Docteur de l’universit´ e de Reims Champagne-Ardenne

Optimisation des codes LDPC pour les communications multi-porteuses

TH` ESE

pr´esent´ee et soutenue publiquement le 29 juin 2004 pour l’obtention du titre de

Docteur de l’universit´ e de Reims Champagne-Ardenne

Sp´ ecialit´ e G´ enie Informatique, Automatique et Traitement du Signal

par

Val´erian MANNONI

Composition du jury

Pr´esident : M. Philippe LOUBATON Rapporteurs : M. Maurice BELLANGER

M. Emmanuel BOUTILLON Examinateurs : M. Lo¨ıc BRUNEL

M. Didier LE RUYET M. David DECLERCQ M. Guillaume GELLE Directeurs de th`ese : Mme Inbar FIJALKOW

M. Georges DELAUNAY

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(N = 9, t _c = 2, t _r = 3)

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ê õ

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P A=!9Ag"S!MpvaA=’<^A=gXf

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ï#îí … 7R Xf s  ê7ð M<GDA<M7a =!@AE HLLBv!rH# #a###a####a###a###a# ô ó

ï#Zï ‰DA=!‹“A!:L:"ASy A=BH B! =!M"SM:Ln#KLY#K Y#ILA„>:

u _04k+1

^{P A=!}

Im { X _k } = 1

^l

C _04k+1 = 0

=IGHA=B F A!

Im { X _k } = − 1

^l

C _04k+1 = 1

^#-‰DA=!

:“=A!BLBL=! lWU!7 !MQMByLY#KLY#K n#LA>?

u _04k+1

^{F A!}

Im { X _k } = 3

l

C _04k+1 = 0

^{nGHAa F A!}

Im { X _k } = − 3

^l

C _04k+1 = 1

#g#a###a####a### ðì ï#Kõ ‰DA=!?‡“A!ELE"A=SE "AyQ "y! !M"ME:Ln#KLY#K n#^L=Aß>:

u _04k

^{P A=!}

C _04k = 0

nGHAB P A=!

C _04k = 1

#P‰DA=!“=A=!LL! l6! =!M"SMa7Ln#KLY#K Y#

LAE>?

u _04k+2

^{P A=!}

C _04k+2 = 0

^{YGDAa F A!}

C _04k+2 = 1

^# ###a####a### ðê ï#îô

…

LUS!>:

…

caŽ+TU >: M<M7aL=amJZL= SR7A!@L=a7Ym#Km#KLn###a### ð í

ï#

ð

TC#KLY#K n#L^ "ƒ! = P !^L=’!7>W[= HMuAD;GHAR!7^[lS^ =L=M!M( P A=!

mJKM>: "EL ì ?k =!a"AS@g<6 P A=!@mJKM>? EL ê ?k !qL! "lI@<#n#a###a### ð ï

ï#> f6 = ! 7;>:RS =!‹A=‡_` O M<<M‡L=\ _Œ <S L!=_`!qELnJKA 7

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ï#Zë t+Q ADS yL3aL=q<L=3A=[=.=

D _K f _{N pmax} , f _n

:_Π<S :L=A: >C[!

n

^L=

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L7@L=lM7

f _{N pmax}

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f _n

^{P A=!}

n = 2

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ó

w =!M"<R XL=6L=lM

f N pmax

<

f n

P A=!

n = 10

?k“=A=!BLBQAS=I@@<U F A!

n = 100

?k“=A=!aLL! lSI@<#:#a###a###a###a####a###a###a####a###a###a# ð ï#

ê7ì

f6 = ! 7;>:RS !@A=a_` =< EM<<MaL=a?!M F aE_`!MGDA=aLAV=m#:###

ðó

ï#êê ‰D F<S!aLYJZ>: lA=LaL=A7nfUTa‰b„<6](! "K=D3eU# #a####a###a###a###a### í

ï#ê í f6 = ! 7;>:RS Ô !A=‹_` O ÔMOSML‹\!M F ‹ß_`!MGHA=7‹LAÔV=p<

!7 !MQSR EL =O 7"!r>W[F O3= HAL=@L=aL "M77#‡### ï

ï#ê ï TC#KLY#K n#L@>?7@ ! 7Q"L=A‹7=- !GHAnJKA "RS f

s  ê7ð ADSl

MBnGHAa X =! 7;>yR E !@AE>?M=aL=aGHAR!aL=lSM6vaA=7# ë

õ#

ê

HLMRS L=A ! [=€>:aLYJK >?RS L=

N _p

ž Aƒ7 HL7ž6bIT] … P "A!

N _p

^7R

=AD;‹E =!€#g#a###a###a####a###a###a###a####a###a###a###a### óì

õ#îí ](! “3LYJK!!MA!lMX "[DDA| =!š>?>?R LXç ! [[—MXLYJK!!A!š P A=!:

7YfT‰b„L Q3a F<!LYJZ>: lA=La>CAl =Ma !3ôC7q6M"7>?Q3! =!MQM# ó ô

õ#Zï ](! “3LYJK!!MA!lMX "[DDA| =!š>?>?R LXç ! [[—MXLYJK!!A!š P A=!:

7Y

nm

L= "3 P<S!aLYJZ>? —AL=a>WAlS M =![3M7">?7"3! !M77"M#ç# óð

õ#Kõ … >: =! XL=6 P!_` !>y73A!3W7=IfUTa‰b›L73 DLUbITU] … D>:MB =!

minP e

N/ >:M‹ P A=!C†7pegfUe@v <?!M7A!7#+bIES†LAÔ>? ?L=† HL†7q

N = 1024

#a###a###a####a###a##a####a####a##a###a####a###a###a## ó õ#îô … >: =! ÔL=? P7!_` !>y7?A=!‹V=@

nm

L=? HL7:bITU] … D>:M‹ =!

minP _e

N/ >:M‹ P A=!C†7pegfUe@v <?!M7A!7#+bIES†LAÔ>? ?L=† HL†7q

N = 1024

#a###a###a####a###a##a####a####a##a###a####a###a###a## ó ë õ#ð ](! “FLnJK!!M7A!lM [D7HA‹ =!p>?=>?R "‹LA‹A- P "A!ga7=-fUTa‰b‡L Q@

P7<!aLnJZ>? =lA=La>WA=l Ma =!3íï?3M"7>?"6!7 !MQM#@##a###a###a### ó ë

õ#> ](! “LnJK!!M"A!lSM@ [D7HA !^>?=>?R :LA?7A F A!^@7](! "!=Hue›L "

F<S!aLYJZ>? lSAL>WAlS M =! ê íC3M7">:"6! =!M"M"#¢###a###a### óó

õ#Zë ]/!_` !>:=BLA„7 HLybIT]

…

>?M: F A!:V=(fTa‰b|<WLAç HL: " D>:M

F A!a?7=(egfepv²AM !LnJKACS!>: ›cŽ+T A!a:7=^fTa‰b(#

bnW!L=>?7"UL=B7 HL="B

R = 1/2

NP?SWLA>? L=B7 HLW

N = 65536

^<

>W[=!aLnJKlM! 7q

L = 500

#ç##a###a####a###a###a####a###a###a# ê7ìê õ#ó ]/!_` !>:=^LA7 HL@bIT] … DS>?Mp F A!+pV==]^! QK=H(e›<(LA7 HLp " D>:M

F A!6y7/e’fe@vŠADMU !ULYJKA=!=>? ‡caŽ/T£A!6WV=/](! "K=D

eU#PbnB!L=>?7"6L=B HLB

R = 1/2

NSBL=AX>? UL=B HLB

N = 65536

<6a= >W[=!aLYJKlSM!R 

L = 500

##a###a####a###a###a####a###a###a# ê7ìê

minS N = 1024

õ#

êê

… >? ! yL7( F!_` "!>:=u"S!3 HL3 D>?M !

minP e

<gA== DS>?M

=!

minS

P A=!@a7nfUTa‰b(#=bIWSaL=A>? 3L HL=a

N = 16384

^{#Ô#a###a# ê7ì ï}

õ#ê í ]^! “=’LnJK!!MA=!lM† [DSHA› =!W>:>:R LA›Au F A!WšV=’fTa‰b L= "

F<S!LYJZ>: lA=L>WA— M !@ï ì M"7>?Q! !M"M"#bIC =qSR

ADM3A=f

s  ê7ð #’##a###a###a###a####a###a###a####a###a###a# ê7ì õ

õ#ê ïœ]^! “=LYJK!!MA!lM@ [HA? =!^>:>?RS ?LA?A P "A!+@V==]^! "K=Hue›L= "

F<S!LYJZ>: lA=L>WA— M !

