Computationally Efficient Error-Correcting Codes and Holographic
Transcription
Computationally Efficient Error-Correcting Codes and Holographic
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A6J$@H";^9]AJZH"J$@:;?BI?>G>eU^68d8d>GI4UhH";« e >k`b`n¬ H"7$AIkH";?AI^9]>#U ;?B>eiA%Aj@+H@%|g9_Sk>dH"J$@riA%Aj@2`b9_U^;?>GJ%9]Ji+A"M 8VYH@%|g9_U^AI#ZO+9_S=B$H">k`5"9]F$U^>G!I |HW`]6$H"7%`]>dH@%|g9TSk>Z@g9]I?>#Sk;^9]AJZH"J$@r>GJ$SkA6I4HNi>G8d>GJ%;hMI=A" 8 "AKHNJ >k9]i>GJ%7$H"6# 8 8VeigAKA@MI^9]>GJ$@UO>GI=>#@g9<;=B q I^9]iBK;WZCeH#|g>lCE6$ A;#Z"OyHgH"I?;?>GJ LA%>G²ZOQH"IkSkAKU 9]h9$Z "AJ `]>k9<Ju 7>GI?i$Z 5B$H"Jiu 6$H+H"J$@56k9<>r>GJi$ZH"J$@QhB>2¹I?>G% ;?I?6>MI^9]>GJ$@UeH"I?>D;=BAKU^>DBA;4HN>DFI^9_@%>9]J VAg6I{HS=B%9<>G|g>G8d>GJK;kU HdS=`]AKU^>HNJ$@2`<A"|g9]JidMH"89b`]V+;?B$H";hi"9]|>#U8d>DU^;?I=>GJi;?BH"J$@r8HN>#Uh8d>eB$H"FF%Vw & AOA8ZjCeHg@Z H"J$@+CeH"I?I=>GJ \ ¥ ¥ w ¥ nI?I?AI?uSkAgI?I?>#Sk;^9]JidSkAj@%>#U ¥ w hB>eF6I=F$AKU^>A"M FI?AKANMU ¥ w ¼ 5;?I=6$Sk;?6I?>eA"M ;?B%9_Uh;?B>#U&9_U! ¥ w «EJr9]JK;?I=Aj@%6$Sk;^9]AJ;?Ad>GI?I=AI?uSkAI?I=>#Sk;^9]JiSkAj@%>#U ¥ w w ¥ ¬9]J>#H"IeSkA@%>#U ¥ w w «eU^V%8dF;?A;^9_SE7A6J$@U" ¥ w X hB>DSkAg8dF%`]>kK9];^V+A"MSkAj@g9]Ji # $&%('*)+*,-.*,0/ w ¥ JK;?I=Aj@%6$Sk;^9]AJ;?A>kjF$HNJ$@%>GI{iIkH"FB$U3 w ¥ w ¥ hB>>kjF$HNJ$U&9]AJA"M/I4H"J$@%A8igI4H"FB$U w PjF$H"J$@%>GIESkAj@%>#U w ¼ «-U&9]8dF%`]>>k[H"82F%`<>4 w ¼ w ¥ 5>#fg6>GJK;R9TH`n@%>#SkA@g9]Ji w ¼ w >#Sk>#U?U&9];^V+A"M /F$H"J$U&9]AJ! w ¼ w ¼ m H"IkHW`b`<>k` @%>#SkAj@g9]Ji5 w PjF%`b9_S=9];DSkAJ$U^;?I?6$Sk;R9<AgJ$UhA"M >kjF$H"J$@%>GIEiI4HNFB$U6 w w ¥ PjF$H"J$@%>GIEiI4HNFB$UhA"M>G|>GI=VYU&9]G>7 w X PjF%`b9_S=9];DSkAJ$U^;?I?6$Sk;R9<AgJ$UhA"M >kjF$H"J$@%>GIeSkA@%>#U8 w Xw ¥ «³i>GJ>GI4HW`b9]#H";^9]AJ w A;?>#UhAgJr9<82F%`<>G82>GJK;4HN;^9]AJyH"J$@ >kjF>GI^9]8d>GJ%;4HW`I?>#UR6%`<;kU9 w w ¥ 5A82>e>kjF>GI^9]8d>GJK;kU: ;<=+,>) ?@BAC,D,2)EGFH,I)+J*,-*)E(F,I,- K ?LK ,0=+M8*,-/ ¼ w ¥ OA;^9]|"H";^9]Jid;?B>DSkAJ$UR;?I?6$Sk;^9]AJ ¥]¥ #¥ ¼ ¥X ¥ ¼ #21 X \ ¼%¥ ¼¼ ¼ ¼$& ¼\ =¥ X X¼ X X N0O nI?I?AI?uI=>#@%6$S=9<JirSkAj@%>#U ¼ w ¼ nI?I?AI?uSkAgI?I?>#Sk;^9]JidSkAj@%>#U ¼ w PjF%`b9_S=9];¹AJ$U^;?I=6$Sk;^9]AJ$U ¼ w X 5A82>e;?BA6iB%;4UAJr9]8dF%`]>G8d>GJ%;4H";^9]AJzH"J$@2M6;?6I?>lAgI? KFMK )' + @/ w w ¥ xL6;^`b9]J>A"Mh¹B$H"F;?>GID w ¥ ¹B>#S=HN7%`<>H"J$@+|g>GI^9ba²H"7%`]>SkA@%>#U w n9PH"J$@+ I^9]|HNI^9_H";?>SkA@%>#U w w ¥ hB> 9]I4U^;E5;?>GF w w h>#UR6%`<;kH"JK;kU w w ¼ mnI?>#U^>GJ%;4H";^9]AJS=B>#S?g9]Ji;=B>GAI?>G8U w w U&9]Ji >GGA6; ULhB>GAI=>G8 w w X 567uFI?>#UR>GJK;4HN;^9]AJ$U{HNJ$@+|>GI^9ba²SGH";R9<AgJ w ¼ «-U&9]8dF%`]>BA"`]AiI4H"FB%9_SFI?AKANMnU^VU^;?>G8 w ¼ w ¥ ¹BA%AKU&9]Ji+H>GVrFI?A7%`]>G8 w ¼ w «`]i>G7I4HW9_SGHW`b`]VzU&9]8dF%`]>DiI4HNFB$U w ¼ w ¼ «EI^9];?B8d>G;^9]k9]Ji+;?B>eiIkH"FBSkA"`]AI^9]JirFI?A7%`]>G8 w ¼ w « LAN`<AgiI4H"FB%9_SDmnI?A%A"M7 w h>#Sk6IkU&9]AJ: w w ¥ nJ$SkAj@%>#@Y9<JF6;kU7 w w U&9]Ji;=B> HU^; A6I^9]>GI{ I4H"J$UMAI=8 w X PtYS=9]>GJK;LFI=AKA"MS=B>#S?>GIkU w hB>DSkAN`<AgI^9]JiFI?Ag7%`<>G8 A"M hH"7$HW9Z AgI?;?JA"DZK¬¶>G|9]JZH"J$@5G>Gi>#@%V N *(+,0H*/ Xw ¥ «EI?>DS=B>#S?"H"7%`]>:SkA@%>#UhJ>#Sk>#U?U=H"I?VrMAIhBA"`]AiI4H"FB%9_SFI?AKANMU EGF H2MK ) ' ,% ¼w %¥ $& & && 10O %¥ ¼ \ \ \ ¥ ¥¼ ¥ ¥ ¥]¥#¼ ¥]¥ ¥]¥ ¥ X ¥ X ¥ $& ¥#¼%¥ #¥ ¼ >N ¥#¼$& >2O *N OQH";?B>G8HN;^9_SGHW`PU^;?6$@g9]>#UEA"M FI?A%A"MU{H"J$@ >GI?I?AI=uSkAI?I?>#Sk;^9]Ji+SkAj@%>#UEF%`_HWV>#@H"J+9]8dFAI?;4H"J%;EI?A"`]>e9]JQ;?B> @%>G|>k`]AF8d>GJ%;A"M{;?B>GAgI?>G;^9_SGHW`¦SkAg8dF6;?>GIdU?S=9]>GJ$Sk>w J.;?B%9_U@g9TU=U^>GI?;4H";R9<AgJZ h>Y@%>G|>k`]AF9_@%>#HUMI?A8 ;?B>GAI=>G;^9_SGHW`nSkAg8dF6;?>GIDU?S=9]>GJ$Sk>H"J$@H"FF%`]VQ;?B>G8¤;=A;?B>SkAJ$U^;?I=6$Sk;^9]AJQA"M>GI?I?AI=uSkAI?I?>#Sk;^9]Ji+SkAj@%>#U H"J$@FI?A%A"MUGw xL6I iA%HW`9_U/;?AeSkAJ$U^;?I?6$Sk; 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X §k2>GI?I?AIF$H";?;?>GI=J$UeA"ML;?B%9_UU&9]G>6J%9bMAI?8`]VsHN;DI4H"J$@%Ag8w+m/A"9]JK;4U¸9]Js;?B>riI4HNFBQ9]J$@g9TSGHN;?>+;?B> J%687>GI{ANM>GI?I?AI4U¦;?B$H";eHdSkAj@%>dSkAI=I?>#Sk;?>#@r9]JzHW`b`X§4g;?>#U^;4UGwExLJ>SGH"JzU^>G>D;=B$H";L;?B>>kjF$H"J$@%>GI SkA@%>#U{SkAI=I?>#Sk;?>#@+H"JH"8rH"k9]Ji"`]Vr`THNI?i>eJ%687>GI{ANMP>GI?I=AI4UGw Some expander codes of length 40,000 1800 1600 number errors corrected 1400 1200 1000 800 600 400 200 0 0.2 0.25 0.3 0.35 rate 0.4 0.45 0.5 M , #>?L ´687$>GIeA"M>GI=I?AI4U>kjF$HNJ$@%>GIDSkAj@%>#UESkA6%`_@zHW`]8dAKUR;¸HW`]LHWVjUSkAI?I?>#Sk;#w AI>k[H"82F%`<>Z ;?B>rF$A"9]J%;9]J;?B>r6FF$>GI`]>kM;dB$H"J$@SkAI?J>GIA"ME;?B>da$ig6I?>d9]J$@g9_SGH";?>#U2;?B$H";;?B>SkA@%>A"MEI4H";?> ¥ SkAI?I=>#Sk;?>#@+HW`b`A"M ;?B>DX§kF$H";?;?>GI?J$U¦A"M ¥ § & >GI?I?AIkUhAJ+LB%9_S=B+9];LLHUh;?>#UR;?>#@w < J 9]i6I=> u Zh>¸SkAg8dF$H"I?