Joint ChannelNetwork Coding for Star Networks Christian Koller Martin Haenggi Jrg Kliewer - PDF document

JointChannel/NetworkCodingforStarNetworksChristianKoller,MartinHaenggi,JörgKliewer,andDanielJ.Costello,Jr.DepartmentofElectricalEngineering,UniversityofNotreDame,NotreDame,IN46556,USA,Email:{ckoller,mhaenggi,costello.2}@nd.eduKlipschSchoolofElectricalandComputerEngineering,NewMexicoStateUniversity,LasCruces,NM88003,USAEmail:jkliewer@nmsu.eduAbstract—Channelcodingaloneisnotsufcienttoreliablytransmitamessageofnitelengthfromasourcetooneormoredestinationsasin,e.g.,letransfer.Toensurethatnodataislost,itmustbecombinedwithratelesserasurecorrectingschemesonahigherlayer,suchasatime-divisionmultipleaccess(TDMA)systempairedwithautomaticrepeatrequest(ARQ)orrandomlinearnetworkcoding(RLNC).Weconsiderbinarychannelcodingonabinarysymmetricchannel(BSC)andq-aryRLNCforerasurecorrectioninastarnetwork,whereYsourcessendmessagestoeachotherwiththehelpofacentralrelay.WefocusonniteblocklengthsandcomparetheexpectedthroughputsofRLNCandTDMA.ForatotalmessagelengthofKbits,whichcanbesubdividedintoblocksofsmallersizepriortochannelcoding,weobtainthechannelcodingrateandthenumberofblocksthatmaximizetheexpectedthroughputofbothRLNCandTDMA,andwendthatTDMAismorethroughput-efcientforsmallKandsmallq.I.INTRODUCTIONRandomlinearnetworkcoding(RLNC)hasrecentlybeenshowntoimprovenetworkperformanceinseveralbroadcastandmulticastscenarios.Forexample,consideringpacketera-surechannelsonthelinklayer,RLNCisknowntoimprovethroughputandreducedelayforwirelessbroadcast[1]–[4].Incontrast,weconsiderthejointdesignofchannelandnetworkcoding.Weassumethatthesizeofablockisnotpredeterminedand,foranitemessagelength,thesourcesinanetworkmaychoosethenumberofdatablockssothatthethroughputoftheoverallsystemismaximized.Jointerroranderasurecorrectingcodingfornitemessagelengthswasanalyzedin[5]–[7].In[5]theauthorsboundtheperformanceofrandomcodingonthephysicalandlinklayerusingerrorexponentstotradeoffsystemthroughputanddelay.In[6]thecombinationofRLNCandcontinuous-timeorthogonalwaveformchannelswasinvestigated.Bothpapersaimtomaximizethroughputgivenamaximumdelayconstraint.Bycontrast,inthispaperwedonotenforceamaximumdelayconstraint,butfocusinsteadontheexpectedthroughputforreliablecommunication,assumingthesenderscontinuetotransmituntilthereceivershavecorrectlyreceivedtheentiremessageasin,e.g.,letransfer.Bymaximizingtheexpectedthroughput,wealsominimizetheexpecteddelay.Inthispaper,weextendtheresultsof[7]whichconsideredbroadcastfromonesourcetomultipledestinationstoastarnetworkasdepictedinFig.1.Withthehelpofacentralrelay,ThisworkwaspartlysupportedbyNSFgrantsCCF08-30651,CCF08-30666,andCNS-1016742. Fig.1.StarnetworkinwhichYsourcescommunicateovernoisyBSCswiththehelpofacentralrelay.Ysources,S1;:::;SY,communicatewitheachotherovernoisybinarysymmetricchannels(BSCs).Weassumethereisnodirectpathbetweenanyofthesources,i.e.,theyareonlyconnectedtothecentralrelay,whichreceivestransmissionsfromallsourcesandcanbroadcasttoallsources.WeconsiderthecasewhereeachsourceShasamessageofnitelengthKbitsthatisintendedforalltheotherY1sourcesSj