ê

íWM"7>?Q! !M"M"#bIC =qSR

ADM3A=f s  ê7ð #’##a###a###a###a####a###a###a####a###a###a# ê7ì ô

õ#ê õ ]9!_` !>y^L+ HL=+ D>:M+ =!+6>:>?RS CL=ACA P A=!9p7DfT‰b

L=a?7a ¡ˆA=A=GHA? 7R „Xf s 

êVð

AMOL=aA=+ ¡ˆA

>W[=‡L=‡ =qSR =L=‡L FM!Q†SAELYJKA=ˆ HVR xLˆ[l

QAM7#-bnW!7L7>?"aLC HLW7q

R = 1/2

NF:SCL=A>: L=W7 HLW7q

N = 65536

<3a >C[!LYJKlM7!R =

L = 500

#\###a###a####a###a###a##

ê7ìð

õ#ê ô ]9!_` !>yuL=( DLu D>:M( !(a>?=>?R yLA:A= F A!(7]^! "R

=HpeßL==’67g ¡šA=6A=GHA6 7R ‹Xf s  ê7ð pADSM6<pL==’7AP ¡

AE7>W[=L=7 RS 6LBL— FM7!"S@S73A73LYJKA=B H7R "XLB[l

QAM7#-bnW!7L7>?"aLC HLW7q

R = 1/2

NF:SCL=A>: L=W7 HLW7q

N = 65536

<3a >C[!LYJKlM7!R =

L = 500

#\###a###a####a###a###a## ê7ì õ#

ê7ð

]^! “=HLYJK!!MA=!lMg [DSHAW =!/>?=>?R CLAWAQL/’7L=!’LYJKA=(!>?

"ˆcŽ+T Q H7R \A!UC7/fUTa‰bZ?k ¡\mJZR;La[! !M"

! " P ! L7@= HA=LLaL= =M.@<#ò##a####a###a###a####a###a###a# ê7ì ë

õ#

ê ]^! “=HLYJK!!MA=!lMg [DSHAW =!/>?=>?R CLAWAQL/’7L=!’LYJKA=(!>?

"CcŽ+T²Qg D7R WA!Iu7=D]^! QK=H/e\?k ¡BmJZ;uL=9[=79! !M"

! " P ! L7@= HA=LLaL= =M.@<#ò##a####a###a###a####a###a###a# ê7ìó

õ#

ê

ëœt/Q "AD ?L+ F!_` "!>:=+L=AC7 HLp >?Mp =!+U>:>?RS LpU ! [=[=lM

LYJK!!A!@[==!a F A!@7=nfUTa‰bˆ F A!@ =AA=!pA=!LAE!R

r p

#B##a###a# êê7ì

õ#ê7ó t/Q "AD ›LBA=BLy QQ!7y P "A!B:7ufUTa‰b|L=A„7 HLy DS>?Mš

GHAaL=A DLegfUe@v‘7_` =< ELAE!R

r _p

#”####a###a###a####a###a###a#

êêê

õ#îí ì t/Q "AD :L^A^L@ "H!=@ P A=!+@7=](! QK=Hue›LA7 HL@ >?M

GHAaL=A DLegfUe@v‘7_` =< ELAE!R

r _p

#”####a###a###a####a###a###a# êê í

ê #ê

‰DA3LB7 "Q!B [DA6 =!3M<Q "AD XL=BL=—ML=37 HLUbITU] … LB!D

L>:"

R = 1/2

DS>?My !y ! !>?>yR |MV!X AšM<Q ADS |LXL

lSMN" = ! 7;>:R švaA=@ =!>?MO!M =!(>: 7rQ=@<’ Au = ! 7;>:RS

vaA=a =!>?M<S!M =!@mJK=_` !>yR E>CADA=#œ###a###a####a###a###a# í ð

ê #îí ‰DA/L=g "Q!7g [DSHA7/ !/M<Q A CL=gL=lMgL+ HL=^bnT] … >?M

A=’ ! 7;>:RS †vaA==3 =!>?M<S!M3 =!umJK=_` !>yR š>WADSA3 F A!

t _cmax

8 ê7ì Nï ì OUí ìì #ˆ###a####a###a###a###a####a###a###a####a###a###a# í!

ï#ê ‰DAL "Q7!= [DSHA !M<Q A ‡L=CL7lMLLA; DLabIT] … !M<

A!@L!7L7>?Q

R = 0.5

ADMA=!p7=YfUTa‰bˆ A@A!pa7Y](! "K=D eœ !GHA=a7  XXf s ƒõ?7q6ADM7#œ###a###a###a####a###a###a# ê

ï#îí ‰DAL "Q7!= [DSHA !M<Q A ‡L=CL7lMLLA; DLabIT] … !M<

A!@L!7L7>?Q

R = 0.5

ADMA=!p7=YfUTa‰bˆ A@A!pa7Y](! "K=D eœ !GHA=a7  XXf s  ê7ð 3ADSM#D###a###a###a####a###a###a# õ

ï#Zï ‰DA(L= "Q7!= [HA=( !(M<Q "AD yLL=lML=LA;? HL=’bnT] … !M7AD

!gL=6!=L>:"

R = 0.5

AMgA!’U7PfT‰b\ AgA!’U7=F](! "K=DpeU#

bY@ =RS @ADM7p!MAlQLYJKA= D7R L[=lpL=U>y€7!a8C [DS!

›>? 7rQ=šõ[lC =!W Aƒ F !qSA0?8EQ "!WGHA=yš ">W[!šL=y[=lW>yR;>WA=>

P A"6~<! AMBM<SM“;M8 ð @<# ##a###a####a###a###a####a###a###a# õ

ï#Kõ ‰DA9L=’ "Q!7’ [D7HA/ =!/ = ! 7;>:R "vaA=’L’L7AD;B HL=/bIT] …

!MA!+L=g!=L>:"

R = 0.5

AM+A!+g7DfT‰b† =A+A!/p7](! "K=D eœ !GHA=a7  XXf s ƒõ?7q6ADM7#œ###a###a###a####a###a###a# !

ï#îô ‰DA9L=’ "Q!7’ [D7HA/ =!/ = ! 7;>:R "vaA=’L’L7AD;B HL=/bIT] …

!MA!+L=g!=L>:"

R = 0.5

AM+A!+g7DfT‰b† =A+A!/p7](! "K=D eœ !GHA=a7  XXf s  ê7ð 3ADSM#D###a###a###a####a###a###a# ó

ï#ð ‰DA9L=’ "Q!7’ [D7HA/ =!/ = ! 7;>:R "vaA=’L’L7AD;B HL=/bIT]

…

!MA!@L=!7L7>?"

R = 0.5

AM@A!@a7=-fUTa‰b„ A@A=!pa7=Y](! "R

=HaeU#-bn =R "AM!MAlQaLYJKA= D7R ‡L=C[lSLC>y€!8

[D=!^?>? 7rQ=@õa[l( =!^ "Aƒ F !qA=a?k@= >W[!pL@[lS^>:;>WA=>‘ F AD""

~<!B AMBMOM“;M8 ð @<#²#a###a###a###a####a###a###a####a###a###a# ó

õ#ê ‰DA+Lp ""Q!"p "[DDA+ !+M<Q A L=gL=lSMgL7+ DL(bnT]

…

Lp!=L<

>?"

