4] errors.

• Generalized minimum distance (GMD) decoding:GMD decoder was presented by Forney in [29]. It uses soft information to de-code binary block codes while only utilizing an algebraic decoder working onhard information only. It was later shown in [30], [31], [33], that GMD decoderscan be used to decode a whole class of concatenated codes which also includesproduct codes.GMD decoding for product codes utilizes two separate decoders for the row codeand the column code. It first decodes the row up to half the minimum distanceand then the column up to its minimum distance and stores bothresults. It thenstarts to eliminate the rows, two at a time, until the number of erased rows is lessthan the minimum distance of the column code. Each time it decodes the columnagain and stores the result. In the end it chooses the codeword that is closest tothe received matrix, from the various results. Hence it can decode up to half theminimum distance of the code

• Maximum likelihood (ML) decoding:ML decoding on product codes uses viterbi decoding [37], since it was shownin [38], that linear block codes can be represented by a trellis. However, thecomplexity of using viterbi decoding on product codes risesexponentially withthe size of the code and hence is only suitable and practical for very short codesor product codes with really high or really low rates.

• Turbo decoding:Turbo decoding was first introduced in [39]. It involves using a Maximum Apos-teriori Probability (MAP) to iteratively decode a parallelconcatenation of twoconvolutional codes. In [38], a modification was suggested for the MAP decoderto make it directly usable for trellis decoding, hence making it possible to use thismethod on product codes, however this also resulted in a hugecomplexity over-head. However various methods have been suggested to reducethe complexityof the MAP decoding with [34] and [35], being the major ones.

Chapter 4

Proposed scheme: Combinationof Network and Product coding

4.1 General Scheme

We propose the combination of product coding and network coding in a way such thatrelays employ product codes to produce extra redundancy, and we transmit only theredundancy of the product codes thus formed from the relays.Thus when all the datafrom the users reaches the end destination, the data will be combined with the redun-dancy from the relay and a product code will be formed which can then be decoded, inaddition to the decoding of individual data from each user, to achieve a good error per-formance.Since we are going to use product codes which are powerful error correctioncodes, our scheme allow us to combine more than two users at a relay. It can infactbe used to combine a large number of users thus significantly reducing the number ofrelays required.

Consider the uplink case shown in figure 4.1,

We see that the data from N users, or N packets of any number of users, each ofwhich was coded using (n, k) block code, is received at the setof relays,as well as thebase station.The relays can either simply decode and forward the data to the base sta-tion or perform network ccoding on it.We propose to combine the data from N users toproduce a product code as illustrated in figure 4.2. Each column is coded using a (l, N)block code. The novelty of our scheme is that we only send the redundancy (indicatedin figure 4.3) of the product code through the relay, instead of the whole product code,to increase throughput. The Base station also receives one copy of each user’s data.It can then combine all the users’ data with the redundancy ofthe relay to form the(n,k)∗(l, N) product code as shown in figure 4.3.Note that our scheme only requiresone relay to accomodate N users.

If on the other hand we used the two user per relay (optimally robust) version of XORbased network coding, we would have had to use N/2 number of relays to accomodatethese as shown in figure 4.4. Note that by using our scheme witha (n,k)∗(l, N)code weare reducing the number of relays required in a network by (N/2)-1.

19

20CHAPTER 4. PROPOSED SCHEME: COMBINATION OF NETWORK AND PRODUCT

CODING

Figure 4.1: Multi-user communication using relay

4.2 System model

For this thesis we will look at uplink case in case of four users connected to a basestation while employing relaying as well, shown in figure 4.5.

Each user will encode its own data into packets of n bits using(n, k) block code, acopy will be sent to the relay over an ideal channel and the base station over a Rayleighfading channel. We will investigate the uplink case for bothcases using Hammingcodes. At the relay, we will either use XOR coding or product coding. As shown infigure 4.6, the product code will be formed by employing an (l, 4) block code on thecolumns. The redundancy matrix of the order(l − 4) × n will be transmitted to thebase station over the Rayleigh channel.