,j=1;:::;Y,j=i.Inthissetting,channelcodingaloneisnotsufcienttoguaranteereliablecommunicationanditmustbecombinedwithratelesserasurecorrectingschemes,suchasatime-divisionmultipleaccess(TDMA)systempairedwithautomaticrepeatrequest(ARQ)[8]orq-aryRLNC.Wedenethetimethatittakestotransmitonebitasatimeunitand,whenmaximizingtheexpectedthroughput,weminimizetheexpectednumberoftimeunitsittakestosuccessfullytransmitmessagesfromYsourcestotheotherY1sources.OurgoalistojointlyndthenumberofblocksmandthechannelcodingrateRthatmaximizesystemthroughput.Choosingastarnetworkasamodelallowsustocombineseveralprominentfeaturesofmoregeneralnetworks.IntheRLNCcase,thestarnetworkmodelincludesamultiple-accesschannel(MAC)phasefollowedbyabroadcast(BC)phase.Werstanalyzethesetwophasesseparatelybeforecombiningthemtomaximizethethroughputofthestarnetwork.II.SYSTEMMODELA.StarNetworkSetupAsshowninFig.2,asourceS,i=1;:::;Y,splitsitsmessageoflengthKbitsintombinarydatablocksDij,j=1;:::;m,oflengthK=mbits,orequivalentlymq-arydatablocks~Dij,j=1;:::;m,oflengthK=(ml)q-arysymbols.Weassumethatqisapoweroftwo,i.e.,q=2landthatK Fig.2.CombinedchannelandnetworkcodingatsourceSi.isdivisiblebyml.AsourceSthenperformsRLNConitsmdatablockstocreateanetworkcodedblock~Bbychoosingavector~aoflengthmofcoefcientsfromGF(q),wheretheindexbdoesnothaveaxedrange,sinceasmanyblocksarecreatedasarenecessarytoachievereliablecommunication.Thecodedblock~Bisthenthelinearcombinationofthemdatablocksmultipliedbythecorrespondingcomponentsofthecoefcientvector~a,i.e.,~B=Pmj=1~a(j)~Dij,whichcanalsoberepresentedasabinaryblockoflengthK=mbitsusingthenotationB.AheaderofconstantsizehbitsisthenappendedtoeachcodedblockBtoformachannelinputblock^Boflengthk=K=m+hbits.Theheadercan,forexample,containacyclicredundancycheck(CRC)todetectdecodingfailures.Finally,eachchannelinputblock^BisprotectedbyabinarychannelcodeofrateR,formingthechannelcodedblockv.a)MACphase:DuringtheMACphase,allsourcestransmittotherelaysimultaneously.Wemodelthechannelfromthesourcestotherelayasabinaryadderchannel[9],[10],sothattherelayreceivesavalueequaltothe(real)sumofthebitssentbythesourcesplusanoiseterm.Therelaythenquantizeseachreceivedvaluetothenearestintegerandmakesaharddecision.Ifthequantizedvalueiseven,itdecidesareceivedzero,andifthequantizedvalueisodd,itdecidesareceivedone,sothattheresultingreceivedbitcanbemodeledasthemodulo-2superpositionofthebitssentbyallthesourcesplusanoisebit.Equivalently,thereceivedsuperimposedvectorattherelayisgivenbyr=ve=v1v2:::vYbe;(1)wheresymbolizesmodulo-2additionandeisabinaryvectorwhoseelementsareBernoullii.i.d.randomvariablesthatareonewithprobabilitypMAC.Iftherelayisabletodecodev,itbroadcastsvtothesources.Shouldtherelaynotbeabletodecode,itdoesnottransmit.Weassumethesourcescansensethechannel,soiftherelayfailstodecodeanddoesnottransmit,thesourcesimmediatelytransmitanotherchannelcodedblockandwehaveanotherMACphase.b)BCphase:DuringtheBCphaseweassumethattherelayisconnectedtoeachofthedestinationsviaindependentBSCs.Wealsoassumethatthesourcesareataboutthesamedistancefromtherelay,andthusexperiencethesamepathloss,sothattheyshareacommonchannelcrossoverprobabilitypBC.EachchannelcodedblockvsentbytherelayduringtheBCphaseisalinearcombinationofYmdatablocks,multipliedbyacorrespondingsetofYmnetworkcodingcoefcients.WeassumethatthesourcesandthereceiversuseYsynchronizedpseudo-randomnumbergenerators,eachsourcewithadifferentseed,togeneratethesequencesfor~a,sothatanysourceknowsthenetworkcodingcoefcientsofallsources.ThecolumnvectorofYmnetworkcodingcoefcients~a=[a1;:::;aYb]0correspondingtoablockbisthebthcolumninthegeneratormatrixGemployedbytheRLNCinthestarnetwork,and~B=PY=1~B,thesuperpositionofthenetworkcodedblocks,canbeviewedasacodesymboloftheRLNC.WhenasourceSreceivesavfromtherelay,itrstdecodesthebinarychannelcodetoobtain^B=PY=1^B.Ifdecodingissuccessful,theheaderisremovedand,afterbinarytoq-aryconversion,weobtain~B.SourceSthensubtractsitsowncontributionto~B,whichis~B,andstoresthesuperpositionoftheotherY1networkcodedblocks~Bjb,j=1;:::;Y,j=i,asanelementinavectorofreceivedRLNCsymbols.Italsostoresthesubsetof(Y1)mnetworkcodingcoefcientsin~ainvolvedincreatingthesuperposition~BjbasacolumninitscoefcientmatrixG,theperceivedgeneratormatrixoftheRLNCfromthepointofviewofsourceS.AftertheendoftheBCphase,anotherMACphasebegins.OnceasourceShasreceivedenoughblocksfromtherelaytoformamatrixGwith(Y1)mlinearlyindependentcolumns,itcanrecoverthe(Y1)mdatablocksfromtheothersourcesbyinvertingthematrixGandmultiplyingitbyitsvectorofreceivedRLNCsymbols.Thenitsendsasingleacknowledgment(ACK)totherelay.OncetherelayhascollectedYACKsfromtheYsources,itbroadcastsanACKtothesources,terminatingtransmission.AllsourcescontinuetotransmituntiltheyreceiveanACKfromtherelay.WeassumethatthetransmissionofanACKisinstantaneousandreliable,i.e.,itdoesnotconsumeanyresourcesanditisneverreceivederroneously.AsareferenceschemeweconsiderTDMAtransmissionofthesources,pairedwithARQ.Wealsoassumeasourcesplitsitsmessageintomdatablocks,butnonetworkcodingisused.TheMACphaseinFig.1isreplacedbyaTDMAphase,whereonlyonesourcetransmitstotherelayatagiventimeandtheindividualdatablocksareagainprotectedbyabinarychannelcodeofrateR.ThetransmittingsourceSrepeatsthetransmissionofachannelcodedblockasmanytimesasisnecessaryfortherelaytoreceivethedatablockcorrectly,atwhichpointtherelaytransmitsanACK.Aftertherelayhasreceivedthedatablockcorrectlyitbroadcastsittoallsources.Whenasourcereceivesthedatablockcorrectly,itsendsanACKtotherelay.TherelayrepeatstheBCtransmissionasmanytimesasisnecessaryuntilallY1sourcesSj,i=1;:::;Y,j=i,receivethedatablockcorrectly.AfterthestepsdescribedabovehavebeensuccessfullycompletedforsourceS,itistheturnofthenextsourcetotransmitadatablocktotherelay,andthesourcesarescheduledinaroundrobinfashionwithmrounds.Aftereachsourcehassuccessfullytransmittedmdatablocks,thetransmissionends. B.ChannelCodingTheblockerrorprobabilityofrandomcodingontheBSCwithcoderateRcanbeboundedaboveasafunctionoftherandomcodingerrorexponentE(R).Usingtheunionbound,wehaveE(R)=R0R,whereR0,thecutoffrateofthechannel,dependsonthecrossoverprobabilitypoftheBSC[11].Thenweobtain2n(R0R);(2)wheren=k=Ristheblocklengthofthecodeandk=K=m+hbits.UsingtheunionboundtoboundchannelcodingperformanceallowsustoobtainanalyticalexpressionsfortheoptimumchannelcodingrateandoptimumnumberofdatablocksinSectionsIII–V.C.TheExpectedOverheadofRLNCConsideringasinglesourceonitsownandRLNCoverGF(q),theprobabilitythatm+xindependentlycreatedcolumnvectorsofnetworkcodingcoefcients~aformanm(m+x)matrixofrankm,i.e.,theprobabilitythatm+xnetworkcodedblocksaresufcienttodecodetheRLNCofthatsourceisgivenbyPsuccess(m;x;q)=m=11qx[12].Inthestarnetwork,ablockbroadcastbytherelayisalinearcombinationofYmdatablocksand,sincethenetworkcodingcoefcientsarechosenindependentlyatallsources,theprob-abilitythatallYsourcescanconstructaninvertiblematrixofrank(Y1)mfrom(Y1)m+xcorrectlyreceivedblocksisgivenbyPsuccess(m;x;q;Y)=(Psuccess((Y1)m;x;q))Y.Wecannowmakeuseofaresultfrom[12]toderivethefollowingupperandlowerboundsontheexpectedoverheadX(q;Y)ofRLNCinthestarnetwork(see[13]fordetails):YXj=1Yj(1)j+1(q2q)jqj (q1)j(qj1)2X(q;Y)YXj=1Yj(1)j+1q2j(q1)j (q1)j(qj1)2;(3)wherebothboundsareindependentofthenumberofdatablocksmandtendtozeroasqgetslarge.ModelingtheexpectedcodingoverheadofRLNCasaconstantfractionalnumberofblocksleadstoopposingop-timizationcriteriaforchannelcodingandRLNCwhenamessageofnitesizeKbitsisdividedintomdatablocks:Moredatablocks,andthusshorterchannelcodedblocks,leadtoasmallercodingoverheadofRLNCinbits.Longerchannelcodedblocks,andthusfewerdatablocks,leadtomorepowerfulchannelcodes.III.THEMACPHASETooptimizethroughputfortheMACphase,weassumethatthechannelsfromtherelaytothesourcesareerror-free,i.e.,pBC=0,sothattherelaydoesnotneedachannelcode,andthattherelayremovesthehheaderbitspriortobroadcasting.ModelingtheexpectedcodingoverheadX(q;Y)ofRLNCasaconstantfractionalnumberofblocks,onaverageeachsourcemustcollect(Y1)m+X(q;Y)networkcodedblockstobeabletodecode.Using(2)andlettingn=k=R=(K=m+h)=R,weobtain(see[13])NMACRLNCk((Y1)m+X(q;Y)) R12k(R0=R1)(4)fortheexpectednumberofbitsthatmustbesentbythesources.ForTDMA,atotalofYmblocksmustbetransmittedtotherelaybytheYsources,andweobtain(see[13])NMACTDMAY(K+mh) R12(K m+h)(R0 R1)(5)fortheexpectedtotalnumberoftransmittedbits.(4)and(5)dependontheBSCcrossoverproabilitypMACthroughthechannelcutoffrateR0.A.TheOptimumChannelCodingRateTakingthepartialderivativeof(4)withrespecttoRandsettingittozero,weobtain(see[13])R R0=ln(2)k 1e(ln(2)k+1)+1(6)fortheoptimumchannelcodingrateasafractionofthecutoffrateofthechannel,whereW1(x)representsthelowerbranchoftheLambert-Wfunction.From(6)weseethattheoptimumchannelcodingrateratioR=R0isonlyafunctionofthechannelinputblocklengthkandisindependentoftheexpectedoverheadX(q;Y)ofRLNCandthenumberofsourcesY.ItisthusalsotheoptimumchannelcodingrateforaschemeemployingTDMA.B.TheOptimumNumberofBlocksNowtakingthepartialderivativeof(4)withrespecttomandsettingittozero,weobtain2z(K m+h)= 1+ln(2)zKK m+h(X(q;Y)+m(Y1)) KX(q;Y)hm2(Y1)!;(7)wherez=(R0=R)1.Ingeneral,aclosedformsolutionof(7)cannotbefound.However,forh=0andY=2wecanagainusetheLambert-Wfunctiontosolveform,andtheoptimumnumberofblocksm,givenaconstantR=R0andthemessagelengthK,ism=ln(2)zK 1+ln(2)zK X(q;2)+1e(1+ln(2)zK