R = 0.5

>?M^ F A!^p7fTa‰b(NpV=](! QK=H(e›<upV==egfUe@v#

bn?7 =lML7=AD;E>?7"M3 =!@A=f s õ?3=LGHA=Ma"S!a !Q€7# ê7ìì

õ#îí ‰D7AgL6 "Q7!=6 [DSHAg =!’M<Q "AD †L6L=lM6L’7 HLpbITU] … L=3!L=<

>:"

R = 0.5

DS>?Mu F A=!(V=fUTa‰b(N"7=P]^! QK=H’eß<’37egfUe@v

!GDAaC7 R AMaA=Xf s 

ê7ð

#=bIC7 lMaL=@7AD;8C7"!M7

f s  ê7ð 3=LGHA=MaQ!a !Q€7# #a###a###a####a###a###a###a### ê7ì õ

õ#Zï ‰D7A6LB "H!=B [DSHA73 =!3M<Q ADS XLBL=lMBL6 HL=6bIT] … LB!

L7>?Q

R = 0.5

DS>?M? F A!7pfUTa‰b^N^‹7=]^! QK=H:e <y‹V=

egfUe@v#IbY7 7R =ADSM!M7A=lQBLYJKA=: H7R "„L:[lL=?>y€!

8\ [=!Ô>: 7rQ=‹õ‡[=l? =! A=q P !A?m8‡H !GDA‹‹= >W[=!‹L‹[=l

>yR;>WA=> F AD"@~<! AMaWM<M6“;M8

ð

@<#bIW7 =lMUL=gV=A;š@=LGHA=M

Q!a !Q€7#-#a###a###a####a###a###a###a####a###a###a###a### ê7ì

/ / Æ eg!A=l3f@L=LlS_/egWvaA

/ ägfih fU !l=>?LM<Q7 FMa !epmN … H !="N5=j=PNO6w3llkKôm

/ È ä + eglz"!7QL … DLLXX HL=ARS

/ égnpo eg=!qr](=W‰D=_6<r

ä#4 èq+ … HL7LXcU! " =IŽ!GHA7<rTUl XAl =<;=

è / TUlSnf6AL= :eg! "L=7

èr / ¶ TUlS-–L yeg! "L=7:h9!!!

èq+ ¶ TU!<SAlSlƒh9 =

»ts—»(s—æts L=lMLa =! [=[lM

1 é h " t+;H7L7LE]+!qIw P "aõ

1<u È ¶ t+;H!azq_` !>yR Eh9!_`!

4 é Æ

Ž/LX](! !>?>:[WvaRf6!!r

vEÈOé 1 h  xw 6zE]utuw_` !>y3bn D7YfU!7?i6<zyp !{=

È 1 n zQ!_`M7!73t("!W‰Qr>W[F

à sÃs»(s =LM FL"<6L"SGHA7>?"6L![=AM

à s¿(s—»(s =LM FL"<6A_` "!>?M>:"6L![=AM

’èé ä bn y@qT6=lrE]/!lrH … .=

+ é \R;>WA=>¨f¯]/ qS! !

+ |, HLA=R ELYJKfU>? =lAL=a s A=L=!RA=!

+¯È+ # Al =az= A6XAlS BcAD =AD

minP e

!lS€!LYJK >?R !@?>?=>?R "L=a? ! [[lSMaLYJZ!!A=!@[!

minS

!l€!LYJK >?RS !@>?=>?R L=AA

+‘Â ä AlOQ

…

HL

+}+~n 1 >:˜7R X7‰DGHA!t(!! !

#|4 èq+ c!q= =IŽ=!GHA=<rET6— EAlS <;

# n n º¹ºDÊÄ cU!L!3‰HSRS6‰D7<

hyä èq+ w3=L > … HL=TUl XXA=l af@7

h n / w3 = P !U‰Dn8:e’!Al

¶ ä + h9! … HLLX HLA=R

¶ 4 è h9!_` !>:MLŽ= A=!!TU!€<

€q+¯é / é ¸9·¼ºD»|·P¹¾PÇÀ ÃÄ å d3_Œ !>?lrEX U]9 yp7!_`AIeg_+](! ="R E[=LX "!l>

Å ènn TUlS9‰DA[=7![P7!bY=

‚ 4 ƒ ! :Ž= !

ä Ç É Ä7ºQÅYÄ7º

b^JK7 !‹L‹ >?>CA=7R =†DA>?M7!GHA=† =lA=A=\L‹ != =‹!M<Q A šSS= 

"GHA7‹L‹L!€!7†LM7=‹GHA\‡ "lEAœ—Q7A L\„MM "\>: [NpL\mJK"S!O

O?[Q™?L=šXMM< DA>?M7!GHAyS!!!# … <S†!M<Q "AD ÔYJZA!l? =AÔQ !AÔ=

L=M<Q = P7>?"y<y‡7 7 D ßL=‹7!A=l: >? =<;:<y>?=RA!M? F!>:<S":LnJK>? ="7!

j:77>?Q3L7 !l=>?3LaL=M HLLa A =A P7!_` !>y"7#

… [P D > S= GDAaCM=M!ML "ADQ7A;†[P =pš!>?@LLM[=l@LnJK_` !>:R ‹ADS

OaLW“[lMA>:"SQa =!8š>?~>:NFš >: <;lMCL=qrqS€>?LBS!>: "U>yM"

7>?QylH7A !GHA=?L= P7!_` !>y7#(d6L=CSS= y P!>?<<"yL!M P "L!E8ˆ

[F yšˆ>? HL=ARS cŽ+T ? c!q " =3Ž=!GHA=OrßTUl XA=l O;T@O#’‰DA = P !S"

L=’L=M[lS’LYJK=_` !>yR :S!€g>? F !qSQ’ !gL=@S!>: †A=!uV=PM7<_Yš_Œ!MGHA= A

7† >C[=R<"@@"!_`M!=p"S!Ur>W[F 7ND7<UL=!=€!U7qL<QDAUmJKA=UL=’S=

=!lM7M(LpU!7S!<<A=#"bnp ! FpL@[=@L@U>: HLA=R ycŽ+T (L=pLM7 A F!

C!A=!gL=[=L=UL=A†V=-‹A=†7>W[=UL= A=ƒ[==Lp !€7NSQSM7p<L=M FD

L"7pA!g7GDA7pmJK=_` !>yR ‹p!>?#b^JKAR "‹LYJKA=† =!MO“;U<rGHA=U F!>:< !

LnJZA!!pJK !q= =lML=p=_` !>yR =’HMA=M@ =!pS=7AL7p A=q P !Ap F!>:<S"

-LYJK "[D=!pA=y7lMa P<S!>: P !S"S6 AD3‹M<<">: QMj:77>?Q =!Ž+Ž/ha#

TU3 A=NQmJKM<S P3LYJKMQR yp !g>? =>?7"g>: —“M =AGHAnJK3 6>WAlS !uS=

7AL7 A= F !q7A !AE>: Hj:QV#Ž= !qSL= A36QS7N3>? HL=AR "

caŽ+T­ Q:M<M‹S ? P "A!CL=lQ!?S=L=!L=WGDA‹\!L HL— FA= ÔHA>:M!GHA=?mT@fe@<N/

L=— FA XS!!7qS!WL=W:M7M< \DA>?M7!GHA5?mTp–aeuƒh<@<NFU!M7AD; H7AD;6“+8yADaLM[=l

? 3zq]utuwbnfi@qí!@ON A7pS= p;Ta‰b(N=<?###

… FL="@LpL=U >C[!A=@ =7R "7ND7p7=AD;†LU ! =QR ‹ !M"7"@L7p "A

[L=8„!€‹_Œ[\! F !q=8 [!A=lV#p]/ A!†7‹7AD;-Ng‹ P!_` !>y‹L‹r€>:

caŽ+TŽ?k7ES!>?7UL<AD;LYJK7!!A=!6[!I@3! Q6!€ F7<M =!U[lUM7>?a8?!Q!

A=ƒ[==La_` !qS>?QBRSMHA=Ma>? F "B !amJKAR ˆLnJKAˆ HL= !!<A=!LnJK!!A=! P7!q

_Œ !>:"?