>L;=B>{F>GI^MAgI?8H"J$Sk>EA"MH|>GI4U&9]AJrA"M;?B>eU^>#f6>GJ%;^9_HW`@%>#SkA@g9<JirHW`]iAu I^9];?B8 9<JrLB%9_S=Bh>LHW`b`]AJ>GiKH";^9]|>FI?AiI?>#U=U 9]F$U h9];?B+HE|>GI4U9<AgJ9]JLB%9_S=Bh>L@%ADJAg;#w q >{S=BAKU^> HI4H";?> ¥ >kjF$HNJ$@%>GI{SkAj@%>eANM`]>GJi;?B §kw AIh>#HS=BrJK687$>GIA"M>GI?I?AgI4UGZh>EF$>GI^MAI?82>#@ §4g ;?>#U^;kUEA"Ml>#HS=BsHW`]iAgI^9];?B8 H"J$@zSkA6J%;?>#@Q;=B>J%687$>GIA"Ml;^9]8d>#U;?B$H";e>#HS=BzU^6$SGSk>#U?U&M6%`b`]V SkAI=I?>#Sk;?>#@ ;?B>E>GI?I?AIkUGw q B%9b`]>DHW`b`]A"h9]JidJ>GiKH";^9]|>EFI?AigI?>#U?U 9]F$UL@g9_@rJA;hU^>G>G8;?ADB$HW|>E86$S?B29]8dF$HSk;hAJr;?B> J%687>GILA"M>GI=I?AI4U;?B$H";;?B>¸H`<igAI^9];?B8ULSkAg6%`T@.}GHW`]8dAKUR;¦H`<LHWVU?~SkAI?I?>#Sk;#Zg9];{@g9_@+igI?>#H";^`]V9]J$SkI?>#HUR> ;?B>DS=B$H"J$Sk>D;=B$H";h;?B>HW`]iAgI^9];?B8¤SkA6%`_@SkAgI?I?>#Sk;{HiN9<|g>GJJ%687>GILA"M >GI?I?AIkUGw G , ? 1 0.9 0.8 probability decoded 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1700 < 1750 1800 1850 1900 number of errors 1950 2000 M , #>? ´2« IkH";?> ¥ >kF$H"J$@%>GIdSkAj@%>+A"Mh`]>GJi;?B Z w ¹A8dF$H"IR9TURAJsANM¦;=B>FI?Ag7$H"7%9b`c9];^V ;?B$H";;?B>U^>#fg6>GJ%;^9_HW`n@%>#SkA@g9]JiHW`]iAI^9];?B8 SkAI?I=>#Sk;4ULHi"9]|>GJJK687>GIeA"MPIkH"J$@%A8`]VS=BAKU^>GJ>GI=I?AI4U LB>GJQh>D@%AYHNJ$@y@%ArJA;eHW`b`]A³J>GiKH";R9<|g>DFI?AigI?>#U?U 9<F$U^HJ>GiKH";^9]|>FI?AiI=>#U?U 9]FQASGSk6I4ULB>GJ H|"H"I^9_H"7%`]>e9]J82AI?>DU?H";R9TUa$>#@+;?B$H"J+6J$U=H";^9_U&a$>#@SkAgJ$U^;?I4HW9]J%;4U9_U 9<FF>#@4w 4 J;=B%9TUS=B$H"F;?>GI#Znl>276%9c`_@ªAJs;=B>+;?>#S?BJ%9_fg6>#Ud@%>G|>k`]AF>#@.9]J¹B$H"F;?>GI ?; AySkAJ$UR;?I?6$Sk;dHU^V%8dFu ;?A;R9TSGH`c`]V.iA%Aj@vMHN89b`c9]>#UANM{>GI?I?AI=uSkAI?I?>#Sk;^9]JizSkA@%>#UD;=B$H";B$H#|g>d7$A;=Bz`b9]J>#H"I?u;^9]8d>>GJ$SkAj@g9]JiH"J$@ @%>#SkA@g9<JizHW`]iAgI^9];?B8UGw q >SGH`c`h;?B>#U^>SkA@%>#U 0 8: *$)1$ 1#*#)*!$);3&E7$>#SGH"6$UR>;?B>k9]I>GJ$SkAj@g9]Ji S=9]I4Sk6%9];4UI?>#U^>G87%`]>¸mP9]FF$>GJi>GI UlSkAgJ$U^;?I?6$Sk;^9]AJ2A"MU^6F$>GI4SkAgJ$Sk>GJK;?IkH";?AI4U m/9]F & & w J+5>#Sk;^9]AJ ¼ w ¥ Z"h> >kF%`TH9<JQLB%Vr;?B%9_UhI?>#U^>G87%`_H"J$Sk>9_UhJ>#Sk>#U?U?H"I=Vw AyH9T@v9]JsAg6I:SkAgJ$U^;?I?6$Sk;^9]AJ.A"M{U^6F>GI4SkAJ$Sk>GJ%;?I4H";?AgISkA@%>#UGZ h>9]JK;=I?Aj@%6$Sk>r;?B>rJA;^9]AJ.A"M{H"J **)* *340($;"1 $) 3WwD«EJQ>GI?I?AI=uI?>#@%6$S=9]JiySkA@%>dB$HUE;?B>dFI=AF$>GI?;^V;?B$H";9cM{H+@%>#SkA@%>GIDI?>#Sk>k9]|>#UH F$H"I?;R9TH`c`]VSkAI=I?6F;?>#@+SkA@%>GlAgI4@Z;?B>GJr9];{SGH"JQSkAI?I?>#Sk;R`<V+SkA8dF6;=>DH`_H"I?i>EMI4HSk;^9]AJ+ANM9];4Uh8d>#U?U=H"i> 7%9];4UGwEhB>eMI4HSk;^9]AJA"M ;?B>D8d>#U=U?H"i>7%9<;kUL;?B$HN;{;?B>@%>#SkAj@%>GIDSGHNJJA;{SkA8dF6;=>:URBA6%`_@7>U^8rHW`b`<>GI ;?B$H"J(;?B>MI4HSk;^9]AJ(A"M;?B>Q7%9];4UrA"M;=B>vSkA@%>GhAI4@;?B$H";+h>GI?>QSkAI?I?6F;=>#@9]J;?I4H"J$UR89_U?U&9]AJw J 5>#Sk;^9]AJ ¼ w Zh>{SkAJ$UR;?I?6$Sk;h>GI?I?AI=uI?>#@%6$S=9]JiSkAj@%>#U;?B$HN;B$HW|>E|>GI?VMHU^;n`c9]J>#H"I=u;^9]8d>e>GJ$SkAj@g9]Ji+H"J$@ >GI?I?AgI?uI?>#@%6$S=9]JiHW`]iAIR9<;=B8UGwhxLJ>DSkA6%`_@29]8H"iN9<J>e6$U9<JirU^6$S?BHSkA@%>E9bMAJ>E9_Uh;?I4HNJ$U^89];?;^9]Jid@H";4H ;?B$H";9_ULHW`]I?>#Hg@%Vd>GJ$SkAj@%>#@ 7KV+H"Jr>GI?I?AI=uSkAI?I?>#Sk;^9]JidSkA@%>w JU^6$S=BHU&9];?6$H";^9]AJZj9<;¦9TU¦JA;hJ>#Sk>#U?U?H"I?V ;?A+I?>G82A|g>HW`b`l;=B>>GI=I?AI4UeSkI?>#HN;?>#@ 9<J.;?I4HNJ$U^89_U?U&9]AJZ76;9<;89]iB%;e7$>H@%|"H"J%;4H"i>GA6$UE;=Af6%9_S=`]V SkAI?I=>#Sk;{U^A8d>eANM;?B>G8w Jd5>#Sk;^9]AJ ¼ w ¼ Z#h>h@%>G8dAJ$U^;?IkH";?> BA"s;=A{I?>#Sk6IkU&9]|>k`]VSkA87%9]J>h>GI?I?AI?uI=>#@%6$Sk;^9]AJdSkAj@%>#U ;?AE76%9c`_@ A6Ih>GI?I?AgI?uSkAI?I?>#Sk;R9<JiU^6F$>GIkSkAJ$Sk>GJK;=I4H";?AIhSkAj@%>#U±whB>eSkAJ$U^;?I?6$Sk;R9<AgJ$U 9]JQ5>#Sk;^9]AJ$U ¼ w H"J$@ ¼ w ¼ H"I?> X\ 5 9 4 9 ( ? "H J$HW`]AigA6$U/;?Ae;?BAKUR>9]J5>#Sk;^9]AJ w ¼ whB>AJ%`]VLHWVD;=B$H";h>L%JA"A"M²Ag7;4HW9]J%9]JiD>kF$H"J$@%>GIiIkH"FB$U A"M ;?B>Dfg6$HW`b9<;^V+J>G>#@%>#@YMAI;?B%9_U{SkAJ$UR;?I?6$Sk;^9]AJr9_U;?AdS?BAKUR>e;?B>DiIkH"FB$ULH";hI4H"J$@%Ag8w A76%9b`T@>kF%`b9TS=9];DSkAJ$UR;?I?6$Sk;^9]AJ$UhANMnU^6F>GI4SkAJ$Sk>GJ%;?I4H";=AILSkAj@%>#U¦9]Jz5>#Sk;^9]AJ ¼ w Z%l>e6$UR>DHdSkAJu U^;?I=6$Sk;^9]AJA"M>GI=I?AI?uI?>#@%6$Sk;R9<AgJvSkA@%>#UEH"J$HW`]AiA6$U;?Ad;=B>:SkAgJ$U^;?I?6$Sk;^9]AJA"M>kF$H"J$@%>GIDSkAj@%>#U¦9]J.5>#Sku ;^9]AJ w XwhB>DSkAgJ$U^;?I?6$Sk;^9]AJ$UH"J$@rFI?A%A"MUn9]JQ5>#Sk;^9]AJ ¼ w HNI?>¸HNJ$HW`]AiA6$U;?A;?BAKUR>E9<JQ5>#Sk;R9<AgJ ¼ w ¼ w !_* B5 /$!s(! ; 9_U/JA;H"JdHSGS=9_@%>GJ%;;=B$H";Ag6I`b9]J>#H"I?uU&9]G>L>GJ$SkA@g9]JiS=9]I4Sk6%9];4U `]A%AE`b9<g>{U^6F$>GIkSkAJ$Sk>GJK;=I4H";?AIkUGwhB>GV B$HW|>;?A$w Jz;?B%9_UU^>#Sk;^9]AJZ h>dh9b`b`l>kF%`TH9<JBKVw q B%9b`]>r;?B%9_UEMHSk;DLHUE;?B>28dA;^9]|HN;^9]AJz7$>GB%9]J$@ A6IDSkAJ$UR;?I?6$Sk;^9]AJZAJ>dU^BAg6%`T@v7$>H"7%`]>;?A+6J$@%>GI4UR;4H"J$@A6IDSkAJ$U^;=I?6$Sk;^9]AJQh9];?BA6;eI?