X(q;2)):(8)ForothervaluesofhandYwesolve(7)and(6)jointlyusingnumericalmethodstoobtaintheoptimumnumberofblocksmthatminimizestheexpectednumberoftransmissionsandmaximizesthethroughput.Forh=16,Fig.3showstheoptimumnumberofblocksmgiventhetotalmessagelengthK,thenumberofsourcesY,andRLNCoverGF(q).AsKincreases,weobservethatthemaximumthroughputisachievedforalargernumberofblocksm.SincetheexpectedcodingoverheadX(q;Y)inblocksincreaseswiththenumberofsourcesinthestarnetwork,theoptimumnumberofblocksmincreaseswithYforaxedmessagelengthK.Ontheother 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 5 10 15 20 25 Message Length KOptimal Number of Blocks mMAC phase, h = 16 Y = 2 Y = 3 Y = 4 q = 64 q = 4 q = 16 Fig.3.OptimumnumberofblocksmgiventhemessagelengthK,thenumberofsourcesY,andheadersize=16forRLNCoverdifferentGaloiseldsizesq.hand,sincetheexpectedcodingoverheadX(q;Y)decreaseswithincreasingGaloiseldsize,theoptimumnumberofblocksdecreaseswithq.IV.THEBCPHASETooptimizethroughputfortheBCphase,weassumethatthechannelstotherelayareerrorfree,i.e.,pMAC=0.Further,sincepMAC=0,weassumethatduringtheMACphasethesourcestransmittotherelayuncoded,i.e.,R=1,andthatnoheaderisused.Aheaderoflengthhisthenappendedtoeachblockattherelay,andtherelayusesachannelcodeofrateR1toprotecttheblocks.A.TDMABCPairedWithARQConsidertheTDMAscheme,wheretheexpectednumberofblocksthattherelaymustbroadcast,MBCTDMA,isgivenin[3].Using(2)andNBCTDMA=kMBCTDMA=R,weobtainfortheexpectednumberofbittransmissionsbytherelay(see[13])NBCTDMA=Y(K+mh) R1X=0112(K m+h)(R0 R1)Y1;(9)whereR0isthecutoffrateofaBSCwithcrossoverprobabilitypBC.ForanyxedcodingrateR,thefactorY(K+mh)=Rin(9)aswellastheBCchannelblockerrorprobabilityBCarestrictlyincreasingwithincreasingm.SothethroughputfortheTDMAsystempairedwithARQismaximizedform=1andachannelinputblockofsizek=K+h.Toobtainthechannelcodingratethatmaximizesthrough-put,wetransform(9)intoanitesumandusethepartialderivativew.r.t.Rtoobtain(see[13])YX=1(1)Yi12izkikln(2)R0 R2izk (12izk)2=0;(10)wherez=(R0=R)1.ForTDMAandY=2,thechannelcodingratethatmaximizesthroughput(10)intheBCphaseisthesameastheratethatmaximizesthroughputfortransmissiontotherelay(6),obtainedinSectionIII.Inbothcases,messagesaretransmittedfromonesendertooneintendedreceiver.For 102 103 104 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1Block length kOptimal rate ratio R/R0 Y = 2 (Broadcast & Jointly) Y = 3 (Broadcast) Y = 5 (Broadcast) Y = 6 (Broadcast) Y = 3 (Jointly) Y = 5 (Jointly) Y = 6 (Jointly) Fig.4.OptimumchannelcodingrateRforBC(solidlines)andconsideringtransmissionfortheMACandBCphasejointly(dashedlines)asafractionofthecutoffrateR0fordifferentchannelinputblocklengthskwhenTDMAisemployed. 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 2 4 6 8 10 12 14 16 18 Message length KOptimal number of blocks mBroadcast from the Relay, h = 16 Y = 2 Y = 3 Y = 4 q = 4q = 16q = 64 