…

DLLšcaŽ+T@<#

…

JZ’L==(3 "S<;HGDA7JK!l’pS!LpS€JKq8RƒL!A=

L=AL=M<Q = P>:"3LnJKA=S!A<SA!†ž<L= HDž3L=a HLaV=n P "A!@p!>? =6cŽ+Tç#

T67 A/GHA=p‰D== B“;3 ê7ó õ"ë3/ F!_` !>:79>:l/LYJKA=(!>? WA=!IV=H[=!AlSMN

aS=aLS!SA=!@3LU! AQ!@L7@ HL==YGHA=aL3 !lS>?7@LaLM7 HL= 

7MGHA=/ F!>?OQaLYJZ=L!<SC>:l#-f6= NFL7AD;\_Œ>?7L HL= !!<SA!LnJK!!7A!

=AL= M7RS !3 Q6L=AM73 =!3A!3 F!_` "!>:=6R7=Q6:7 =7lML=W‰D= YN

8Q !’(A![F ƒ7 HLuQ! HL=Alg =!’eg!! AYNDvAD;:Ogh@lS>:oq>:-k”•mP ê7óó ïB<g’ HL7

bITU] … ?mbn y@TU=—rE]+!lrH… 7.=&@p"! DLAlS !6va7!3

ê7óð íkõ ê

mI<6!L=M AQ!q37Al

=!3.=r

et al.

^{kKô ð mn7 ê7óó} ô<‰D 7>:

et al.

^{k” ð mI êVóóð #}

bn( HL7(bIT] … QuL( HL7^ !!<SA!(LYJK!!A!( A=L M7R !^[M^A!^L7^>yR!7

8QM!—“=7RS šL3 =!lM3L3_k[6L=—M#Hd36L’ "S![A ’’ =A’—“7RlQgL =A ê7óó ô

/7uL=’bYA=["r

et al.

kKôô•mHGHA=Q "/Q! HL=Al/<+ =!>?M<S!Mu9 DL/bITU]

…

ÿ*.{“–QKYùîÿúFSü # …

9L7!!

"+ F A!9 != =’7!<M7!GHA=uL=’ !MQ!9L’>?7A!/ P!_` !>y79GHA’/ HL79!M7A!

8š7 LlS ‡GHAmJK!!M"A!lSMC l@oqA=L7A>?QaS ""#Ff6“=‡LC[\S= !<!!M7A!lMN

wS=!L YNYd6![=—=

et al.

^{k” ì N ê} m^ ">: "!MCGDAYJK+M<<l P "[=L !M7L! ">? F !q>:"

LC DLbITU] … L:Sš“=š "!CLšmJKM<< PšL=šLM7 HLš<GDAšA=lƒ’MOSl?_` <S ›L7

=!>?€<S!’LM“="g6 HLUbnT] … # … <p "!l>:NQ= >?>:M

“˜•Dù™YþkÿGDKPXûQúaûQúPüSÿŒþ–“

NQ F!>:<p !

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›

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7A="3@ S @O@ ADS !lS>?GDA@=L P7[8CA=[P " >: !M= "‹LA

L DA>:"V#fU !€7(Q !^"S! HLA=l(7^ HL=(bITU]

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L BL‘D‘Nuù9ûpœÿ*F{“Q!Yù—÷K<ÿŒþ–“ # …

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…

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7 >?>:† ADSgL‹ =!>?M<S!R ÔL7 HLq?kQ! HL=Al?L‹S= =l!

ê

@<#+b(JZlr‹L HL=

bIT]

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! HSND38WLYJKA=a>?M7 LA7=I8mJKM>: "Y#

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LL H7A>?7"3<3"! DLA!M">:"6 F! F<lQ7@L=U!"AD;_`ADA=!7#

¬ e£¬­«¯® ] ­ ® ¬ «x± ±„µÔ¦ ‡ ®‡° ³^®ˆ®„³We ‡ ®

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bY7 DL6bnT] … ?mbn y@qT67lrE]+!lrD… 7.=&@<N=Q! HL=Al3 =!6va!3

ê7óð íµkõ ê

m¶kõQímI Q

L=+ HL=+ !!<7A!+LYJZ!!A=!+ A=L M7R !+[=M+A!+L+>yR!^83QM!—“7R Lp =!lMgL=

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7 >?>:†C!AD; L= ƒ rH[ 7

et al.

^{këëKmp7 ê7ó} "ô<:L‹h/=!µk”

ó

m@

ê7ó ë ê

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7 >? =<;lMyLnJK7 HL=":OWL=B>: 7rQ=B>yRM!7B=M!7B F A!BWLM DL!# … Oy =!"S€

L=A!7!poqAGHAnJZ8amJK"S! HLA=< yL7+A=![F HL7( !ueg!! "AYNHvAD;:<’h@=l>yoq>y|k”•m#QTUA:_Œl

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et al.

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et al.

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L=†!7S!<7#/fI"?7N+h/=!CLµk”

ó

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7 HL6 "A!>:>?Q FM6 HL=6LBh/=!7#zI ! P NP !7NA=B<;H7 XL=BmJZ !l>:BL=

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7 "!"#^ [F!¸kë"ô•mN(7JZ =ADrHQšA!:?S!A;ßLEh+7!N(LMOQ ˆL7y !l=>?yL=

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mJZ !l=>?:ž>?DA>\žR#QTU3>?~>:NQ>? "S!GHA3mJZ" !l=>?6Ž= !¹y@!Lƒ[=.=Ty@!Ly<gmJZ" !l=>?

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va!’Jî =!"<lp8mJZ !lS>?6L3 ! =" yL3! 7rH6L6]/7!ºkð ô•m#HTUH<r:Og\.=r

""A=lW"! HLA=lAQ! " "[=!WLU HL=ULva"!UQ7WGHA=UU [D77"

L=>?7A!† P!_Œ !>:†GHAnJZQ7† HL=š[==!kKí ì mƒ#gbnA["r

et al.

kKôô•mU "‹GDA="E8çAD;

Q! HL=AlWB>:R!BLQM!—“7R çL: =!—MyQ:A=:L![=AD „LA„= >W[=!:L‡ž ê žWA!

"š< • A|A!yy7 =yLX„>:R!XLX =!lSME= DƒA=_` !>?# … y DL7Nu P7My HL=

bIT]

…

!!MA=!7N’ "‹>? Q!Mš A=š!q<† L=l =šGHA A=šLM<Q7 F! =šL=š7

S lS!U A=g F!_` !>:"SgGHA=6p HL7g!M7A!#TQOr

et al.

^{kKí ì} m- "VN !7ND !M77"MUL7g HL=

L=†va!W!!MA!W< ›[!QšGHA=WC " [DSHA›Ly>:A=!C P!_Œ !>:

GDAa3>?A=!pA=![F ? HL=@7 HA=7#

‰D A=3!S737 L=l 6GHA=B A=3LA! 3MQ>?7"UL6BS= =l!"N6 HL76bIT] …

O;[F"^AW =M= >?€=’L=’Am# … gAmN [DSHAC P A=!9A=(SgLg[ Hg“==N! !M"g’ =A

_k[a! = P !q3"=n8C[!A=l

E _b /N ₀

ADƒL=A=pL=AGHA=-C =! [=[lMaLYJK!!A!p[= H7q3!<S>?Q DA#(]+!? MGDA"RN+A@LYJKAß HL=bIT]

…

F A!!\! =!M"S!A=‹>:A! P!7"‹L=

GDA=lSMš› =LM!" ›MV!q !W! = P !?8XX7 lM†L‹‰== Y#

…

š =M= >?€7† A

LnJZ[P "!L†M<MU [!qHMU =!@va!pL==akõQí•mF F A!gA†7F[=!Uqr>?M<S!GHA=6 !GDAYJKF"! HLA=l

3 HL=!M7A!OU?AlSaMOM SRSM =!3bnA["r

et al.