>#Hg@g9<Ji;?B%9_U U^>#Sk;^9]AJw ¹AJ$U&9_@%>GI¸HrS=9]I4Sk6%9];¥;?B$H";;4H">#UEHU¦9<JF6;DURA8d>D7%9];4U % ® § ?*§ %EZ²H"J$@ FI?A@%6$Sk>#UeHULAg6;?F6; 7%9];4U ® § ?§ U^6$S=B;=B$H";h;?B>{hAIk@U % ® § ?*§ %E§ ® § § MAI?8HiA%Aj@2>GI?I?AgI?uSkAI?I?>#Sk;R9<JidSkA@%>w hB%9_U28d>#H"J$U;?B$HN;d;?B>GI?>r9_UrHzSkAJ$U^;4HNJK; 3QU^6$S=B;=B$H";d>G|>GJ.9bMDh>+>GI4HU^>"®dH"J%V 3 A"Me;?B> 9<JF6; 7%9];4UeH"J$@QH"J%5 V 3"¢zA"M;=B>DA6;?F6;e7%9];4U±Zl>SGH"JyUR;^9b`c`I=>#SkA|g>GI{;?B>>GI4HgU^>#@ 3 9]JF6;e7%9<;kUGw q >h9b`c` U^BA";=B$H"; ;?B%9_U8d>#H"J$U ;=B$H"; ;?B>GI?>L86$U^;7$>EiKH";?>Gu@g9_U·^A"9]J%;F$H";?B$UPMI=A8-;?B>L>GI4HgU^>#@9]JF6;4U;?ADURA8d> U^67$U^>G;A"MP;=B>e6Ju>GI4HU^>#@ A6;?F6;4UGw «eU?U^682>{;?B$HN;lh>DSGH"JJAg;a$J$( @ 3 |>GI=;?>kju@g9_U·^A"9]J%;LF$H";?B$UnMI?A8;?B>e>GI4HUR>#@9]JF6;4U;?A;?B>E6Ju >GI4HUR>#@rA6;?F6;4UGwhB>GJZ$O>GJi>GI UhhB>GAgI?>G89]8dF%`b9]>#UE;?B$H";L;?B>GI=>E9TUU^A8d>DU^>G;A"M 3 @¥ i%H";?>#Un9]J ;?B>{S=9]I4Sk6%9];LU^6$S=B;=B$H";HW`b`²F$H";=B$U/9]Jd;?B>eS=9]I4Sk6%9];MI?A8-;=B>L>GI4HU^>#@9<JF6;kU;=Ae;?B>{6Ju>GIkHU^>#@A6;=F6;4U 86$U^;hiA;?BI?A6igB;?B>#UR>{iKHN;?>#UGw hB%9_ULSkAJ%;?I4H@g9_Sk;4UA6ILHU=U^68dF;^9]AJ+;?B$HN; 9];h9_UhF$A%U?U&9]7%`]>D;?AI?>#SkA"|>GI ;?B>E|HW`]6>#UhANM¶;=B>{>GI4HgU^>#@9]JF6;4U¦7$>#SGH"6$U^>E;?B>GI?>eH"I?> 3 7%9];4UA"M9]J%MAI?8H";R9<AgJd9]Jr;?B>{>GIkHU^>#@:9<JF6; 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7>H"J+6J%7$HW`_H"J$Sk>#@7%9]F$H"I?;^9];?>>kjF$H"J$@%>GIiI4H"FB+h9];?BQ¢JAj@%>#UA"M@%>GiI?>G> $QAJ.AJ>U&9_@%>H"J$@¢ JA@%>#UA"M¸@%>GigI?>G> $QAJ.;?B>+A;=B>GI#w q >rh9b`c`E;?6I?J;?B%9_UiIkH"FBz9]J%;?AzH S=9]I4Sk6%9];e7KV@g9]I?>#Sk;R9<JiH`c` ;?B>>#@%i>#UhMI?A8 ;?B>e`_H"I?ig>U&9_@%>DANM;?B>DiIkH"FB+;?Ad;?B>U^8HW`b`U&9_@%>Z`]>G;?;R9<Ji ;?B>JA@%>#UEAJ;?B>`_H"I?ig>U&9_@%>7$>d;=B>e9]JF6;DiKHN;?>#UGZ$H"J$@ `]>G;?;^9]Ji+;?B>JA@%>#UEAJ;=B>:UR8HW`b`nU9T@%>7> F$H"I^9];^VriKH";?>#U U^>G> 9]i6I?> ¼ u ¥ 4w hB>#U^>eF$HNI^9];VriKH";=>#ULH"I?>e;?B>eAg6;?F6;4UhA"M ;?B>DS=9]I4Sk6%9];#w ) B+M K ;("I p [ fI# 3} 6( }(I"I;(= z 6#Bz% #Bo 3# z;("I @ ("#B8D# o*# ~ (Tz'0#7%#7&x(~ f 6o 3;( l("II(*L#B mK ;("I O KB%9+-#9"03 #&=8AK76)#&,@#&*)=,&67 (76I4- ;G+-#&,+-27K)67#&,N 6m*=oU) 6C@# 8+ ;(";*# *=)$ < 5 9 4 9 ( 6 6 HM* ,D0? ´ +*?+z 3 +*}+: 6 @# 8+ ;(";* ? $ "*$ 0" ( 1$);3& 1 **)* *340($;"1 $) 34 + #BI;(T 6# @:8+("I}I @*L 6@# 3# + 5# z$ 3 $a 6 +*?+0 6 & $ ;(";*f# *= 1 #BI;(T 6# z8+(I"};(z;("&B 6;( ?(T : 3# l(I K * @%#7* ': 6 & $ (*f5#x* 1 q >JA"-6$U^>2;?B%9_UDS=9]I4Sk6%9];;?A@%>ka$J>YH+SkA@%> A"MlI4HN;?> ¼ 7KVQF%`_HS=9]JiQ;?B>8d>#U?U?H"ig>d7%9];4U H";;?B>r9]JF6;4U;=Ay;=B>S=9<IkSk6%9<;+H"[email protected]$U&9]Jiz;=B>+A6;?F6;4UANM{;?B>+F$H"I^9];^VziKHN;?>#UHU;?B>S?B>#S=.7%9<;kUGw hB>SkA@%>+h>+;?B>GI?>G7%VsA7;kHW9]Jz9_UrHBAgI?I^9]7%`]>>GI?I?AgI?uSkAI?I?>#Sk;R9<JizSkA@%>´r9];dB$HUdHhAI4@A"M{h>k9]iB%; $ ¥ ^U^>G> 9<ig6I?> ¼ u 4w Ah>G|>GIWZ9bMl;=B>igI4H"FB9_UeHriAKA@+>kF$H"J$@%>GIiI4H"FBZ;?B>GJz;?B%9_UeSkA@%>9TU 0d 0 1 1 0 1 0 0 1 0 < M ,J0? # ´ 0 , % )45 6 5 " $) 3;56)*3 7)1% + )417"1 8 0( " "" ( *3 "" " * /%%)(;*!* + 13 )1% + () 9 8*"#+ 4#& 2H iAKA@>GI?I?AI=uI?>#@%6$S=9]JiySkA@%>w ;E9_UEA7%|9]A6$UE;=B$H";{h>dSGH"J>GJ$SkAj@%>d;=B%9TUeSkA@%>9]J`b9<J>#HNID;^9]8d>+&l> J>G>#@8d>GI?>k`]VySkAg8dF6;?>;?B>D|"HW`]6>#UeA"M;=B>DF$H"I^9];^Vi%H";?>#UGZ>#HgS?BQA"M LB%9_S?BB$HU{HrSkAJ$U^;kH"JK;EJ%687>GI A"M/9]JF6;4Uk4w q >e¦9c`b` JA" U^BA";?B$H";¦9<;9_ULFAKU?U&9]7%`]>;?AdF>GI^MAgI?8¤>GI=I?AI?uI?>#@%6$Sk;R9<AgJAJ+;?B%9_UESkAj@%>9]J `b9]J>#H"I{;R9<82>w 6 ¦ B>:@%>#SkA@%>GIJ$H";?6I4H`c`]VzHU?URAjS=9_H";?>#UE>#HgS?Bz7%9];A"Ml;?B>2lAgI4@+;?B$HN;E9<;I?>#Sk>k9]|>#Ue¦9<;=BzAJ>dA"Mh;?B> iKH";=>#U9]Jy;=B>D>GJ$SkA@g9<JiS=9]I4Sk6%9];#w q >h9b`b`nU?HWV+;?B$HN;{HF$H"I^9];^V+iKH";=>E9<JQ;?B>S=9]I4Sk6%9];E9TU #" 3E7%VH hAI4@29bM;?B>e7%9];eHU?U^AS=9_H";?>#@+¦9<;=B+;?B>DiKHN;?>E9_Uh;?B>DF$HNI^9];VrA"M ;?B>D7%9];4UHU?U^AS=9_H";?>#@+¦9<;=B+9];4U9]JF6;4UGw xL;?B>GI?¦9TUR>Z$h>eh9b`b`nU=H#Vr;?B$H";¦9<;9_,U 01 #" 3UR>G> 9<ig6I?> ¼ u ¼ 4whB>@%>#SkAj@%>GIEh9b`b`nU^6$SGSk>#U=U&9]|>k`]V 1 0 1 1 1 1 0 0 (b) (a) M,D ? ´ < "", #"" 3 8*"#+ #+ 8 ",0(1 94#"" 3 + 9]Fd;?B>7%9<;kU HU?U^AS=9_H";?>#@h9];?B;?B>9]JF6; iKH";?>#U¶9]J:HNJ>AI=;;?Ae@%>#SkI?>#HU^>h;?B>hJ%687>GI A"M$6J$U?H";R9TUa$>#@ F$H"I^9];^VriKH";?>#UGw , G+,2H)F<,0+ ? ,0 )FMK (A H[ ML;?B>GI?>9_UH"JQ9]JF6;;?B$H";E9_UH"JQ9]JF6;;?A+8dAgI?>d6J$U?H";R9TUa$>#@z;?B$HNJsU?H";^9_U&a$>#@vF$H"I^9];^Vyi%H";?>#UGZ ;?B>GJ 9]F;?B>|HW`]6>DANMP;?B$HN;9]JF6;#w h>GF$>#H";E6J%;^9b`JAU^6$S=B+9]JF6;4UI?>G8HW9]Jw h B%9_UEHW`]iAI^9];?B8 9_UL>#U?UR>GJK;^9_HW`b`]V;=B>DU?H"8d>HUh;?B>DU^>#fg6>GJ%;^9_HW`P>kF$H"J$@%>GI?uSkA@%>d@%>#SkAj@g9]JiHW`]iAgI^9];?B8 A"M 5>#Sk;^9]AJ w ¼ ZKU^A9];9_Uh>#HgU^V;?Ae9]8dF%`]>G8d>GJ%;h9];LU^A;?B$H"; 9];h;4H"g>#USkAJ$U^;kH"JK;;^9]8d>MAI>#HS=Bd9];?