Fig.5.OptimumnumberofblocksforBCfromtherelayfor=16.broadcasttoalargernumberofsourcesY,wecannumericallyndthesolutionof(10),andtheoptimumrateratiosR=R0fordifferentnumbersaredestinationsareshownasthesolidlinesinFig.4.Weseethat,while(6)doesnotdependonthenumberofsourcestransmittingtotherelay,duringtheBCphasetheoptimumchannelcodingrateRforTDMAasafractionofthecutoffrateR0decreasesasthenumberofBCdestinationsincreasesand,forY�2,issmallerthan(6).TheoptimumnumberofblocksfortheTDMAscheme,however,ism=1forbothtransmissiontotherelay,consideredinSectionIII,andtheBCphase.B.BCUsingRLNCUsingRLNC,theexpectednumberofnetworkcodedblocksMBCRLNCthattherelaymustbroadcastisgivenin[3],andtheexpectednumberofbitsthattherelaymusttransmitisgivenbyNBCRLNC=kMBCRLNC=R.Wesolvetheresultingmultidimen-sionaloptimizationproblemusingnumericalmethods.FortheBCscenariousingRLNC,Fig.5showstheoptimumnumberofdatablocksmforh=16.ComparingtheoptimumnumberofblocksinFig.5totheMACphasedisplayedinFig.3,thenumberofblocksthatmaximizesthroughputisgenerallysmallerfortheBCphase.ThemostprominentdifferencebetweenFigs.5and3isthat,whilefortheMACphasetheoptimumnumberofdatablocksincreaseswiththenumberofsources,fortheBCphasetheoptimumnumberofblocksdecreaseswithanincreaseinthenumberofBCdestinations 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.8 1 1.2 1.4 1.6 1.8 2Message length KThroughput ratio TRLNC / TTDMARLNC over GF(4) h = 0 h = 16 h = 32 Y = 2 Y = 3 Fig.6.AveragethroughputratioTRLNC=TTDMAforGF(4).Y,thusputtingmoreemphasisonthechannelcodingbeingabletoprovidemorereliableindividualblocks.V.JOINTOPTIMIZATIONFORTHESTARNETWORKFromSectionsIIIandIV,weseethatthenumberofdatablocksandthechannelcodingratethatmaximizethroughputdifferfortransmissionfromthesourcestotherelayandforBCfromtherelay.Inapracticalsystem,however,itwouldbedesirabletohavethesamechannelcodingrateandthesameblocksizefortransmissiontoandfromtherelaysowenowjointlyoptimizethethroughputofthestarnetwork,keepingmandRconstant.Werefertothetimeittakestotransmitoneblockasatimeslot.FortheRLNCscheme,theexpectednumberoftimeslotsthatareoccupiedbytransmissionsinthestarnetworkisgivenbyMRLNC=MBCRLNC(1+1=(1MAC)),whichreliesonthefactthat,foreveryblockthattherelaybroadcasts,onaverage1=(1MAC)transmissionsfromthesourcestotherelayarenecessary,forMACblockerrorprobabilityMAC.Similarly,fortheTDMAscheme,theexpectednumberoftimeslotsthatareoccupiedbytransmissionsisgivenbyMTDMA=MMACTDMA+MBCTDMA.Inthiscase,sincethethroughputforboththetransmissionphasetotherelayandtheBCphasefromtherelayismaximizedform=1,onechannelinputblockoflengthk=K+hbitsforeachsourceSisalsooptimumwhenconsideringbothphasesjointly.Inthefollowing,weconsiderthesymmetriccase,whereMAC=BC.Inthiscase,thechannelcodingratethatmaximizesthethroughputforTDMAcanbeobtainedbytakingthederivativeofNTDMA=kMTDMAw.r.t.thechannelcodingrateR(see[13]fordetails),andtheoptimumchannelcodingratethatjointlymaximizesthroughputfortheTDMAschemeisalsodepictedinFig.4.ForY�2,theoptimumchannelcodingrateforthestarnetworkdecreaseswiththenumberofsources,similartotheTDMABCcase.However,comparingtheoptimumratefortheBCphasealonetothejointlyoptimumrateforthesamenumberofsourcesY,wendthatthechannelcodingratethatjointlymaximizesthroughputforthestarnetworkishigherthantheonethatgivesthemaximumthroughputfortheBCphasealone.Fig.6showstheaveragethroughputratioTRLNC=TTDMA=MTDMA=MRLNCofRLNCoverGF(4)toTDMAandtheasymptoticthroughputratiosareplottedashorizontalblacklines.ForsmallmessagelengthsK,weseethattheaveragethroughputratiorisessteeplybeforethecurvesattenoutandslowlyapproachtheirasymptoticvaluegivenbyTRLNC=TTDMA=Y=(Y1).Astheheadersizehincreases,theaveragethroughputratiodecreases,andalargermessagelengthKisneededtoobtainagivenaveragethroughputratio.ForsmallmessagelengthsKandlargeheadersizesh,TDMAismorethroughput-efcient.Forexample,forh=32andY=6sources,werequireK�900bitsforRLNCtobemorethroughput-efcientthanTDMA.VI.CONCLUSIONSWeanalyzedthejointdesignofchannelcodingonthephysicallayerandrandomlinearnetworkcodingonthelinklayerforastarnetworkwhereYoutersourcessendxedlengthmessagestoeachotherwiththehelpofacentralrelay.ForRLNCoveraniteGaloiseldofsizeqandamessageoftotallengthKateachsource,weobtainthenumberofdatablocksandthechannelcodingratethatshouldbeusedtomaximizethethroughputofthestarnetworkusingRLNC,assumingbinarysymmetricchannelsbetweenthesourcesandrelayandabinaryadderchannelmodelattherelay.WealsoobtaintheoptimumnumberofblocksandtheoptimumrateforareferenceTDMAsystemandcomparethethroughputsofthetwotransmissionschemes.Wendthat,forsmallmessagelengthsKandRLNCoversmallGaloiseldsq,TDMAismorethroughput-efcientthanRLNC,whileRLNCismorethroughput-efcientwhenthemessagelengthKgetslarge.REFERENCES[1]D.S.Lun,M.Médard,R.Koetter,andM.Effros,“Oncodingforreliablecommunicationoverpacketnetworks,”PhysicalComm.,vol.1,no.1,pp.3–20,2008.[2]A.Eryilmaz,A.Ozdaglar,M.Médard,andE.Ahmed,“Onthedelayandthroughputgainsofcodinginunreliablenetworks,”IEEETrans.Inf.Theory,vol.54,no.12,pp.5511–5524,Dec.2008.[3]M.Ghaderi,D.Towsley,andJ.Kurose,“Networkcodingperformanceforreliablemulticast,”inProc.IEEEMilitaryComm.Conf.,Orlando,FL,Oct.2007.[4]Y.E.SagduyuandA.Ephremides,“OnjointMACandnetworkcodinginwirelessadhocnetworks,”IEEETrans.Inf.Theory,vol.53,no.10,pp.3697–3713,Oct.2007.[5]M.VehkaperäandM.Médard,“Athroughput-delaytrade-offinpacketizedsystemswitherasures,”inProc.IEEEInt.SymposiumonInform.Theory,Adelaide,Australia,Sept.2005.[6]M.Xiao,“Cross-layerdesignofratelessrandomnetworkcodesfordelayoptimization,”inProc.IEEEInt.Conf.onComm.,CapeTown,SouthAfrica,July2010.[7]C.Koller,M.Haenggi,J.Kliewer,andD.J.Costello,Jr.,“Ontheoptimalblocklengthforjointchannelandnetworkcoding,”inProc.IEEEInformationTheoryWorkshop,Paraty,Brazil,Oct.2011,pp.528–532.[8]S.Lin,D.J.Costello,Jr.,andM.J.Miller,“Automaticrepeat-requesterrorcontrolschemes,”IEEECommunicationsMag.,vol.22,no.12,pp.5–17,Dec.1984.[9]N.T.GaarderandJ.W.Wolf,“Thecapacityregionofamultipleaccessdiscretememorylesschannelcanincreasewithfeedback,”IEEETrans.Inf.Theory,vol.21,pp.100–102,Jan.1975.[10]T.KasamiandS.Lin,“Codingforamultipleaccesschannel,”IEEETrans.Inf.Theory,vol.22,no.2,pp.129–137,Mar.1976.[11]A.BargandG.D.Forney,Jr.,“Randomcodes:Minimumdistancesanderrorexponents,”IEEETrans.Inf.Theory,vol.48,no.9,pp.2568–2573,Sept.2002.[12]G.Liva,E.Paolini,andM.Chiani,“PerformanceversusoverheadforfountaincodesoverFq,”IEEECommun.Lett.,vol.14,no.2,pp.178–180,2010.[13]C.Koller,M.Haenggi,J.Kliewer,andD.J.Costello,Jr.,“Jointdesignofc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