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… <: [=7!qRS ç=Al:M<SM:M=M!M: F A=!A„!=L„= >W[!:L:7AD;„8E"!M7:[!N

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8[=!AlULL=l_9[aQAl?megfUe@vÁ@ak” ì m #

bI7 "![=AD C=M !GDA@a A=^=—“=7RSlQ@L= A=(GHA3.=r =! AD"GHA@7( HL=ubnT] …

P7ADQQ =! HS=!3y7 =7lMBL=C‰D "QWAXL=M HLB " D>ymN7"UL=WwS=!L YNFd3!q

[=—=

et al.

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Am#i3 S pGDA<S = ! H<C F!>?pLYJK [!pA!pUV=YegfUe@vNLp DL@ !M77"S"

LU F!_Œ !>:=3!€76 =! HS=ULWš>?lSWL=‰D= Â?k P "A!6A\!L=>?7"aw<; ê •

íDNPC>?A!

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… WS= =l!B!: =!M"SM A=3y_` !>:BA=lQ#Pi3 "A3Q! HL=A! "7N ADULnJZ[P !LYN6 HL7

bITU] … N A@ PM7—“=! "@ "!@@L— FM7!"7p SR =p !>:M<!"6r PL HL=S?… _#O

ê #ê

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ê #ê #ê

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… <M<SAL"N FMM<H AD "‹L=L7lMN7q3[=MaA!pmJZlrL=@L7lM@La ! [[lSML7

L— -M!"Sa>:? =! =7"B !aLAˆL=M DL=#

…

? =! HML=M? P!>?<S!†L=L=M<!>?7!! l

L>:"UA-LYJKA‹ HLabnT] … N<=-LYJK >?7!gA!pqS!A<SA!"#Di6 Ap !M"! 7ND=A=lN

mJZ = 7RS E<3mJîL= SR EL>?MO DL3AD;7=AD;8?>?M>: !#

… 6S= =l!"NQ ! >?7"g[[ ! GDANHB P A=!u"M7!~<gLnJK"! DLA!@ A=u7’ ADgM7q

!^8UU[P =g >: !M= LgmJZ D>:R L+7 HL^bIT] … F A!/79!=>? (caŽ/T

A!/7DM7<_=7C_`!MGHA="#i6 A+H +M"7>?Q^ !MQMgAgr"S€gL/L=— FM!"9!"AD;

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8š>:M>? "!5?k lMO_(LL >y P<S!@3L= " A =!M"S! aAl!RlQL==

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Ê

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ÉÕÔ¸Ö

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(N, t _c , t _r )

^A HL=g[ Dg=M7!p[!gL=g "A7A!

N

^LM“=

=!A=>:RS!LHM!—“=VR EL= =!lM

H

N=!7AN=LL>:

(M × N )

^Q

M = ^t _t ^c _r N

^#bY

>? S3LB HL

m

=qS"UEA=MGDA=L=

N

[lS3_Œ"UAX=7>W[BL

M

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L? "S!™L? =!lM# … ^C!L=Al !a†!R „AlQ: FM?MGHA= ˆL? "S!™?L

=!lM?Ÿ

Hm = 0

^{? ê #ê @}

bn?= >W[=!?L=?[lSLYJK=_` !>yR „

K = N − M

<W?!L=>?Q

R

^{LA„ HL=?}

R ≥ 1 − ^t _t ^c _r

^#

b(JKM"lMQ7="pLAš_Œl@GHA63!WL6B>:!

H

YJKp =gM7!>?Qg =Y#DbIB>:RS!

H

@ >? F M6L=

t _c

^{ž ê žp !’ =N}

t _r

^{ž ê} ž@ =!’U<pL=6˜M! g P A=!( A=’7pADS!gMM>:"7#

]+!g MGDA"RNHS=GDAUMM>:"@LA†>? @LU HL=6 =!q FU8

t c

MGHA= gL=67 "!™"UL6 !lMU<

S=A@Lp^M7GHA=RS ^( ">? P "M7pL

t _r

^[l7# ’r FpL=p HL=pu = PM@ HL=@bnT] … {“Q!Yùÿmú # d3ˆ HL=?bITU] … W !aA=!M7R ç =!q"B8‹mJK=>C[?La DLbIT] … L=MO“==a =!

3 =!>?€<S!

(N, t c , t r )

#=f3“=ELnJK [D7!A=a>:!7!AN=3 =!>?€O!

t c

O

t r

L= lQ"U~<!

POl@L<"@ A=A!pL=A†>: L HL=

N

AD!>?Q@L=lVN >C[!Lšž ê žL=@C>:RS!

H

L<!~<S!aL=amJK !L=!aL=

N

ADS™3GHA=aLmJK !L=!aL=

N ²

^#

… lƒL= "A7NI A=WQ "W! =!M"SMyE>:!šLy "S!™šLy =!lMyLnJKAç HL=šbIT] …

(N = 9, t _c = 2, t _r = 3)

#‰ !L7>?Q3q

R = 1/3

O3 = >W[=!aL[lS@LnJK_` "!>:RS 7q

K = 3

^#

H =



 

 

 



1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0



 

 

 



? ê #îí!@

d3? DLbITU] … FADu~<S!@M"7>?"’! !M"M@! =GHA>:"RN7<@!7 !MQSR :’ F<

M7U l@"! =ULUh+7!ak”

ó

m- A† lRND A=g!M>:>?"RND! =U_Œ< !7Okõ ó m #d3‹! =U_ŒO !

7qWAˆ! ?[= =!qSl? Q="BL7AD;r FaL=? DAL:ŸY DAL=aL?L= M7aOB H7AL

_Œ <S 7# … 3LA;†r P7L= HA=L Q6!M3Q!AD;E =!L=[=!S=7#TU373LYJKA=

7 HL6bIT] … NH’ DAL’L6L =M7u! !M77"Qg3>: gL=3 HL=3<p’ DAL=u_` =< =7NHGHA=Q

8A;-N! =!M"S"3gHM!—“=VR =@La !lM"#T=@AlSN@= HAL=@_` =< =7! "6 !

= >?>:MB HA=LL=: Q!™7 "A„ H7ALBL: !lM7#Ibn

i ^{eme `}

= HA=L„L:L =M:OW

j ^{` eme}

^HA=L

L=‹ Q!™ ": "7<M? =!?A‹[=!S=‹p<:A=>:":

H _j,i

:= ÔHA=m#^bI“=A=!

ê #ê

! !M77"a! =a_Œ< !7YL=A HL=bIT]

…

!M"A!LMO“Y !@>yR!7

H

^{L=MO“==al? ê #îí!@O#}

vérifications de parité mot de code

c c c c

c c

c

c ₁ c _{2 3 4 5 6 7 8 9}

M N

×tØÙÛÚ ê #ê ›

! =X_Œ< !3LnJKA| DLbIT] … !M"A!

(N = 9, t c = 2, t r = 3)

LX!L=>?Q

R = 1/3

^#

bY"! =_Œ< "!+LnJKA‡ HL=?bnT] … L "A=C! !M77"SRS ‡! =GHA=W!€aLL=<GHA=

L=š!A<SA!šL=š7C DL7#/z’7q?MQ>?7"?>? F !qSQL=† !CGHA=š!7 !MQSR =C!

=GHA=@ Q F!>:@aLM<H F>?7"OC>: DAD!LnJZ !lS>?7@L=L=M HL|kõ ó m #=i3 "A

L=M<S7! 3a F "3L? ! DS==a< n#

d3@@ A^"MM!@L^7 HLubnT] … U=Al@M<M@"! DLA—p !^bnA=["r

et al.

^{kKôô•m N7JZl}

L= HL=abITU] … ÿ*.{“–QKYùîÿmúSü # … <SC!!M7A!—MCW!L=Ala !UA=CL![ADS \= \A=_` "!>?CL=A

= >W[=!‡L= ž ê žXA=!‹E<

•

AxA=!‹E =L=‡Ô>yR!ˆLˆ "S!™ˆLˆ =!lM

H

^#

… 9!O"8yL![A7!3!!MA=€!>:"aW >C[!WL=W[!S76 "7<MAD;L— FM!"3r P

L=‹ H7ALA=!C! =†_k< !m#(‰g\L=![=AD ÔL=‹mJK!!MA=!lSM‹:[ßS N^ !?