>GI4HN;^9]AJw «L;>#HS?B29];?>GI4H";R9<AgJZ9];L@%>#SkI?>#HUR>#U;=B>{;?Ag;4HW`$J%687>GIhA"M6J$U?H";^9_U&a$>#@rF$H"IR9<;^Vdi%H";?>#UGZU^A9];LSGH"JrI?6JdMAI H";L82AKU^;LH`b9]J>#H"I{J%687$>GIEA"M9];?>GI4HN;^9]AJ$UGw ; ,2AIAC)D # = ¢ ~ c i #BI @ (* ( & ( % ( }% "M ` & E $zd 7 ¢ ¢ ¥ I;(=# ' b 1 (z+*QaO* )$§ [$ *=# *W$MT+%.(D*8U E 8 ( 05C 6;( (&B&| @o=$R( 0%z %z) 3;( : 6)k 6# ("&aT *m; 6# + $ =I $Y 3:8 ( @:;( T *m; 6# +o("& 8@# 6"%oK+( ?%z 6;( ?( %z ) @I * 1}p E ¢ $a 6 E ¢ #7I;(= 6# 8+(I" ¢ #BI @ ("& 8" @# 3% $ 3#7"I N;( QaH(LM $ I;(";* # 08+# '(f$( Y$ (fQaO*n 6;( o"I * (";* K~({$O @a 6 @%#7;( 6# +$ 3k 3# ("& #7I" zQR#7&7& "I * 1 4 5 9 4 9 ( ? + q >rh9b`b`L`]>G; @%>GJAg;?>+;?B>SkAI?I=6F;?>#@Q9]JF6;4U±Z ;?B>+U&9]G>A"M 2Z ;=B>+JK687$>GIA"M 6J$U?H";R9TUa$>#@QF$H"I^9];^ViKHN;?>#UEh9];?B9]JF6;4UE9]JrZH"J$@ :;?B>JK687$>GIA"M¦U=H";^9_U&a$>#@F$H"I^9];^ViKHN;?>#UEh9];?B 9]JF6;4U9]J w q >¦9c`b`|g9]>G ;?B>F$HW9]I §HU;?B>dU^;4HN;?>eA"M;?B>HW`]iAI^9];?B8w q >D¦9c`b` a$I4UR;¸U^BA" ;?B$H";9bM ¢ ¢/Z;?B>GJz;?B>GI?>9_UeU^A82>D9]JF6;9<J8dAI?>6J$U?H";R9TUa$>#@Q;?B$H"JzU?H";R9TUa$>#@F$HNI^9];V iKH";=>#UGw hB>D>kF$H"J$U&9]AJA"M;?B>iI4H"FBr9]8dF%`b9]>#UE;?B$H"; [ E E d HS=Byi%H";?>dh9];?B.H"JQ9]JF6;9<J e ¼ ¥ $zd ¼ w¥ HSGSkA6J%;4UMAgIH";E`]>#HU^;AJ>dh9]I?>:`<>#HW|g9]Ji rw ;e9_UF$AKU?U9<7%`]> ;?B$H";DHgUe8H"J%VyHgU ¢ A"ML;?B>U?H";R9TUa$>#@QF$H"I^9];^VQiKH";?>#UEh9];?Bv9<JF6;kUe9]J ¤HNI?>U?H";R9TUa$>#@Q7$>#SGH"6$UR> ;?B>GVQB$HW|>AJ%`]VQAJ>dh9]I?>MI?A8 rZ 76;D;?B>GVQB$HW|>7$>G>GJSkAI?I?6F;=>#@whB>I?>#U^;86$U^;eB$HW|>;hA h9]I?>#U¦MI=A8 w VSkA6J%;^9]Jid;?B>$ ¦9<I=>#Uh`]>#HW|9]Ji dZh>eA7;kHW9]J E $ e d?d ¢ !E $ Nd ¢ E ed ¼ w ¹A87%9]J%9]Ji>#fg6$H";^9]AJ$U ¼ w ¥ PH"J$@v ¼ w 4Zh>Ea$J$@ ¥ $ 8 ¥ e ¥$ }d }d ¢ § E !E ¢ HNJ$@ ¼ w¼ q B >GJ ¢ ¢ Z[l>EB$HW|> $ Z%U^Ad;?B>GI=>e86$UR;L7$>DU^Ag8d>E9]JF6;L9]J82AI?>e6J$U?H";R9TUa$>#@ ;?B$H"JU=H";^9_U&a$>#@+F$HNI^9];VriKH";=>#UGw ADU^BA";=B$H"; ;?B>{HW`]iAgI^9];?B8µ86$U^; ;?>GI=89]J$H";?>Lh9];?B ¢ Zgh>¦URBA;?B$H";86$UR;lHW`]LHWVjU 7$>`]>#U?Ur;?B$HNJ ¢/w q >yHU=U^68d>Q;?B$H";dB>GJ;=B>vH`<igAI^9];?B8 7$>Gi"9]J$U ¢ H"J$@ª;?B>GI?>kMAI?> $ ¢ ¢ we«eUE;?B>dHW`]iAI^9];?B8 FI?ASk>G>#@UGZ v86$UR;¸U^;=>#H@g9b`<Vz@%>#SkI?>#HgU^>w A"h>G|>GI#Z[9cMh;?B> HW`]iAIR9<;=B8 9TU>G|>GIE9]JsHrU^;4H";?>r § [ 9]JzLB%9_S=B +¨ ¢/Z;?B>GJz>#fg6$H";^9]AJ. ¼ w ¼ lhA6%`_@ 9]8dF%`]Vz;?B$H"; $ ¢ ¼ ¢ Z%LB%9_S=BhA6%`_@+7>HdSkAJ%;?I4Hg@g9TSk;R9<AgJw ¦BK6$UGZg;?B>{HW`]iAgI^9];?B8µ86$U^;HW`]LHWVjUn8HW9]J%;4HW9]Jd;?B>{SkAgJ$@g9<;R9<AgJ+;?B$H"; d ¢/whB%9_Un9]8dF%`b9]>#Uh;?B$H"; ;?B>DHW`]iAgI^9];?B8¤SGH"JJA;;?>GI?89]J$H";?>6J%`]>#U?Uh9];h9_U¦9]JyHdU^;kH";?>9<JLB%9_S? B 2 ¢ w q >LB$HW|>LFI?A"|>#@;=B$H"; 9_UhH"Jd>GI?I?AgI?uI?>#@%6$S=9]JidSkAj@%>EA"MI4H";?> ¼ HNJ$@>GI?I?AI=uI?>#@%6$Sk;^9]AJ ¥ w E We e E E E d E E E E Le E d E E E 6 ;h9_U{H`TURAdF$AKU=U&9]7%`<>;?AF>GI^MAgI?8>GI?I?AgI?uI?>#@%6$Sk;^9]AJAJ+;?B%9_U{SkA@%>E9]JF$H"IkHW`b`<>k`w ) )F F,0F ,- K ?L ,-H*8)FM+ GA AIh>#HS=Br9]JF6;#ZSkA6JK;;?B>eJK687>GI{A"M 6J$U?H";R9TUa$>#@+F$H"I^9];^VriKH";?>#U;?AdB%9TS=B+9];h9_UEH"Jr9]JF6;#w AIe>#HgS?BQ9]JF6;;?B$H";9_UHNJQ9]JF6;;=A+8dAI?>d6J$U=H";^9_U&a$>#@;?B$H"JsU?HN;^9_U&a$>#@zF$HNI^9];VQiKHN;?>#UGZ 9]F.;?B> |HW`]6>A"MP;=B$H";9<JF6;Ww ;h9_UL>#HgU^V;=AD9]8dF%`]>G8d>GJ%;e;?B%9_ULHW`]iAgI^9];?B8¤HULH2SkAJ$U^;4H"J%;?u@%>GF;?BS=9]I4Sk6%9];eA"M/`b9]J>#H"I¸U9<G>w ; ,2AIAC)D # # ~ c G #7"I & ( (";* s d 7 ` ¢ b )*=# *f$+% )$§ [$ (f*8+T I;(=# 'f8 ( z; 3;( :(&B&| @:$?( 0%z $ QaO*N 6;( ?T% Kc$ ("' & e $ 1 (0m*=Qa *0$ @I *mK (";*f( l%z E ¢ E ¢ #7I;(= 3# 8+O("I #7"I I;(";* QaC( ' ( }$ ¢ $(; @Y$l( %z 6;( } 6I(O("&B& & @ T*; 3# +)("& 8" @# 3%,# ?8+# )( =I 6;( $ 38 ( z( ( %z E ¢ )"I * 1?p $~ 3#7I" z( K~({$O @ 3: 6# + I;(("&7& &@ T*; 3# +o("& 8# 3%Km( }%z $a 6 ¥ Nd $a 6 #7I" }Q?#B&7& d d ¼ w o@I @* 1 + q D> h9b`b` `]>G; @%>GJAg;?>D;?B>U^>G;EA"MlSkAI?I?6F;=>#@r9]JF6;4UGZ ;?B>U^>G;EA"MlSkAI=I?6F;?>#@r9]JF6;4U ;?B$H";2MHW9b`¸;?A 9]FZeH"J$@(¥ ;=B>yU^>G;+A"Me9]JF6;4Ur;=B$H";h>GI?>AI^9]i"9]J$HW`b`]V S=`]>#H"JZL76;+B%9TS=B(7$>#SkA82> SkAI?I=6F;?>#@r7KVr;?B>eF$H"IkHW`b`<>k`/>GI?I?AI?uI=>#@%6$Sk;^9]AJHW`]iAI^9];?B8w q >eh9b`b``]>G;{©:4Z%;?B>eJ>k9]iB%7$AI4U¦A"M2Z @%>GJA;?>r;?B>U^>G;A"M{F$H"IR9<;^Vzi%H";?>#Ue;=B$H";dSkAJK;kHW9]Jz9]JF6;4U9] J 2w q >¦9c`b`L`]>G; .¨ bZ .¨ bZ ¨ ¥ bZH"J$( @ 3$ :¨ © bw q >e7$>Gi"9]JQ7KV+Ag7;4HW9]J%9]JiH7$A6J$@AJ 9<JQ;?>GI=8UhA"M 3%w |>GI?V29]JF6;E9]J 9_U{H"J 9]JF6;{;?A+H"; `]>#HU^;{HgUh8H"J%VYU=H";^9_U&a$>#@HgUh6J$U?H";^9_U&a$>#@ F$H"I^9];^Vi%H";?