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134bad0bd1 as0 dibabCHAPITRE7.CODESDENSESMAXIMAUX u dj+1=w abab dk+1 aba 2etn'admettantpasaacommesuxe,onnepeutdoncavoirp0=b. impliquequeaaestsuxeded0.LesmotsdeUetanttousdelongueuraumoins p0=b.Onaab:u:ab=d0:b,comme,pardenitiondeu,aaestsuxedeu:acela Onadoncp06=".Montronsquenousavonsdemêmep06=b.Eneet,supposons Fig.7.8{IinduituneU-interpretationpourlefacteurab:u:ab. dedj+1.Ainsi,sij6=k(resp.j=k)ilexistev2A+telquelemotdj+1estegal etab2U,onau2U.Nousavonsdejavuaucasp0="quececasnepeutse produire. Commep0estprexed'unmotdeU,lesconditionsp06=",p06=betp06=ab impliquentjp0j>2,c'est-a-direp02P(UnU1).Nousrappelonsquep0estprexe Enn,nousmontronsquep06=ab.Supposonslecontraire,onaalorsab:u:ab= a(resp.estunprexede)b(ab)jvj:v:(ab)jvja.Nousposonsw=b(ab)jvj:v:(ab)jvja. d0:ab,doncab:u=d0.Ainsionaab:u2U.CommeUestunensembleprexe equationsb(ab)juj:u:(ab)juja=d0:::djdj+1::::dk+1etab:u:ab=s0:di+1:::dj:p0 estfacteurded0.Ainsip02S(u:ab).Deplus,commep0estprexededj+1,les Puisques0=",leprexeabdeab:u:ab,quiappartiental'ensembleprexeU, Sij+1=k+1alorsdj+1=p0(ab)1:(ab)juja,d'oucommepardenitiondewon w2P(p0(ab)1:(ab)juja). adj+12P(w),ilvientp0(ab)1:(ab)juja2P(w). Parconsequent,sij+16=k+1alors,commenousavonsvuquew=dj+1,ona impliquent Lesmotswetp0(ab)1:(ab)jujasontdoncP-comparables.Deplus,onap02P(w) p0(ab)1:(ab)juja=dj+1:::dk+1: Lelemme7.2.4assuredoncquep0:(ab)1=b(ab)jvj:v,c'est-a-dire Onaab:u:ab=d0:p0doncab:u:(ab)jvj1:a=d0:p0:(ab)1:(ab)jvj1:ad'ou doncp0:(ab)12P(w)nfwget,pardenitionde(s0;d0;p0),onap0:(ab)12S(u). ab:u:(ab)jvj1:a=d0:b(ab)jvj:v:ab:(ab)jvj1:a p0=b(ab)jvj:v:ab:
7.2.CODESCIRCULAIRESMAXIMAUXDENSES etdonc Commejd0j6juj,onapardenitionde(Un)n>1,d02Un1.Deplus,ona Onobtientdoncab:u:(ab)jvj1:a2Un1.Onaab2UetUestprexe,d'ou p0(ab)1=b(ab)jvj:vetp0(ab)1estsuxedeu,onadoncjvj
Page 1:
U.F.R.DESSCIENCESETDESTECHNIQUES UN
Page 4 and 5:
ii2.1Denitionsetpremieresproprietes
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iv CODESETINTERPRETATIONS
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partagelebureaupendantdeuxans.Pourt
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viii engenetique[AM95].Lebutdelathe
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admettentchacuneuneversiondutheorem
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egala1:lescodesnon-interpretes. xii
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21.1Monodes ciativeetquipossedeunel
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4primitifsitousseselementssontprimi
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61.3.2Ensemblesrationnels Lafamille
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8Proposition1.4.3Unsous-monodeMAest
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10 codesadelaidedechirageborne,code
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12 jacentess'ilexistei2[0;n]etj2[0;
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14 CHAPITRE1.PRELIMINAIRES
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16 erreurdetransmissionappara^tdans
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18y1 CHAPITRE2.CODESADELAID'INTERPR
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delaid.CommeXestuncode,onax6=". 20
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22 Preuve.SoientYBetZAdeuxcodesadel
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24(ii)Ilexistei>0,j>1,2S(X),x1;:::;
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26 {Supposonsyju2S(xi).Posonsalorsv
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28 demême,onpose Posons CHAPITRE2.