7 HLUbnT] … !!MA7!3 =!M"S"ULB>?7A!6 F!_` !>:=3GHA=3!M7A!N<U7N= F A=!A=

7 >? =<;lMLaL=M HLaM7GHAl""S#

6[—A7>?Q†mJK!!M"A!lSMXLYJKA= DLbITU] … † PM7—“=MX !:L=AD; F lr™>:Ÿ

λ(x) = P tcmax

i=2 λ i x ^{i − 1}

^<

ρ(x) = P trmax

j=2 ρ j x ^{j − 1}

^¡

λ i

?k!7 n#

ρ j

@n+3 =! F !q WL=/[!=S/ =<SM

AD;= HAL=nL^L= =M(?k! n# DAL=YL=^ "S!™.@-L^L=!M

i

JK8RL!( =<M7nH

i

^[=!=S

?m! Y#pL!M

j

^@<#

t _cmax

^{?m! Y#}

t _rmax

@š! =!M"S\SA; L=\7 =<; >:;>:L=‹ DAL=†L=

L =M7?k! n#p7 "!™".@<# … ELA;œ F lr=™>?E "XME =!ˆ!=L>:"

R

LAx DLN@<X

! CQ7JKM7!lŸ

(1 − R) P tcmax

i=2 λ _i /i = P trmax

j=2 ρ _j /j

#/z’M7">:"?AA7’LYJKA!

A=6! =!M"S †LA=6L6mJK!!MA=!lMUGHA=P =U†mJKADRS †L6L=AD;šAD!7’ F lr=™>?UŸ

e λ(x) = P _tcmax

i=2 λ e _i x ^{i − 1}

^<

ρ(x) = e P _trmax

j=2 ρ e _j x ^{j − 1}

¡YN7<S^_`

e λ _i

^{?k!7 n#}

ρ e _j

@-! !M"(p =! P "!q L@= HAL=pL=L =MÁ?k!7 n# "!™.@’L=L!M

i

^{?m! Y#}

j

^@<# @L7AD;†! =!M"<R Á?

λ _i , ρ _j

^@’<

?

e λ i , e ρ j

@p "3M7 =!@!7R =A=lQŸ

e λ _i = λ i /i P _tcmax

k=2 λ _k /k ρ e _j = ρ j /j P _trmax

k=2 ρ _k /k

bn HL=3bnT]

…

Q6Y !>:M<!M =!Á?

N, λ(x), ρ(x)

@?k "A?

N, e λ(x), ρ(x) e

@{@’! !M77"SQVN

! F<lQ7>?QVN’ç =AA=!šLA >: ‹L\ HL"Nu7†L![=AD †LySAD;|L= =<; A!š7

DAL=’L6L =Mg<pA!’g HA=L’L=37 "!™"ž?k6 HL=3!MA!’M<S"/oA6Aš7g =!A=!È@<#

ÜÝzÜÝ–Ü ÕšÖ^×uÒØXÙ6ڄ͋ÕÛØÞ(ØRÐKßFÝ|ÎFÐ"Ñà^Þ(ÒØ

b(JK"! HLA=< L6 HL=6bIT] … rM>:GHA=3 !6.=rˆkKôëKm/:M<MB ! >:"U>? SlQM

=!:ˆ!MLA=< LEˆ ">? O;lMELnJK DL=#ui6 AH!! =:L:7<E<S GHAnJK A=:L

!=L! =!€<mJK>? ="<R ‹L=p DL@bIT] … N<SU!A=<A=!U F!>:<@LYJK7 DL!p†7 !q""

La[F A3[F AD6!A=<A!aLA7 HLbIT] … L=MO“==a =!@>yR!L =!lM

H

^#

bn ! [€>:aL=mJK DL=L= DL3bIT] … 6LLM<S!>?=!>? 6L DL

m

^{8? =!q!LA}

Q<SA!

b

Q="CW[lSBLnJK_` "!>:RS ç<CL=yE>yR!yL=?QM!—“=7RS çL=y =!lSM

H

^AW

Q!"SULMO“U !gmJKMGHAR ? ê #ê @O#‰D

G

7qW>:RS!U"MM!R!LA† DLbITU]

…

8B7 HL=!N

A=3Q BŸ

m = Gb,

^{? ê #ZïT@}

i3 S GHA=aYmJK E"S! HL=Al3OaMGDA=R "LU? ê #ê @<N= A [ "!7R EAl":Ÿ

HG = 0,

^{? ê #KõR@}

bI?>:M<= HLGHA=n>W[=~<!? =A3>? B P A=!7 HL!3qUL "!A=!A=>:RS!LnJKD

HL

H ⁰

!=A!WA PM7!A=!B !U =7R "LA lQ aLvaAUA=!6y>:R!

H

^# >:>?

! =!M"SM@A!^“=A=! ê #îíN>? ’L@ DL

m

’ !(L=M A FM@:A =!qS@rM>yRGHA=@ Q<

=QU6[l6LYJK=_` !>yR

b _n

^N

n = 1, ..., K

N=OUXA=B =!qS7 ""U3[=l6LB!L= L7

r m

N

m = 1, ..., M

N(8‡LM<7!>?=!#(t(Al !?L"S—“=7RS ßL=‹[?ßADRN+@L=<QyMEL 7A=!@@[lS@L!L =L==# … 7 P=L="RNmJKADRS LA‹ =lQ 3LvaA=@A!p>:RS!7!A

H

P7!>?<A=>?QLYJK [DS!pA=>:!LnJK DL=>:lm!A=!A FM!7A!

H ⁰

^QaA

>? F !q<"L=lSMULšž ê ž!=L="3F ">W[!ULYJK FM!RS @8CO F7<A7!g !@LmJK DL=L

mJK !L=!6L

N ²

!gGHA=37A!g 7A!!7"’L=!<S’"JK8RƒL=!6uA![F ƒ7 HL’ ""pA6 ">? O;lM MV!

O (N )

^# <SaM<S PaLnJK DL=! !M" "!mJK7 "QM="6 ! n8mJK>: QSR !RGHA=aL DL3bIT]

… #

]/ A!C!MLA=!š<Sš >? =<;lMšL=AD;› = ! DSW ""M<MšL=M<Q = PM7#/bIX !>?€!š †8

Q!=L!a>:RS!LpQM!—“7R ":L =!lSML@>y€7!38!MLA=!@7< >? =<;lSM3 !uGHA

=LW7qa[=MCA!Uy "!A=< \LYJKA=W>yR!CLYJK7 HLC7>?lmS!"A!Cš AU7!A

P "[=#

t(›yGDAu "!yX !>?€7!š = ! DSNI= AW F ADH ACLnJZ[P !Lç !B7!"AD;çL

‰D =>:

et al.

k”!DNFôï•mIGHA=n =7"U8: 7M7!L3 DL6bITU]

…

L_Œ[=3 =AA=!g?k A

HL=.@<#ubnX HL=E!M7A=lSQš‹ !š DL=[=XQAX >: <;lSME=M7!# … P7L=QVN(M<<"

1

1

0

r ₁ r _M b ₁

b K

=

0

0

M K

x

×tØÙÛÚ ê #îí

›

t(7 HL=L3 HL7bIT] … #

L= =MUGHA=UBSUL=A†[= HLU<=GHA=U A=ƒ HL=UgS!€p F<lSUL<""<LA‹[ H!M7A=lSQVN

7 P7!_` !>y7LUqr PL HL "6>? [P "7@GDAa73 [DSHA73QAE HL=bIT] …

SL=!L LE>?~>: =A7A!#ud3=XAD!E = ! DSN("! HLA=lE =!y\.=r

et al.

^{kKôëKmN( =}

8ˆ_` !7!?‡>yR!L‹QM!—“7R "ßL= =!lME8‡~O!rSM>:RSGHA?m —yL!<>:"š DL=[=I@

"JK8RL!3GHA=mJK=>W[=L=u7 HL’’!!"g !(3_ŒlgGDA3>:R!