>#UGw/«L;L82AKU^; EA"M ;?B>#U^>DU?H";R9TUa$>#@+F$H"I^9];^VriKH";?>#U H"I?>U?H";^9_U&a$>#@ 7$>#SGH"6$UR>;=B>GV+B$H#|g>e7$>G>GJzSkAI?I?6F;=>#@wLhBK6$U±Z²H";¦`<>#HgU^; eA"M ;?B>h9]I?>#Uh`]>#HW|g9<Ji >GJ$@9]JHrF$H"I^9];^VQiKH";?>;?B$H";DSkAgJK;4H9<J$UeH h9]I?>MI?A8 H"JAg;?B>GIe>k`]>G8d>GJK;A"M dwdhB%6$UGZ ;?B>U^>G; SGH"J+B$HW|>DH";8dAKU^; $ $ [ / / / / d A 5 9 4 9 ( ? J>k9]iB%7$AI4U±w >#SGH"6$U^>eh>eB$HW|>DU^>G;;?B>eJK687>GI{A"M J>k9]iB%7$AI4UA"M;?A27$> 3$ $ $ d ¥ 3g)$ ¥ 3g }d >kj;#Z[l>e¦9c`b` 7$A6J$@ $ $Z%h>eA7;4H9<J 3$ ~d $ ¼ w Xj +9]J;=>GI?8UhA"M D 3 7%VU^BA"h9]Jid;?B$HN; 3 ® s 7 d 7 ` d b d ¼ w «eU?U^68d>L7%VdLHWVDA"MSkAJ%;?I4H@g9_Sk;^9]AJr;?B$H";;?B%9_U/9_UnMHW`_U^>Z%H"J$@:`<>G;h¥ 7>¸HU^67$U^>G;ANM ¥A"MU&9]G>e>k[HSk;R`<V ¯° ¯ w HgS?B9]JF6;h9]J¥ %9TUhH"J29]JF6;L;?AH"; `]>#HU^; $ 6J$U=H";^9_U&a$>#@rF$H"I^9];^VdiKHN;?>#UGw«;l82AKU^; ¯° A"M ;?B>#U^>iKH";?>#ULSGHNJ7$>6J$U?H";^9_U&a$>#@7>#SGH"6$U^>D;=B>k9<IeF$HNI^9];VriKH";=>eB$HUh7$>G>GJzSkAI=I?6F;?>#@wLhB>eA;=B>GI4U 86$U^;D7>6J$U=H";^9_U&a$>#@.7$>#SGHN6$U^>+;?B>GVQB$HW|>YHNJQ9]JF6;9]J rwhB%6$UGZ/;?B>rJK687>GIANM¦F$HNI^9];ViKH";?>#U SkAJ%;4HW9]J%9]Jid9]JF6;4U¦9]J;?B>DUR>G; ¥ 9_U{H";h8dA%U^; s \ $ $3 }d ¥ d q >d¦9c`b`{U^>G;d $$Z LB%9_S?B.h9b`b`l9]8dF%`]Vz;=B$H"; ¥ M HSk;?AIhA"MH";h`]>#HU^; 7 whB%6$UGZh>eA7;4H9<J ` d / ( E ¢/Z U^A;?B%9_UUR>G;D86$U^;e>kF$H"J$@z7%VzH b ¼ ¥ d $ $ 3 }d ¥ "§ d LB%9_S=BySkAJ%;?I4H@g9_Sk;4U¦A6I{HU?UR68dF;^9]AJH"7A6;L;?B>U&9]G>DA"M¥ w VzSkA87%9]J%9]Jiy>#fg6$H";^9]AJ$U: ¼ w Xj¦H"J$@ ¼ w kZ$h>a$J$@;?B$H";e;?B>2JK687$>GIA"M¦SkAgI?I?6F;?>#@9]JF6;4U HWM;?>GILH"J>k>#Sk6;^9]AJA"MP;=B>eF$H"I4HW`b`]>k`P>GI=I?AI?uI?>#@%6$Sk;R9<AgJHW`]iAI^9];?B89_UEH";L8dA%U^; d / 3 ¨ ` Nd s d 7 b d ® ® ¼ 3 7 ` ¥ d 7 d d d k d d b ® s 7 d ¥ 3g ® 7 ~d d ` d b $ 6 ( message bits parity bits Good Code < M , ? ´ «³iA%Aj@+SkA@%>D>G87$>#@@%>#@ 9]JyHNJ+>GI?I?AI?uI=>#@%6$Sk;^9]AJySkA@%>w q >SGH"J9];?>GI4HN;?>d;?B%9_UDH`<igAI^9];?B8 H+SkAJ$U^;kH"JK;eJ%687$>GIeA"Mh;^9]8d>#UE9]JzAgI4@%>GID;?A2M6I?;?B>GIDI=>#@%6$Sk>d;?B> %J 687>GI{ANMPSkAI?I?6F;=>#@r9]JF6;{7%9];4U±w !&$#]"!!'(!_* p-K/ hB>8HW9]Jr9_@%>#H29<JA6I{SkAJ$UR;?I?6$Sk;^9]AJA"M`b9]J>#H"I?u;R9<82>>GJ$SkA@H"7%`]>dH"J$@+@%>#SkA@H"7%`]>>GI=I?AI?uSkAI=I?>#Sk;^9]Ji SkA@%>#Ue9_U;=A6$U^>r;?B>2`c9]J>#H"I=u;^9]8d>>GJ$SkA@H"7%`]>H"J$@.@%>#SkAj@H"7%`]>>GI?I?AI?uI=>#@%6$S=9<JizSkA@%>#UI?>#Sk6I4U9<|g>k`<Vw 8rH"i"9]J>LB$H";LB$H"FF>GJ$UELB>GJh>;4H">DH2iA%Aj@r>GI?I=AI?uSkAI?I=>#Sk;^9]JiYSkA@%>DA"M `]>GJi;=By¢.H"J$@+6$U^>;?B> hAI4@UA"M ;?B%9_U{SkA@%>HgUh;?B>e8d>#U?U=H"i>e7%9];4UhA"MH"J+>GI?I=AI?uI?>#@%6$S=9]Ji+SkAj@%>2?5>G> 9]i6I?> ¼ u 4w q >eJA" B$HW|>LH"J>GI?I?AI=uSkAI?I?>#Sk;^9]JieSkAj@%>A"M²iI=>#H";?>GI `<>GJig;?Bdh9];?BdHEB%9]iB>GI>GI=I?AI?u;?AN`<>GIkH"J$Sk> "BA"l>G|g>GI#ZW9]; B$HU ;?B>U?H"8d>JK687$>GIEA"M 8d>#U?U?HNi>D7%9];4UEHU;?B>DAIR9<iN9<J$H`nSkA@%>wLA29]J$SkI?>#HU^>;?B>JK687>GI{A"M 8d>#U?U=H"i> 7%9];4U9]JQ;?B>SkA@%>Zh>Dh9b`b`nSkI=>#H";?>H2J>GµU^>G;EA"M 8dAI?>8d>#U?U?H"ig>D7%9];4UGZH"J$@>GJ$SkAj@%>;?B>#UR>D8d>#U?U=H"i> 7%9];4U¦9<JzH"J>GI?I?AI=uI?>#@%6$Sk;^9]AJySkA@%>w q >;?B>GJ6$U^>;?B>S=B>#S=+7%9<;kULA"M;=B%9TU>GI?I?AgI?uI?>#@%6$Sk;^9]AJQSkAj@%>HU ;?B>e8d>#U=U?H"i>e7%9];4UA"M;?B>>GI?I?AI=uSkAI?I?>#Sk;^9]JiSkA@%>eh>h·6$UR;{SkAJ$U^;?I=6$Sk;?>#@Q?5>G> 9]i6I=> ¼ uX%4w q B>GJh>yHNFF%`<V;=B%9TUrSkAJ$U^;=I?6$Sk;^9]AJI=>#Sk6I4U&9]|>k`]VZLh>ra$J$@;=B$H";dJA;dAgJ%`<VB$HW|>+h>Q@%>ka$J>#@ ;?B>>GI?I?AgI?uSkAI?I?>#Sk;R9<Ji+SkA@%>Z76;el>B$HW|>SkI?>#HN;?>#@QH`b9]J>#H"I?uU&9]G>+S=9]I4Sk6%9];D;?B$H";>GJ$SkAj@%>#U;?B>>GI?I?AI?u SkAI?I=>#Sk;^9]Ji SkA@%>wOAI?>GA"|>GI#Z ;=B>A6;?F6;2A"M¸>#HgS?BiKH";?>+A"Me;?B%9_UrS=9<IkSk6%9<;r9_UrHsS=B>#S=7%9<;rA"Me;?B> >GI?I?AgI?uSkAI?I?>#Sk;R9<JiSkA@%>wµhB%9_Urh9b`b`D7$>Q;?I?6>Qh9];?BAJ>zU^8HW`b`e>k[Sk>GF;^9]AJ´.h>yURBA6%`_@S?BA%AKU^>QH 7$>G;=;?>GI7$HU^>+SGHU^>2MAIA6IDI=>#Sk6I4U&9]AJwzhB>rF$>GI^MAI?8rH"J$Sk>ANM{A6ISkA@%>+h9b`b`L7$>@g9_Sk;4H";?>#@7%Vz;?B> 5 9 4 9 ( message bits parity bits C k-1 Rk ? R’k+1 Good Code < M , ? N ´ (a) ¥ (b) &( * $ 0*C9" 7$)41 #*0$ #")12) + " (7 * * )4** 30$ " 1 $);37)1 1 5 !&19 "$ + "", ( ** )4** 30$ "17$) 3 8 % $3 )1 # )8 )- ;**)* $ ) **$;#"1 $) 4 3 + gf 6$HW`b9<;^VdANM¶;=B>¸SkA@%>{;=B$H";h>{S=BAKA%U^>¸HgU A6I7$HU^>whB%6$UGZgl>E8HWVDLH"J%;;=AS=BAKAKUR>{HF$H"I?;^9_Sk6%`_H"I^`]V iA%Aj@+SkA@%>DANMPU^8HW`b`7%`]AS?r`]>GJi;=Bw«EJA;?B>GI{SkAgJ$U^;?I4HW9]J%;9TU;?B$H";hh>D@%AJ ;KJA"(;?B$H";Lh>eh9b`b`7> H"7%`]>e;?Aa$J$@riA%Aj@r>kF$H"J$@%>GILigI4H"FB$UA"MPU^8rHW`b`PU&9]G>w Ah>G|>GIWZ;?B>GI?>D@%>ka$J%9];?>k`]V 9TUU^A8d>eSkAJ$U^;4HNJK; U&9]G>HM;?>GIhLB%9_S=Bh>eh9b`b`P7>DH"7%`]>D;?Aa$J$@ iAKA@r>kF$H"J$@%>GI{igI4H"FB$UGZH"J$@rLB%9_S=Bh>eh9b`b`nS=BA%AKU^>e;?A 7$>;?B>e7%`]AjS=r`]>GJi;?BA"MPAg6IL7$HU^>DSkA@%>w q >eh9b`c`/JAFI?