The base station will use the two users’ data and the relay’s data to form the wholeproduct code and decode it as described above.

For our simulations of the XOR case we will use (15,11) Hamming codes as thechannel codes and for the product coding case we will use (15,11)∗(7,4) Hammingcodes, meaning we can combine four users’ data or four packets simultaneously atthe relay.XOR based network coding is shown in figure 4.7. Thus we are effectivelyreducing the number of relays required from two to one as shown in figures 4.7 and 4.8.

Figure 4.9 shows how we can combine four packets and form product codes for the

4.3. DECODING OFXOR BASED SCHEME 21

Figure 4.2: Product coding for multiple users for the proposed scheme

case of Hamming (15,11)∗(7,4) product code i.e. using Hamming (15,11) on the rowand Hamming (7,4) code on the column of the product code.

4.3 Decoding of XOR based scheme

At the base station, for the XOR based network decoding, we use joint detection oneach set of two users’ data and their XORed version from the relay by using the truthtable of figure , and choose the Si which minimizes the squared distance equation, i.e.

S = Si : min[|r1 −√

EsU1h1|2 + |r2 −

√

EsU2h2|2 + |rR −

√

EsRhr|2]

where ri is the recieved signal from ith source, Es is the symbol energy, hi is therayleigh fading coefficient of the ith link (h is assumed to beknown due to perfectchannel state information-CSI), U1, U2 and R values are taken from the truth table infigure 4.10.

Once we have the full packets of the two users, we use simple hamming (15,11) decod-ing on them to finally get the source information.We will repeat the same proceedurefor the decoding of the data from users U3 and U4.

22CHAPTER 4. PROPOSED SCHEME: COMBINATION OF NETWORK AND PRODUCT

CODING

Figure 4.3: (n,k)∗(l, N) product code

4.4 Decoding of Product code based scheme

In a similar way we can use joint detection on the column of theproduct coded recievedmessage and then Hamming (15,11) decoding on the row. We now have to use the truthtable of Hamming(7,4) code given in figure 4.11 and the equation

S = Si : min[|r1 −√

EsU1h1|2 + |r2 −

√

EsU2h2|2 + |r3 −

√

EsU3h3|2+

+|r4−√

EsU4h4|2+|rR1

−√

EsR1hr|2+|rR2

−√

EsR2hr|2+|rR3

−√

EsR3hr|2]

4.4. DECODING OFPRODUCT CODE BASED SCHEME 23

Figure 4.4: Network coding for multiple users with XOR

Figure 4.5: four-user communication using relay(s)

24CHAPTER 4. PROPOSED SCHEME: COMBINATION OF NETWORK AND PRODUCT

CODING

Figure 4.6: Product coding for four users for the proposed scheme

Figure 4.7: Network coding for four users using XOR

4.4. DECODING OFPRODUCT CODE BASED SCHEME 25

Figure 4.8: Product coding for four users employing proposed scheme.

Figure 4.9: Product coding for the case of Hamming (15,11)∗(7,4) product code

26CHAPTER 4. PROPOSED SCHEME: COMBINATION OF NETWORK AND PRODUCT

CODING

Figure 4.10: Truth table of XOR based network coding

Figure 4.11: Truth table of Product coding based network coding for Hamming(7,4)code on the column.

Chapter 5

Simulation results

Our scheme presents many benefits through the employment of long powerful blockcodes. We will focus on the improvement in the BER and more specifically compareit to the conventional network coding and calculate the power margin for same perfor-mance in order to calculate the gain of our scheme.

The simulations were carried out in MATLAB. Communication is affected by AWGNnoise and Rayleigh fading.Multi-User interference is neglected. Perfect CSI is assumedat the reciever. The modulation scheme used is BPSK.