H

lgL=_Œ !>?3>?—

S!"A!@=GDAL=M!lg =!(a“=A=! ê #îíD# ñ <"’L= =MGHA3a>:RS!3L =!lM7qg!A="N"<S

!!<S Ô"!"l:A=‹ >? =<;lM‹LnJK DL=‹=M7!#(i3M7=>? =N+†7=L!"N/ÔM=M!mN

GDAGDA P7!q>?>?7A=lQ7AEL F!_` !>:77#

T67@>?M<S HL=@La!MLA=< LaC7 >? =<;lMaLYJK7 HLaA=Aa "S!"SA=!p_` !>:

L=Xç>:RS!LX7 "!™XL=X =!lMX ""‹M<MXM7">?7"‹LM<Q7 FM7#’T=k”ím NgwS=!L |O

d6![=E=ADS"L=p>yR!@7>?lmS!A=!7p F A!p= HL=!#

…

@>yR!7p "3 [D7HA@A=l

GDA>:"a =!UL=W>: P!>WAD<R =UL7U<aL=U " =7NFCGHA/ F!>?OLC =!qQ!U

=! =!M<ML=>:RS!7!A#QbIa7 >? =<;lM3LYJK= HLLM F=Ly !uLYJKA=: =!>?€<S!

g

^!LAQ

mJKM77!qy"!‹\>yR!7‹>?l!=A!‹!MAl<"‹<:A=‹>:RS!š!A=!?k F A=!CGDA

y7 >? =<;lMWMV!I@<#-bny =LC =! HS=NQ! HL=AlWL==JkKíïKmN7 C8y S!A!WA=

>yR!7LYJK7 HLS!"A!BGHAI "l6: A3!7A P [=# … 7n6 F<A=M GDA=QVN

"!pL=A† =lQ LvaA7N!lS€!La!‘=" 7l˜GHA=- a8C lQ S!@ADS A!pLA›ž ê žGHA=F ""Q

A=! JK>W[=?L?‹=?<BL=?‹7 =??>? "L\ž ê žV#Ibn >? =<;lM:!MAlS"?W !

7

O (N t ² _{r moyen} )

^"¡

t _{r moyen}

! !MQCW= >W[!W>? 7rH\LEž ê ža !6# … FL"VNPL=A_ŒlL=

C P7!>WAD<R †L=p =p !pLmJKM<S FUL !A< "†LUW>yR!

H ⁰

NDpL=AD;š>:M< DL =!M"S"3mJK "HQ3L=a F!>WA!@[=l38?mJK"M!A!LA>: 3L=a HL=#

]/!6 =M7GHAQVN A=6Q 6S= ILYJKADS!3:>:M< DLBLW.=rEGDAI W8: =qS!A!

A=>yR!7L= =!—M

H

qrM>:RSGHAª?k>:lm!A!A= PM!A!"N … _#u“A! ê #îí!@<#pT=š

O yAl"NQ A=u"! DLA! uA=M7M! yL= =!>?M<S!R yL=u HL=’bITU] … GHA=

Aa =7>?Q3?qS!A<SA!aL7@7 HL73bnT]

…

rM7>:RGDA7#

ÜpݖÜÝGá

ÍyÏÖpâpà6×[ã`ÑÏ"Ï!ߗä9ޒàkÎYÏQÑkÐKßÔ×uÒØÖ^×uÒØXÙ3Úç͆Õ

bn‹L! D "‡L77 HL7abIT] … ?GHAYJK7?‹M<M =!M"SMlLAaAE:"SL€a !

GDAC7=/aS"S=MC< • "A\SR ==!C<aL= GHA==LX A=[lLA‡>? aL=C HL=C !M<

Q"U73>:~>?73 =! !M<M6SRGDA7# … F=L="RN= !GHABB7=I7q8:>:M>: !B< • AX=

S !N/ F l L=CL FM!QB<AD;›L† =<; Ô8mJK"M!A!CL=A >? ?L† DL†7JZQ€!

~O!:Aç =!>?€<!!€BEå=A7"V#It(„O F<RNn(S=GDA:[=lWLA„>: WL:7 HL:=Q lC =BB>?~>:

=! =!M<MBSRSGHA=BLAç7=(u>W[yQM!"CLyA’ D!BAˆSAD;„Ly =<; çL_

_ŒM!"R#bn\r FˆLˆ7=AD; F A!GHA=E A= =!M"S A=ˆ >: xL=7 HLXbIT] …

L==WCS= =l!ïX<õN/8EQ "!WC7=AD; cŽ+T M7<_`C›_`!M7GHA=N9 "VNnA›L=AçL=M 

LA=!N/ Dq<R ==!7#+]/ A!BS!C >: D†L†<† ƒSR !lM†L=A›V=mN/= AQ "M<M

>?7My8\Q! HL=A!‹A=‹ !>:M<!RS ß A=?M=M!L=? DLybnT] … FM

BL‘D¹Nuù@ûœÿ*F{“æ

Q!Yù—÷K<ÿŒþ–“ kKíõN ðì mƒ# … <† !>:M<!RS A7C F l =

p _i

8mJK"SM!A=!CLA ! =†_Œ< "!

LCJK>W[=L DALLL =MLL!M

i

^{#YbIy“=A=! ê} #Zï† !M"C! C_k< !/LnJKA HL=?bIT]

…

!!MA=!Nn^GHA èçBL‘D¹Nuù(ûpœÿ*F{“Q!Yù—÷K<ÿŒþ–“ ç^ F A!a !>:€<!aAl""?ŸYiž; ð N

e λ(x) = P tcmax

i=2 e λ ^(p _i ^{i )} x ^{i − 1} = ¹ ₆ ⁽¹⁾ x ⁶ + ¹ ₃ ⁽²⁾ x ³ + ¹ ₂ ⁽³⁾ x

^<

ρ(x) = e x ⁶

#YTa7 L— aa HL7 bITU] … ! "" =!>?M<!M7 !

(N, e λ(x) ^p(x) , ρ(x)) e

^A

(N, e λ, p, ρ) e

^NF76QOA!

λ ⁰

^O

ρ e

7 ""

! F<lQ7>?Q@p D:7"pL=p P lr™>?7

λ(x) e

^O

ρ(x) e

O@3H<A=!

p

=LGHA"@7p P l LL— -M!"Sa"! A FL? H7ALL?L =M7#ni3 S ^GHA=?[!=S!"B DAL=aL

L =M7aAD;\= HAL=L ""!™MOS""S!M77N-† P l ‡La H7ALL=C7 "!™aYJKEå=A

=aA=!a P!_` !>y7L=a HL7abIT] … <B+!†L = =M!?LW7!7 >? L

A=!C F l n#+i3 ?M">:"VN^GHA‹<‹ =!>?MO!R "Ô>? =GHA‹GHA=†? H7ALL=†>:~>?

L"!M DA F"U>:~>:Q =BL==3>: UL DLB<U =!7 MGHA="UB>?~>:Q =BLA

7m#D\!M67<6!7qS!< nNH "A’ F =’GHA=6<S3L=! šL=6mJK!!M7A!lM6 "lA=6A

[=U LU P "A!’mJK >?RS †LUWqS!A<SA!6L=g HL=pbITU] … = GHA=Mp8BA=U!L=6UL

7AD;‹O3 =A PM—“GHA7>?Q7AD; DSR ==!7#

(7) (7) (7)

(7) (4) (4) (2) (2) (2)

M N 2

4 7 taux de connexion

mot de code

controles de parite

c 1 c 2 c 6 c 4 c 5 c 3

7 4 2

Entrelaceur

0 1

(1) e λ (2) e λ (3)

e λ

×tØÙÛÚ ê

#îï

›

va! =B_k< !9LYJKA DLCbnT] … !!M7A!ULB!L=>?Qw<;

ê • íD#Fi3 A=UQ =S!M

a =! “YLYJK!!MA!lM !! P =L="3E_` =< EL =! F !q "@L= DAL=@L=aL =M7#

ÅÇÆzé ê}ë{ì

Ì*-.q2iíW')%^4ÎÐB%èÐ

Ê

,XÌaÐgÏ

ì

%èÐW%(4”,XÌaÐB%(4ÓÒ

ÉÕÔ¸Ö

Tu<S@< "YN" A’ u !MQ!(A?L=u "!l>:@ F!>:<S"uLM7 HL="@L7( HL7

bITU] … #TU€gGHA=6a<ULA†>? pL=6 HL=6LO"@>? F !qS"NH76GHA=PpA=6Qr P €6=M!