>#U^>GJ%;{HMAI?8HW`@%>#U?SkI^9]F;^9]AJQA"MAgJ>eMH"89b`]VA"MU^6F>GI4SkAJ$Sk>GJ%;?I4H";?AgI{SkAj@%>#U±w q > FI?A"|g9T@%>;?B%9_UL@%>#U?SkI^9]F;^9]AJQ7%VY@%>#U=SkI^9]7%9<Ji+;=B>D>GJ$SkA@g9<Ji+S=9]I4Sk6%9];4UMAIh;?B>#U^>SkA@%>#UGw ,0/ ' /'G,0 .,2)0 +.*,0/ [ ¹BA%AKU^>DH"7$U^AN`<6;=>DSkAJ$U^;4H"J%;4U eH"J$@($w ¹BA%AKU^>DHSkAj@%> A"M/`]>GJi;?B ZKIkH";?> ¥ ZjHNJ$@+HU`_H"I?i>e8:9<J%9]868 @g9_U^;4H"J$Sk>HUF$AKU=U&9]7%`<>w ¬>G;e¥ 7$>HrS=9]I4Sk6%9];e;?B$H";L;kH">#U 7%9<;kU{HU¦9]JF6;DH"J$@+FI=Aj@%6$Sk>#U ¼ 7%9<;kUeHUhA6;?F6;eU^Ar;?B$H"; ;?B>#U^>E7%9];4U;4H">GJr;?Ai>G;=B>GI MAI?8 HSkA@%>GhAI4@2A"M1 whB> 9]JF6;4U¦7%9<;kUlHNI?>{;=B>{8d>#U=U?H"i>E7%9];4U A"M;?B>SkAj@%>ZH"J$@r;?B>DAg;?B>GI4UhH"I?>e;?B>S?B>#S=r7%9<;kUGwb @ 7>DS=9<IkSk6%9<;kUh;?B$H";h>GJ$SkA@%>DH"J>GI=I?AI?uI?>#@%6$Sk;R9<AgJYSkA@%>eA"MI4H";?> ¼ h9];?B AI "ZN`<>G; H"J$ S S 6 ( d8 >#U?U=H"i>e7%9];4ULH"J$@ S=B>#S?+7%9];4UGZHUL@%>#U?SkI^9]7>#@+9]J5>#Sk;^9]AJ ¼ w w A:MAI=8¤S=9]I4Sk6%9];¥ MI?Ag8¤¥ ® Z$;kH">DHrSkAFKV+ANM H"J$@6$U^>D;?B>9]JF6;4UEA"M HU;?B>e9]JF6;4U A"M¥ wE«¸@@Q¥ ® ;?Ad;?B>dS=9]I4Sk6%9];D7%Vr9_@%>GJK;R9cMVg9<Ji;?B>A6;?F6;EiKH";?>#UA"M h9];?BQ;?B>e9]JF6;4UEA"M ¥ ® w 9]J$HW`b`]VZ²H";=;4HS=BYHdSkAgFKVrA"M ¯® 7%V29T@%>GJ%;^9bMV9]Ji+HW`b`;?B>e9]JF6;eH"J$@rA6;?F6;iKH";?>#UA"M ;?B> SkAF%V+A"Ml¥ ® h9];?B;?B>9]JF6;4UEA"M ¯® wehB>A6;?F6;EiKH";=>#ULA"Ml¥ h9b`b`7$>dHW`b`;?B>9<JF6;H"J$@ A6;?F6;Li%H";?>#UA"M¥ ® HW`]AJidh9];?B;?B>eA6;?F6;Li%H";?>#UhA"M ¯® ?5>G> 9]i6I?> ¼ uXj4w ¬>G; 7>d;?B>dI4HN;?> ¥ SkA@%>dA7;4HW9]J>#@7KVQ;4HN9]Ji 82>#U?U?H"i>27%9<;kUGZM>G>#@g9]Jiy;=B>G8 9]JK;=A¥ Z H"J$@r6$U&9]Jir;?B> ¼ Ag6;?F6;L7%9];4UEHU;?B>DF$H"IR9<;^V+S?B>#S=UhA"M ;?B>DSkAj@%>w \ \ \ \ \ S q >DSGH"J@%>#SkAj@%>;?B>#U^>SkAj@%>#U¦9]J+`b9]J>#H"I¸UR>#f6>GJ%;^9_HW`;^9]8d>w , G+,2H)F /.'G,0,2 )0 K*, *,0(BKM )FM+H GA M [ ¨ " Z%;?B>GJ@%>#SkA@%>d¥ 6$U&9]Ji+H"JH"I?7%9];?IkH"I?V+@%>#SkAj@g9]Ji+HW`]iAI^9];?B8w M "Z;?B>GJzH"FF%`]Vz;=B>:UR>#f6>GJ%;^9_HW`l>GI=I?AI?uI?>#@%6$Sk;R9<AgJsHW`]iAI^9];?B8 ;=A+;?B>dJA@%>#UE9]Jz;?B> ¯® w LA"DZI?>#Sk6I4U9<|g>k`<V(@%>#SkAj@%>Q;?B>JAj@%>#U:9]J(;?B>ySkAF%V.A"M¥ ® 6$U&9]Ji;?B>UR>#f6>GJ%;^9_HW`D@%>#SkAj@g9]Ji HW`]iAI^9];?B8 MAI¥ ® w 9]J%9_U^B7KVzH"FF%`]Vg9]Jiy;=B>U^>#fg6>GJK;^9_HW`h>GI?I?AI=uI?>#@%6$Sk;^9]AJHW`]iAI^9];?B8 ;?A+;?B> SkAF%VA"M w \ \ +,K ,A3 H = ~ $? 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MD6 1?¡> 9 $@#\1/< #? 1\¡> ´ ¦ D9@N3~°/ cND Z># 57 H9³ M´ dD9QID > >@9~ 9CB@ ># 9C ' @#\1/B>9-H 9CM9~sd1@I1@©@#\1/j P#? 9>¡\ > ' /10 - ' - e /6 - 9 / 3*)E' ??+ - .+ 3 " 6 -2- 0?)1+ /I) % - /=!%;6 35 - 09A?A /10376bA!%I))3K 6 46 / F /I)&38/96 K < @7 2 !'!I% '>/10376"6 - 9 / 3*)E' ?!+ - . + 3 6 -2- 0?)1+ ( ; /I)(% - /=!%;6 35 - - 9 /43)*'f<g< ¹ - L 3,'<2 F - 2 53,'3,'?LF/10 - e09F *% 03©A!%d))9K 6/)*'64/ %d=9/ - !%$/ Ku)>= , 09A?A - %#Ku)*%IF A )\L 09A - %#Ku)*%IF A /10 - Í Ú { *% 03A!%I)M)9K 6/)*'64/ %I=9/ - I , 3,' /E0 - 0?) - %0/ 3*)E'6 /100/Z0% - )\A - 6P)9K&)>A - %0/43)*'60 % A 3,'<; - 9 /43)*'Qf<Ug<¥f L 03A /100/P)*' /10 - E 0 - $0 - /10 - %S0 L 35 - 'NA) '?)>F 30 376P¿ - %I)j0/S0# - %1//0 3,'QA)3,'/ ? ¦ 10 - @3,' $0 - /10 - %S0 L 35 - 'NA) '?)>F 30 0 ¦ 10 - @3,' $0 - /10 - %6#)>F - ?Lj+ *'64/ ':/ ?L+ 3 - 'Q510 30'4;@/ ¦ /0 @6E6#=?F - 60L35 - '510 - EL % -2- , A) '?)>F53C0 - )>G 6 - %I5 - )>2 - B0 /S0# - %1/0 30'A)3,'/ ? = / - 590 =0 3 - '4A)3,':/&/0 - 6 B/ ,<0 0jL 5 - ?!0', = $? 10 - @3,' A /) - %Z)9KA/E0 - 0?) - % 10 - @3,' ?Lj+ A 6 H HDg Õ < - %0/43)*'6 3,'4;@/ - ¹ - )>A ,Q2 /E0 - 1 0 - ,N2 ¦ 0L 5 $0 - /10 - %S0L35 - 'QA) ?Lj+ '?)\F53C0 @616 0 - 60L35 - 'F6 - /)3Ks510 =?F (/100/6 - /.376$6#F-0 - % /100' /10 - B30'/;62?+$0 - % - /E0 - 6 3¿ - )9Ks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½ /10 - 6 - 2 2 )3K )\A ,j)>= /= A K«) dF M= À ?) ) H+ )>L /09A 7 , Õ #) 0>, K«) 0 4)9K 0 ?) ) *¿ ?) /) L /= )>L /09A /0 =?F2 2 0 L>F * ©A I))3K )9K `0 ©)9K 092 I) ½K Í Ï × L\F 0 0>,40 /) 5 0 I)A c09A 0 )\A Í )3K /) 0 C, 0N0 , ) L>F 0 c0 0 *) I) Õ ) /= 0 )>A #Ku) IF M=?F2 /= F-0 À I)>=?L 0 2 H)9K ¿ Z)9K 1092 Ï )>L 0@5 P)9K Õ ¿ < I=0 c0 ¹ *¿ Q)9K A I)M)9K ,M) d)>=?L )>L /09A ,q2 0 + )3K I))>2 0 N0 b)9K ¿ Z+ *) Q+ ) À M=?F2 )9K^»\= 0>, F f [ MKu) º[ #+P)BF-0 À /10 - I)M) @64/SV¤)>=!%43 - % V0 / 0 - ' P0920 " 0 ©+ + = d) FBA F % /)?, , ?A?= 0 , %0'6Ku)*%IFF< À 3 ?sV¤)*%1/ '?)9+?!T - 5 3,' ?^0 ',<;>¿ - L - !% - 6 - '/ - 5 , /E0 - /E0 - * ) % A ?L - %#Ku)*%IF - A I)ZA d)15 C, A I)M)9K L\F 0 2 /) 0>,Q)3K Í ¹ , - %0/ 3*)E'6c0'º2 - , - 6©0/056 - / = F , R `V&T^; 8H c] ' 3 / - 30'j0' - % %I)*%IG /)E% % - 9 /43,'?L c0 , C,?0 3,':/ %I):,M= - /)?, , / %0'6#A0 % - ':/ZA!