Figure 5.1 shows the comparison of Xor based network coding,using Hamming (15,11)code, to our scheme for the two-user case while employing hamming (15,11)×(7,4)code for product coding.It also shows the error performanceof a direct link that does-not employ relays at all.The second order diversity gain forthe two network codingschemes employing relays is clearly visible. Also we see that the Product coding basedscheme outperforms the conventional scheme by a margin of 2 dbs.

Figure 5.2 shows the comparison of Xor based network coding,using Hamming (15,11)code, to our scheme for the two-user case while employing hamming (15,11)×(7,4)code for product coding.The relay is also given a gain of 15dbfor both XOR and Prod-uct coding, in order to see which scheme has smarter redundancy. We see that theProduct coding scheme now outperforms the conventional scheme by around 4 dbsconfirming that our scheme infact possesses ’smarter’ redundancy and thus any per-formance margin over XOR scheme can be achieved by giving therelay an increasedgain.

Spectral efficiencies:The spectral efficiencies of the two types of schemes used canbe compared by lookingat the redundancy produced by relay for each for one bit of actual data. If we onlytransmit the redundancy from product coding, the XOR can be considered as a (3,2)code, i.e. a (6,4) code acting on the column as opposed to Hamming (7,4) code actingon the column in our scheme hence our scheme has 6/7 times the efficiency of XORbased coding.

27

28 CHAPTER 5. SIMULATION RESULTS

2 4 6 8 10 12 14 16 18 20

10−5

10−4

10−3

10−2

10−1

SNR

BE

R

XOR based network codingDirect Link without network codingProduct code based network coding

Figure 5.1: BER:Proposed Scheme vs XOR based network coding

2 4 6 8 10 12 14 16

10−5

10−4

10−3

10−2

10−1

SNR (Eb/No) dbs

BE

R

XOR based network coding without relay gainProduct code based network coding without relay gainXOR based network coding+15db relay gainProduct code based network coding+15 db relay gain

Figure 5.2: Effect of relay gain on BER:Proposed Scheme vs XOR based networkcoding

Chapter 6

Conclusions and Future Work

6.1 Discussion

The advantage of employing product codes in network coding are apparent from theencouraging simulation results. As we can see from figures 5.1, we get a gain/Powermargin of around 2 dbs while using our scheme over the conventional network coding.The results are obtained for the case when we have an ideal channel from the user tothe relay and thus avoid error propogation. Error propogation can greatly reduce theefficiency of network coding, so it is better to avoid it, since the ’more intelligent’ re-dundancy of product codes is rendered less effective.

Figure 5.2 gives an even more encouraging result as we see that when the relay canamplify the signal, which is not unusual, we get an even better gain with the proposedscheme relative to the XOR based scheme.With a relay gain of 15db, the gain from ourscheme is almost doubled to 4 dbs compared to XOR case. Thus our scheme possessesmore ’intelligent’ redundancy and relay gain can be used to achieve a much improvedsystem performance over the conventional network coding.

It must be noted that the gain can be significantly increased by employing more pow-erful codes, like BCH codes or even convolutional codes, as the basis of our productcoding scheme.A higher modulation scheme will present better results as well. On thebasis of our simulation results we can conclude:

• Combining Product coding and network coding is a promising option for increas-ing the robustness of a communication system.

• Using relays to transmit only the redundancy of the Product codes instead of thewhole codes is an effective method to improve spectrum efficiency.

• Our scheme results in an improved BER performance while giving additionalbenefits over traditional network coding such as reduction in number of relaysrequired compared to the optimal two user/relay conventional network coding.

• Our scheme possesses more ’intelligent’ redundancy and given a gain at the relayit starts performing even better compared to conventional XOR decoding withthe same relay gain.

29

30 CHAPTER 6. CONCLUSIONS ANDFUTURE WORK

6.2 Future Work

Some future investigations beyond the scope of this thesis are

• employing more powerful codes, like convolution codes, as the basis of our prod-uct coding scheme.

• using a higher modulation scheme

• using a more sophisticated method to decode the product code

• employing ARQ and looking at the throughput

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