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deX+(d). deX(b)ouencorelemotcdestla
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adjacent.Parl'absurde,supposonsX\S(
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34 {Enfait,lescodes(1;2n),(2n;1)-li
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36 {Unautreexempleinteressantestcel
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plementaires: Exemple2.5.1Lecoderat
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40x1g(0)=1g(1)=2 CHAPITRE2.CODESADE
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42 (x1:::xf(q))12Xcequiimpliquerait
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44Leschema2.3reprendlesprincipauxre
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quisontinclusesdanslafamilledescode
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construction: centemaispasd'exhiber
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acelledec,parhypothesederecurrence,
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52Eneet,onaU1(X)=P(X)nf"g=fabab;abc
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ni: Theoreme3.2.2LabaseYdel'envelop
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56 CHAPITRE3.THEOREMEDUDEFAUTETCODE
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Proposition3.4.4UnmonodeNestextrêm
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pretee.SiunmotdeXadmetuneX-interpre
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qu'ilssontenfaitdescodesnon-interpr
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64 CHAPITRE3.THEOREMEDUDEFAUTETCODE
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maximaldanslafamilledescodes?Lefait
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Codesprexes,suxesetbixes Nouspresen
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YFtelqueXYonaY=X.Evidemment,cetteno
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mauxnishormisA,c'estenparticulierle
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estuncodebixemaximal,nonmaximaldans
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coupantestmentionneen[Per84]:fanbnj
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78 CHAPITRE4.SURLESCODESMAXIMAUX
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80 consisteamontrerque,pourtoutcode
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82 {Sinonilexisteunmotz2X00etdesmot
Page 98 and 99: 84 s0 CHAPITRE5.MAXIMALITEETCOMPLET
Page 100 and 101: 86 5.2Unemethodedecompletion d'alph
Page 102 and 103: ni. uniformementsynchronisants(cf.T
Page 104 and 105: 90 {jt0j
Page 106 and 107: 92 {Sijsj>jt2jalorsilexistes02Atelq
Page 108 and 109: 94t0).Soientz002Xetp2P(X)deuxmotste
Page 110 and 111: 96c1 CHAPITRE5.MAXIMALITEETCOMPLETU
Page 112 and 113: 98 Preuve.Nousallonsmontrerquepourt
Page 114 and 115: veriantt1wt2=xt1r0t2y. 100contredit
Page 116 and 117: 102 CHAPITRE5.MAXIMALITEETCOMPLETUD
Page 118 and 119: t0=yxyxetV=(t0A\At0)nAt0Xt0AnAt0X+n
Page 120 and 121: 106 CHAPITRE6.CODESSYNCHRONISANTSMA
Page 122 and 123: possibilite. P(1:::k01yxyx)etdonck2
Page 124 and 125: montrequek=k. 110Nousallonsmontrerq
Page 126 and 127: 112Onajkjjkj
Page 128 and 129: nisantmaximalY.DeplussiXestuncodera
Page 130 and 131: uneintersectionnonvideaveclafamille
Page 132 and 133: lelemmesuivant: preservelasuxitedel
Page 134 and 135: 120{Supposonsx2Xety2U.Pardenitionde
Page 136 and 137: lemme7.1.1assurequeby0n'estpasS-com
Page 138 and 139: 124 V:l'ensembleVestsuxe. Nousavons
Page 140 and 141: 126 (b)Ilestclairquelecasfz01;:::;z
Page 142 and 143: 128Parconsequent,leseulcasquipeutap
Page 144 and 145: doncn>m. P(x).Deplusonab(ab)n2P(x),
Page 146 and 147: 132impliqueb:(ab)jvj:v:a2P(w:a),c'e
Page 150 and 151: 136obtientdonc Ainsi,ona CommeUestb
Page 152 and 153: maximalitedeUdansFcirc. 138 Lemme7.
Page 154 and 155: 1.LRab(z)\Ujzj16=;. Extendab(z),cel
Page 156 and 157: 142etu00=k+2:::h0:((ab)m0a)1,d'ouu0
Page 158 and 159: 144 L'ensembleUestdoncmaximaldansFc
Page 160 and 161: 146 Preuve.Consideronslasuitedes M(
Page 162 and 163: 148 Onaalors: 9n02N;8y2X;8(s;p)2(y)
Page 164 and 165: 1509n2N;8y2X;8(s;p)2(y)\(S(X)P(X))
Page 166 and 167: 152 CODESETINTERPRETATIONS
Page 168 and 169: 154 [Bru91a]V.Bruyere,Codes,Dissert
Page 170 and 171: 156 [NC92]J.NeraudetM.Crochemore,As
Page 172 and 173: 158 CODESETINTERPRETATIONS
Page 174 and 175: 160prexe,8,68,75,122 rationnel,38,1
Page 176 and 177: 162 INDEX
Page 178 and 179: 164P(X) S(X) Ensembledesprexesnontr
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