%d))9K 62?/10 - A %d))9K E 0 - 6#)"/100/ /10 - ! - %+P)>= , ) *' ) 00@5 - /I)-% - *'64/ ':/ 0>,Z0 /) +%I)E%0S0 ',B;01K %0R¥½; )9K!½ + 3 - . + F-0 = 6 - @616 3*LE'?F - ':/ <ν +%d)*%0?>T!=!',?)@/#+°0 '3 ?\;>=,?0' ?M0',H;>¿ - L - ' '6I0/4376K *' - A 3 - - / < ?LQ) ?LQ0 '/ /d) +0 )9K,?0 0 +0 )9K *% 03A!%I)M)9K(376$/E0 - 6I09F - @0 60HA3 - - - ':/ 3,'¼0 0?) L ?F ':/ /I)/)*'64/ %I=9/¶F-0' +°0 0=62?¤+ - J + 3 ' -2- L , 6#)>F - À )>L 09A *'64/ ':/@' /10376+.3/10?)\=/-% - 0>,3,'?LBFB)E% - /100'N0 ,M) R¥½T 87 × ? ' 3] >0 , *% 03ZA!%d))9K(/100/0 E 3,% /=3/00@6©0+6I0B/4376K ,3 '?L )>L 03A *' - % '3,' 3,' @616 3 *' 0 - 0 )' - 616j)3K/10376/)*'64/ %I=9/43)*' 376'/100/ /E0 - 6d0/4376K ,3 '?L 0 616 3L*'?F - '/N09A?A - 0 ;% 6 + /) , E% +`) )\L 09A *% 03°A!%I))3K <¶V?)*%¶)>=!% % - /=!%;6 3)*' ?\+ - . + 3 .3/103,'+/10 - 0?) + 6 0©+0 5 92?]?0 ,½ 3/S0@6P+ - < )9K 0 !% 87 @6 38//0 /I)$/E0 - % - /=!%;6 3)*'6 3,'/10 - 03763C, À 0 0?) - 9 /43)*'f<g?M+ - 6I0@+ 0?)9+ /I)#/% - B0 / - + ' ( <; A I)M)9K 'M=?F2 - % 38/96W23/;62< /0 /) /0 10 - 30' )>L /09A /E00/A0%1/P)9K ?L - e / - % '0 )9K *% 03 0?) / < ¹ - ,?0 /0 - %b)9Ks23/;6b)9KC/10 -&- e / - % '0 =?F2 / < ,?0 0 Theorem: C’ is satisfiable. Holographic proof that C’ is satisfied by E(p(x)) and E(v) check proof Proof-checker E(v) check that satisfying assignment in proof actually is this assignment. m X@i Í 2 p *% - )9K 590 ÍË!Õ E%A< ) E3,% 3/ /= $ 3*L0:/&+0'/C/d)ZA!%I)95@3C, F - ' @6 0 -0 *% /10 - % - 6 /S)9K./10 - E 009A/ - % < Ï ?+ - 3,'6 3764/ /100/./10 - /100/&/0 - 60QA) = %;6 / 1 0?)M):6 - . Ku) + 0 - ' - $ - 510 )>L 03A E'/&+0':/ /I)K«)*% - /E0 - ! A %I)M)9K E 0 - /103762?\+ - +0 *% 03°A!%I))3K/100/0©A) 0?) 0 ,M) ),M) c0 0 - A!%I))3K E0 - - % ' -2- ,6$/d)B2 - #6 =!% - /E00/$/10 - ! A %I)M)9KA640?)9+6./100B/ Í Õ ?>+$03E0 0% - +@%43/1/ - ' - 6 - + 0 - % - < ? / À 00@6 /100/P590 Ë 92?= + 30'L/10 - '?)*%IFj0 0 À 8- ?\+ - e¤09FHA - 0/ = + 3 - . + ' 0 , Í#Ë!ÕIÕ V¤) /10 - ctt E(p(x)) / - =0 '?)>F 30 / ,N0 Ë ' FBA I) '64/ ( 510 3 FB):, 7K = " @6 Í , ÍË!ÕdÕ ' 0 , Í /d)A!%d):,M= - #E3,% 3/ 0 /= $ ÍË!Õ ' 0 , Õ <¶Y)9+? /)*'6 3, - % /100B/ e?A ;: EBJHIcCVa`a2P!Z=2JX=2;iI@9@9<?BE?B;H;-COAHPJHKcCOJ KCBg:? EB8:9<;HJ-C79 JdA-C7JH?!Z]@I<JX[?LM CVP0C7;i[ ?Ga`aC7;H;HI<M?LJHKcCOJX=2JX><8 ?B;BT /0 - 6 - 6 / %43A/43)*' - 6 / %43A/43)*'<)9K , A /;6 37K/100B/A) - @0 6 30'?A?=/j0',_0! - $ ÍËÐÕ , - % %d)*%IG /)*% % - 9 /43,'?L/)?, - ?>+$03E0 L>)M):, @%43/43,'?L /10 - ,Q0 3 6 6I0/ 3 6. '?)\F53C0 ) - %A/I) % - 0\,5/10 -.- ':/ 30% - 0>,)9K+ A % - 6 - '/ - ! 2 0 ,_0 < ¹ - / @/ 3c0 $ '?)>F53C0 - 9 /96 3/;6 @7 2 3,'?A?=/;6 /I)j2 -&- '/):, - L , 3,'>/10 /100/ - '/)?, - )9K @6 0 $ @616 3*LE'?F - ':/Z)3K 0 /) M5 $ % 0 / 3/"00@6 '?) !2?= E0 / " - 1 0?)M):6 - / 0 /) 0 376# :) 6 - /I)<0Q= C»>= *% 03PA!%I)M)9K ? /10 - E 0 - )>L /09A !% - %437K $ < À - /):, 0 - /10 - 6 3¿ - $ )9K J3 A + - @0 6E6#=?F - /100/ ¹ -2e + 1H@H Õ < - % /)>F - 3 6&/10 - 6I03F - @0 6 - %Z+.3 % - 0 - E 0 À - )j2 À - 0>,0K + *% % - 6 E' 30' '10 - >/10 /) - % #A) , ?L - c0 6 - /E0 - ! A % - 6 - ':/0B/43)*'6 3,' /10 - ! A %d))9K - % ' - %437K c0 5 - %©+.3 /E0 - 510 ' = )9K0 / - L0 )9K /):, 3764/0 ' - 1/ 0 - ' ?01K / - %(6#= - 616 , *' 30' - 2 %1/ 6#) < A - 1 0?)M):6 - ( Ks+ 3 - - 0 )9K , E% >376&/10 - 6I03F - @0 6&/10 - *) ' - +`) , 6 - ?&0@K / - %E0 - @3,'?L>/10 - c09= /I)2 - - ?/10 - 'F/E0 - E 3,% /=3/ /)?, $ E'64/ % 9/ /) 2 e / - % '0 - *'/ $ 0 /)?, ' c0 + E% 6 - /)?, +`) , *% -2- . + 3/10L03*L0 09L - 3,'-+$03E0'3/A376 - 0 6 7 ' -0 ,4g = 6 - * ) ' - /I) :64/ /10 - 6d09F - 0 FB) ' - c0 N2 (/10 - 6 - )9K :64/Z0/)*'64/0'/ /43F - 6 /10 - 6 3¿ - / 3,'6 0 .3 00D5 , ( K`+ /) # / ,?0 0 I3,' A 003A/ - %;6 I= G I/100/$/E0 -- e / - % '0 /) M5 0376 376Z2 À 6 = 38/ - , /I)H)>=!%SA?=!%dA):6 - 6=< /E0 - 6 - - '/(/100/ 3/;6 6I0/ 3 6K ,3 '?L0@616 3*LE'?F - ':/ *% G c03= E% 03A!%I)M)9K1< +P) ,H)9K *% -2- J + 38/E0+/10 - f>G )>L 09A % >/10 - 6 3¿ )9K 0 , K 0 S09A0 , 376)E' - 3L>=!% V - %P+.3 " 0 - ' /10 - E 0 - ¹ 09L /E00//100B//)?, - ' - 6 /100B/S+ - /):, 3,' - 0 %&/43F - ?6#)D/E0 - 6 - 0 % - 6 -2Í *% 03ZA!%d))9K1< /) M.8, 0 *' ) /(/E0 - 6 - /100B/ /E0 - % - 376056I0B/4376K ,3 '?L , ?/10 - 1 0 - ÕdÕ +`) , )\L 09A $0 - /10 - %Z)*%@'?)@/0H+P)*%, 376 3,'>/10 + 6 3¿ - @0 6 /10 - E 30% /=38/ 3,' J38/E0-)*' + Í 0 *% 03©A!%I)M)9KW2 - *' % +`) , )3K )>L 09A +P) /,<)9K - % 376 E% 6 - /)?, 3 3/ A I)>2032 5 " ?0',N0 )9K ' 0 , ÍË!ÕIÕ #E0 - 5/10 ' *% - /):, Õ< Í *% 03A!%I)M)9K09/I=0 ,?0 /0Z0 , 3,'/10 - 6d0/4376K ,3 '?L 0 616 3L*'?F - '/ 3,'/10 - 0?) 