“=šLYJK [=!uL’ F!_` "!>:=’ "Q7=[7NQ6LM DL=UAy=uL=Ay>yR;>WA=>ŠL=@!>C[=

!M"UA67 >? =<;lM3! " š>? F !qS"6 F A!g P!>:<S!6A=6!M7RS † !GHA"#D]9 "A!g =7!g8

pr FpL=@ ! [€>:NQva!( !M"SaL=<kõQí•m=A=: !—>:@LLM7 HL=@lM!R_F A=ƒ DS>:

QyA„ >C[!: =q<"WLnJK PM!R = !a[=lV# … <C !l=>?š‹7AlS?M<SM:!<A„ !BVar

kKô ð m-OS=S

et al.

^{kõ ó} m-L=pU7L!UL=UBM7 !Lp! =7#b^JZ" !l=>?!MA—S"@7q !@7 HA A=@ AA!@ >:gYGHA= B‘D.B÷QQ÷þkÿGDKPˆûQúC’‘DKÃ"÷KP’ú ?k[F_9 ! =" A@j?k ! 7H=Q

L=ç ">?>WA==AMLmJK"S=‡!q—“ðk

ð

ôm*@y A [= 7 !

÷"ùQTDK<ÿŒþ

¼

ý:úXüDýCý:úæ BL‘DVûKYDÿkþ

?mA=>C ! HL=A<@<# … Oa !l>:W FAD~<S!A\ >?>:WA\ !l=>?CLW =! =QR \LW>?7"

A=!! =_Œ<S !9 DM#PbY73>?@S!lS"6 !3[=!<3LAX! = FADQ7"6~<!

l:L ! "[=[lM7N/ l? "!l>:GHA7L! F !qLy!>W[#^bY‹ != P†L=š

=! "R ˆL?! 7rH=7qBmJZ =7R "‡L=†!€L=?eprQa D7>:"-?k lA!SGHA=[l@

O?lM!RlQ>:"y“=ÔLYJK>:!C7 ! [[lSM ÷-B&Dü<þƒúSÿDKSÿ L‹S=GHA=†[=lV#+T‹7L!†LYJKA=

L=M HLaA!@A! a=@<r7?ka! =a6 !@AE![!I@g"JK8RL!aL=@a7@ ¡† "A

7^>?^ "uLM7 P=L=Q7NU_Œ< !RS ? H7@L^!€"^L@eprQ7^ L=AluA?7A=<;O

L= ! "[=[lM ÷aBEDü<þƒú<ÿGDK<ÿ L= DAL=@LL =Mgkõ ó m #

TUš!qSyVyA="†A=L=y7 HL†bITU] … NuX"! =E_k< !3 !M77"XLš<r7#

T=7 "LlS 7NnL=Aˆ_klWL:‹L=M FL=:<r7GHA:L=L=— FM!7"B>?7Nny7A=(<;O

L=U =! [[lM7 ÷CBEDü<þƒú<ÿGDK<ÿ YJZqa =AaA=!M# … P=L=QVNF AA\! =W7q!A; AšL=M<

FL7W7"!CL— FM7!"S>?!š_Œ["#-TUC =A7NF F A!U>:>:7!U<SCL=M P=L==W

7quM7!@L@77A!( AA=!^_` "^(>?7"^A!^([!=S(@GHA=>?€78A@ADSR

lSM!R—QaL=amJZ "!l>:La =! =QR EL! 7rH=#

é ÀRÇPæ9·=¾·Ä ÃÇ É »IºÊ"ÀRÇ3P· É Êº

… =GHA=alM!R EL=amJZ !l>:aL=a ! =" ELa7! 7rH=a6 >: P MaL=aL=AD;‹M<< PŸ

›

d6M<S PaL=>?B8oq A!L3>?3 !GDAYJK3 ="6 !@A== HALELL= =M = PM7

ûQ÷þ÷aB÷üü #

›

d6UM<S FULU>?8o A!gL7g>?7g !GDAYJKp ="@ =!gA†= HAL†LU Q!™ = PM7

’ ¼

úȒFñ<B÷üü

#

f6“=‡L_Œ7l!†7<A!CL=C7>?M>: !N-a>?77AMa7! "L7 ! F !qaLW!—

>C[=#pi3 "

v = log ^p(y _p(y ^| _| ^c=0) _c=1) = log ^v[0] _v[1]

\>?7\L\ !q\LYJKA= = HALœL\L =MO

u = log ^p(y _p(y ⁰ ₀ ^| _|c ^{c 0} ₀ ⁼⁰⁾ ₌₁₎ = log ^u[0] _u[1]

a>:aL=a !qLYJKA= DALL=a "S!™#

+jüº0ò-óOÇP¿IÀ†»IºD¼„ºD¼7¼·=¾ºD¼:¼7¿IÀ†¿ É É ÇYºD¿9» »Iº\»IÇ É9É ËQºˆô è ·Ä7·ˆæ9·¼7¼

bn !L=†>??8Wo A!A!= HA=L\L=L =M

k

^L=L!M

i

8†mJKlM7!R

l

^N->?

v

^{QQ 7rQM}

A=!@

q ^ieme

[=!=Sa6L =Ma =!!R EAl""|?… _#“=# ê #KõT@UŸ

v _q ^(l) = u ₀ + X i n=1;n6=q

u ^(l _n ^{− 1)} ∀ q = 1, ..., i

^{? ê #îô!@}

¡

u _n

^N

n = 1, ..., i

N!7 !MQ"U6>:Q"UL=6 H7AL6LB Q!™3Q 6<

u ₀ = log ^p(y _p(y ^{k | c k =0)}

k | c _k =1)

! =!M"S ! F !q’L=p!>C[= [=!qQM3: !q3LA:7=m#"f a !>?€!

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2 = Y j m=1;m 6 =p

tanh v m ^(l)

2 ∀ p = 1, ..., j

^{? ê #ð @}

"¡

v _m

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m = 1, ..., j

"6>?7pQ=Q3L= HA=LLL =MpQ 7#

c y _k

k p(yk/ck=0) p(yk/ck=1) U = ln 0

U U

V 1

2

. . . . . .

×tØÙÛÚ ê #Kõ

›

zA=qS!R ELAEL=RS? =

c

c _i . . . _j . . . c _k

U V

V1 2

×tØÙÛÚ ê

#îô

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f f tv

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f f tv = log f f tv[0]

f f tv[1] = − log tanh(v/2),

^{? ê #>!@}

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f f tv[0], f f tv[1]

= T F D 2

v[0], v[1]

^¾

c

c _i . . . _j . . . c _k

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V

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1 2

2

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ê

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ê #ð

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w

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? >?>:aL=U A3>?6>?QS"6a= HA=LLL =Ma "LM!M?Ÿ

w _k = u ₀ + X i n=1

u ^(L) _n k = 1, ..., N

^{? ê #Zë@}

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ˆ

m _k = 1 − sign(w _k )

2 k = 1, ..., N

^{? ê #ó @}

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¼

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êVð

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L=’S!€^[F ^ >: ! ">?^7"!@A=!^ >: <;lM^O(7A!( P!_Œ !>:7#i6 A( P AQ ^lS!^mJ-.µÀ

- P=ÿD!Sý?ù™Ã . Dü<þ À D M ú.Yù 0/¶À æø÷üúûX÷"ùQTD!SÿŒþ ¼ ý L?Ž= !!

et al.

GHA+ !M""NF F A!a HL=

!MA7!

(3, 6)

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