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ii2.1Denitionsetpremieresproprietes....................16 2.3Codesadjacents..............................29 2.2Untestpourlescodesadelaid'interpretationni...........23 2.1.1Codesetsous-monodesadelaid'interpretationni......16 2.1.2Premieresproprietes.......................19 CODESETINTERPRETATIONS 3Theoremedudefautetcodesnon-interpretes Introduction...................................45 2.5UnecaracterisationdescodesdeFdif..................37 2.4Fdifetlesclassesdecodesclassiques..................32 3.1Theoremedudefautetadjacence....................46 3.2Theoremedudefautetdelaid'interpretation..............52 2.5.1Caracterisationpourlescodesnis...............37 2.5.2Caracterisationpourlescodesrationnels............38 3.4Lecasn=1:lescodesnon-interpretes.................55 3.3Unexempledecodeadelaid'interpretation4.............54 3.4.1Denitions.............................55 4Surlescodesmaximaux 3.4.3Enveloppenon-interpretee....................58 3.4.4Deuxexemplesdecodesnon-interpretes............62 3.4.2Codesnon-interpretesetcodeslimites.............56 4.2Maximalitedanslessous-familles....................69 Introduction...................................65 4.3Existenced'uncodemaximal......................70 4.4Equivalenceentrelesdeuxnotionsdemaximalite...........72 4.1Maximaliteetcompletude........................66 4.5Completions................................74 5MaximaliteetcompletudedansFdif 5.2Unemethodedecompletion.......................86 4.6Tableaurecapitulatif...........................77 Introduction...................................79 5.1Codesadelaid'interpretationnimaximaux..............80 5.2.3Completiondescodescoupants.................101 5.2.2Completiondescodestrescoupants...............88 5.2.1L'approcheclassique.......................86 6Codessynchronisantsmaximaux Introduction...................................103 6.1Completiondescodessynchronisants..................104 6.3L'ensembleYestuncode........................106 6.2L'ensembleVetF(X)..........................105 6.5Codessynchronisantsmaximaux.....................113 6.4LecodeYestmaximaletsynchronisant................113
7Codesdensesmaximaux TABLEDESMATIERES Introduction...................................115 7.1Codesprexes,suxesetbixesmaximaux..............116 7.1.1Codesbixesmaximauxdenses.................117 7.1.2Codesprexesmaximauxdenses................122 iii 7.2Codescirculairesmaximauxdenses...................128 Annexe7.2.3Uestuncodecirculaire.....................136 7.2.2UneproprietedesU-interpretations...............133 7.2.5L'ensembleUestmaximaldansFcirc..............139 7.2.1Un'estpasuncodemaximal..................129 Bibliographie Index7.2.4QuelquesproprietesdesensemblesdutypeU[fyg......137 Notations 145 153 159 163
Page 1: U.F.R.DESSCIENCESETDESTECHNIQUES UN
Page 6 and 7: iv CODESETINTERPRETATIONS
Page 8 and 9: partagelebureaupendantdeuxans.Pourt
Page 10 and 11: viii engenetique[AM95].Lebutdelathe
Page 12 and 13: admettentchacuneuneversiondutheorem
Page 14 and 15: egala1:lescodesnon-interpretes. xii
Page 16 and 17: 21.1Monodes ciativeetquipossedeunel
Page 18 and 19: 4primitifsitousseselementssontprimi
Page 20 and 21: 61.3.2Ensemblesrationnels Lafamille
Page 22 and 23: 8Proposition1.4.3Unsous-monodeMAest
Page 24 and 25: 10 codesadelaidedechirageborne,code
Page 26 and 27: 12 jacentess'ilexistei2[0;n]etj2[0;
Page 28 and 29: 14 CHAPITRE1.PRELIMINAIRES
Page 30 and 31: 16 erreurdetransmissionappara^tdans
Page 32 and 33: 18y1 CHAPITRE2.CODESADELAID'INTERPR
Page 34 and 35: delaid.CommeXestuncode,onax6=". 20
Page 36 and 37: 22 Preuve.SoientYBetZAdeuxcodesadel
Page 38 and 39: 24(ii)Ilexistei>0,j>1,2S(X),x1;:::;
Page 40 and 41: 26 {Supposonsyju2S(xi).Posonsalorsv
Page 42 and 43: 28 dem^eme,onpose Posons CHAPITRE2.
Page 44 and 45: deX+(d). deX(b)ouencorelemotcdestla
Page 46 and 47: adjacent.Parl'absurde,supposonsX\S(
Page 48 and 49: 34 {Enfait,lescodes(1;2n),(2n;1)-li
Page 50 and 51: 36 {Unautreexempleinteressantestcel
Page 52 and 53: plementaires: Exemple2.5.1Lecoderat
Page 54 and 55:
40x1g(0)=1g(1)=2 CHAPITRE2.CODESADE
Page 56 and 57:
42 (x1:::xf(q))12Xcequiimpliquerait
Page 58 and 59:
44Leschema2.3reprendlesprincipauxre
Page 60 and 61:
quisontinclusesdanslafamilledescode
Page 62 and 63:
construction: centemaispasd'exhiber
Page 64 and 65:
acelledec,parhypothesederecurrence,
Page 66 and 67:
52Eneet,onaU1(X)=P(X)nf"g=fabab;abc
Page 68 and 69:
ni: Theoreme3.2.2LabaseYdel'envelop
Page 70 and 71:
56 CHAPITRE3.THEOREMEDUDEFAUTETCODE
Page 72 and 73:
Proposition3.4.4UnmonodeNestextr^em
Page 74 and 75:
pretee.SiunmotdeXadmetuneX-interpre
Page 76 and 77:
qu'ilssontenfaitdescodesnon-interpr
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64 CHAPITRE3.THEOREMEDUDEFAUTETCODE
Page 80 and 81:
maximaldanslafamilledescodes?Lefait
Page 82 and 83:
Codesprexes,suxesetbixes Nouspresen
Page 84 and 85:
YFtelqueXYonaY=X.Evidemment,cetteno
Page 86 and 87:
mauxnishormisA,c'estenparticulierle
Page 88 and 89:
estuncodebixemaximal,nonmaximaldans
Page 90 and 91:
coupantestmentionneen[Per84]:fanbnj
Page 92 and 93:
78 CHAPITRE4.SURLESCODESMAXIMAUX
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80 consisteamontrerque,pourtoutcode
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82 {Sinonilexisteunmotz2X00etdesmot
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84 s0 CHAPITRE5.MAXIMALITEETCOMPLET
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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
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98 Preuve.Nousallonsmontrerquepourt
Page 114 and 115:
veriantt1wt2=xt1r0t2y. 100contredit
Page 116 and 117:
102 CHAPITRE5.MAXIMALITEETCOMPLETUD
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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 148 and 149:
134bad0bd1 as0 dibabCHAPITRE7.CODES
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
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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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