0?) Í - %)9K&23/;62? /10 - E 0 - M=?F2 )\L 09A + 3/10#/)*'64/0'/ % - 0 /435 - 5 F 3,'3F=?F - . /):, *'64/ '/-' !'3 - %43,'?L-0 ' - 6/)*'53,' - /)>F - 9 /43)*'f<H Õ 6#)?/E0 - E 0 - ; /d)Z2 /)>FBA0 ,?0 0 *% 03HA!%I))3K @6 376 /=6E6 - I , 30' /I)B0 - A / <'=!%E0 0 , Í#Ë!ÕIÕ E% 6 - /)?, ' 1/ 03,'?LN092)>=/"/E00/0616 3L*'?F - '/ < )\L 09A Í 0 , Í 37K 38/96.3,'?A?=/;6Z0% - ,<0 Í + :6 - 'H238/96&)9K 38/96 - e / - % '0 $ 6 - c09= ?3/P2 - 0 ?) ( K8+ , *' 3,'?A?=/;6 $0310L3/ZA!% - 6#=?F - 6 /d)-2 - B/ ,?0 0 - %437.8092 5 3ÉK`0 ',N)E' A P) /) 3,'F/10 - E 3,% /=3/ % - ! A % - 6 - ':/ - ' , 3,'F/10 - 0?) Ku= A *% 03A!%I))3K 0 - /100B/`+P)>= = 0 A % - 6 - '/ - 5 , 3,' /10 - 0?) - 6°)9K30'?A?=/96A% - ! = + - /;6 376 E% % - 9/ @/ @ - / - % ' - 3,'?A?=/;6 % - ! A % - 6 - '/ - L , 3,'/E0 - 0?) 510 ' - / .3 0 ,Q0590 )>L 09A ? $ < /100/"/10 - 6I0B/4376K 3,'?L @0 6E6 3L*'?F - '/"3,'/10 - 0?) , 3/;6 - e / - % '0 %0',M)\F '?)>F53C0 )>L 09A /100/./10 - 0?) *' @3,' - - /):, *% 03©A!%I)M)9KC/E00/ *% &0?) K«) P0 E0 - 6 0A) /)?, , - /)?, , - 62?/10 - ' (3/;6 3,'?A?=/;62< 0 SFH) )9K - 6J3,'F/10 - % - 64/)9K(/10376E009A/ - % < / > '-' - ' 6. %;64/ 3,' 3,'0/ 3*)E'QKu)*%(0?)9+/I) % - /=!%;6 35 - /)?, !% 6E0 -+- 5 - % 6 *% - A I)M)9K ¶Ku) /I)M)-23L< 0 L>F B)>A - '/S)3K 5 E'64/ % 9/43 *' 0 *% 03 !% 6 *% 03 - % j3 - 9/ 3,' ' - % % E% *% % - 9 / 30' - ' !% @3 - 0 / /10 - E0 - - % ' -2- 6 / 6 /10 / % 38/E/ - % *% / % - / - 6 !% - '6 3 - / % /10376 E%/10 - 3 %43 / - - /10 / - !% - 6 - '/ - I3,' )\2 09A?A _0 - %0/43)*''/100 *'6 3C, - %Z+$00/ 009A?A ) /) I) IG /) , 7K`+ !% )>= I)A bK«) I= ?L /)?, #Ku) IF 2 510 ) j)9K )>L /09A A I)M)9K 0 ,A I)95 C, 0 C0 ?) A I))3K /)?, 0 ©+ ?) 0 `0 A )>L /09A 0 , 7 - 9 /43)*'Ef< < ; *'/ /) 3,'3,'?L 0 / ' 0 0 - 0 )3K 0 - % - 0 % À Í %E/0 3,'E510 3,':/;6=< A) Õ 00@5 - /I)@/0 , !% - 5 - '/+/10376 - A/10 - A) '?)>F53C0 Í 3L>=!% V - 30' ?!0' - '/)?,3,'?LQ)9KC/10 - Ku) ' = 6 - c0 E2 /10 -5- '/)?, - @6 !% /100/B0' ,q0 0NA d))9K ,<590 = ,M)\F-0 - 6P+$0 - ' 3/A376 - 7 1H f\G *% - 0E0 510 = - V¤) / - 590 / =0 ( + Í ,Q0 P0 Ñ Ñ Õ Õ B0 / /E00/A)B30'/ ? )3K - 6 / %43A/43)*'Q)9K , !%/ K«)\= A Õ 3,' /10376A!%d))9K 6 0?)>= )9K 0 - ! A %I)M)9KE0 - - 6)9K)>A - %0B/43)*'6"/100B//10 - 1 0 - - 60/0jA)3,'/ < À + *% 03ZA!%d))9K(/100/bA!% - 6 - '/;6 )\L 09A /E0 - - '/):, - *' ) 3,'64//0' - /100B/°, - 6 / %432 - 6 /10376 % - 0 / 3*)E' < F *% % - 9/ < 0 ? / % / E3 - ':/ )/ , - %437.8092 - 6#) 5 '?)>F53C0 6+//0 - 6)*' - ,wA) - ,N2 , /d)QA!%I)95@3C, - %Z+.3 " ' -2- > , 6/d)NA - % K«)*%dF - % ' -2- Í 0 *% 30'?L /) 3764/ - ) /&/10 - ,_0 *% 03A!%d))9K(/100/ /10376)>A - %0/43)*' 0062 -2- 'NA - %dG A % - 6 - '/S050?) - ! )>L /09A /) ?) &ab)>A 510 ' I)3K B0 , =0 / B/ - !% - 6 - :' / , A ,<2 ¾ ' - 6 - /B)9KZA)3,'/;6 30'¯0. - 0 #) 092 ' @f ?A009A/ - % <%$9].640?)9+ /100/B0 ' F-0 £R ½a©¾ % - @6 E' -- .E3 - '/"3 % - % / ,º0 -0 / %0 '6K«)E%IFD62<¾ 6 3,'?LF/E0 =0 6 -2- *% - 0E0NA)B30'/ 38/ 3,'/ - %;6 - 9 /96.3,'>/10 - = A) - BL 30'!'3,'?L)9K`; - 9 / 3*)E'Nf<¥f . Í 3,':/;6=?@/10 - '5/10 - 6 - ! A %I)M)9K 6 !% - 6 - '/0/43)*')9K°0j, - *L % -2- 4A 0 A) Õ 30'>/10 - ! A % - 6 - '://0/43)*' ?050?) 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A %I)M)9K?K«)E% - 010)9K/10 +P)>= 0 - E 0 - À 0 - 0?) À - %1/ - 0 /0 !'3590%43C0/ = Õ 3/;6P)9K^,?0B/0< = 2 Õ Í Í - FBA F ' A0 /0 0N0 , % / - 0E0ºA)B30'/'3,' /10 - 3,%N,M)>F-0 30'¯2 ,£0 <g<HH"/I)/)*'64/ %I=9/ /10 - 0?) FBFj0jf /100'L/10 - 6 3¿ - (/10 - ! A % - 6 - ':/0B/43)*'N2 )9K / R ' E3,% 3/;6 /= / 64/S0A) 0 FB) G +6 3¿ -< )\L )>L0 - 376 / % - / + 6 3¿ - Ï , A) < ./10 - 6 - )9K ' c0 , !%43 - %&/ %0'6Ku)*%IF Q)9K *% 03A!%I)M)9K 6=?/10 - /I)/0 K '?)>F53C0 Ku)>= c] ? 1/ 0 - != '3*590%43C0/ ; )>L /09A 6 3,'?Lº0 ,E= - '/0/ 3*)E'_)9K./10 - , 376 /% - / ;0 I+0 A) )95 - 2 KuG !%43 - % Ku)>= - %0H.G '?)\F53C0 6#c0'<2 - 0=6=?.37K+ - = 6 À j3 - 9 /;6J3 6 )>2 %43/10?F53K09/I)*% <°;3*F 3 0 % % - % 0 dL ? 37K+ - = 6 - @7 2 1-y 2+2y 3+5y 2 -1 6 13 1 0 4 8 4x 0 1 2 3 x+1 0 1 2 7x-1 E(1-y) E(2+2y) E(3+5y) Proof: (7x-1)(0,1,2) = (-1,6,13) E(-1) E(6) E(13) E(7x-1) Proof: (4x)(0,1,2) = (0,4,8) E(0) E(4) E(8) E(4x) Proof: (x+1)(0,1,2) = (1,2,3) E(1) E(2) E(3) E(x+1) one check Checker provides: m X@i p Theorem (encoded): p(x)(0,1,2) = (a,b,c) !% - 6 - '/ /43 *''376 % - *% % - !%;6 3 *' j ½¸A ct 0 ) A 0 , Ku) /= ) < : 0 - *) % À F :6 /°0 FB) A) - ¹ <[<H /I)/)*'6 / %I=9/./10 - 0?) f '?)>F53C0 30'L/10 - 6 3¿ - ' '?)9+£A!%I)95 - /10 - c0 !iMo @i *% 03©A!%I)M)9K 62?/10 - ''/10 - /I)@/0 6 3¿ - Í ÕÌ -, )>L Í )\L j3 - 9 /;6 3760/ )>2 C/10 - ! 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