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w w w . l u c e n t . c o m / s e c u r i t y
!"# 22:
$%&'() *+,-µ%,% .'(,/01 # / 2
!""#$%&' ($)%&*' +,$-.%/*0µ%)
+1'21,µµ,, «+"#1)3)1%&0»
-
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w w w . l u c e n t . c o m / s e c u r i t y
3%1)4'% 56'(7'1018%
309'(7678+&+ 54:;?)4µ%,7@AB%µµ'(78 3C9'(D@
!"#$$ & '()*+, -./01(/( 2*+/345 #67 $
-
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4,$5"%, !.%&)%$6$7,8
!
9%,&1%*5 4,$5"%, !.%&)%$6$7,8 :6178 ;$0µ# ! /-%8 +"#1)3)17,8 -%/'?)@ –
-A'?)@
!
(µ)%B,7, +"#1)3)17,
! :61#*%&'*#*, ?%,&1%*C$ &,$,"%C$ =6178 µ$0µ# ! D-µ-"%C?-8 D-C1#µ, +"#1)3)17,8
-
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3%1)4' µD EFB=G7
4,$5"%, !.%&)%$6$7,8 (1)
+#20
46?%&).)%#*08 (.)&6?%&).)%#*08
9>&*#8
ŝ s
t r 4,$5"%
!.%&)%$6$7,8
46?%&).)%#*08 +#208/")
;)$*>")
;)$*>")
s!
;)$*>")
;)$*>")
s!ˆ
;)$*>")
D'1@B)8
!"#$$ & '()*+, -./01(/( 2*+/345 #67 8
-
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-
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4,$5"%, !.%&)%$6$7,8 (3)
KANH"I
J, pX(xi)
.
.
.
.
.
.
2, p Y(y j)
x1
x2
xn-1
xn
y1
y2
yn-1
yn
.
.
.
xi.
.
.
y jpij
pi2
pi1
pin-1
pin
[p Y/X(y j /xi)]=[pij]=[P Y/X]
[P Y]=[PX] [P Y/X]
[PX]=[pX(x1) pX(x2) ... pX(xN) ]
[P 2]=[p 2(y1) p 2(y2) ...
p 2(yN) ]
( ) ( )
N j p x p y p N
i
iji X jY ,...,1,1
==!=
!"#$$ & '()*+, -./01(/( 2*+/345 #67 :
-
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4,$5"%, !.%&)%$6$7,8 (4)
!
26D1K/µ'&+
– $# !$*1).7,
•
•
G(:,H)=G(:)+G(H/:)
– 9-/µ-@µ>$# !$*1).7,
•
– (µ)%B,7, +"#1)3)17,
•
I(:;H)= G(:)-G(:/H)=H(Y)-H(Y/X)•
!7$,% *) µ>2-F)8 -&-7$) .)@ µ-*15-% *) .)/' *#8 µ-76/#8 *#8 ,B-B,%'*#*,8 *#8 : '*,$ 27$-% 2$6/*' *) H.
•
J# µ>/# ."#1)3)17, *#8 : .)@ µ-*,3>1-*,% /*# H.
!!"=#$% x y
y x p y x p
)/(log),()/(
!!"=#$% x y
y x p y x p
),(log),(),(
H(X/Y) I(X; 2) #( 2 /J)
H(Y)H(J)
H(:,H)H(J, 2)
!"#$$ & '()*+, -./01(/( 2*+/345 #67 ;
-
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4,$5"%, !.%&)%$6$7,8 (5)!
!%B%9D8;µ%,% .'%(B',C1 3%1%4'C1
–
!"#$ &' & #( )* +, - /
0'-"12
11
00
1
1
!" !
!" !
J, pX(xi)
x1
x2
2, p Y(y j)
y1
y2
( ) ( )[ ] ( ) ( )[ ]
( ) ( )( ) ( )11
00
10
01
1010
xY
xY
x xY Y
p p
p p
p p p p
=
=
!"#$
%&=
!"#$$ & '()*+, -./01(/( 2*+/345 #67 <
-
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4,$5"%, !.%&)%$6$7,8 (6)!
!%B%9D8;µ%,% .'%(B',C1 3%1%4'C1
–
3(0'-"12 ()*
µ 4 µ 5 4 6 & #* "6
7' µ 4(4/ 48'$2"/
J, pX(xi)
x1
x2
2, p Y(y j)
( ) ( ) ( ) ( )[ ] ( ) ( )[ ]
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )2423
1211
214321
3
2,
3
1
2
1,
2
13
2
3
100
002
12
1
x p y p x p y p
x p y p x p y p
x p x p y p y p y p y p
xY xY
xY xY
x xY Y Y Y
==
==
!
!
"
#
$
$
%
&=
y1
y2
y4
y3
1/2
1/2
1/3
2/3
!"#$$ & '()*+, -./01(/( 2*+/345 #67 =
-
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4,$5"%, !.%&)%$6$7,8 (7)
! !%B%9D8;µ%,% .'%(B',C1 3%1%4'C1
– 3(0'-"15 9-#:2 µ 5+#(;
• {,,B,2,?,...,=,K,6}
%
G
;
L
0
.
.
.
%
G
;
L
0
.
.
.
1/2
1/2
1/2
1/21/2
1/2
1/2
1/21/2
1/2
2/10000002/1
2/12/1000000
02/12/100000
0002/10000
0002/12/1000
00002/12/100
000002/12/10
0000002/12/1
!
!
!
"""!""""""
!
!
!
!
!
!
!
"
#
$
%
& '
(
! " # $ % & ' (
!"#$$ & '()*+, -./01(/( 2*+/345 #67 >?
-
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4,$5"%, !.%&)%$6$7,8 (8)!
!%B%9D8;µ%,% .'%(B',C1 3%1%4'C1
–
!"#$ &' >
-
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4,$5"%, !.%&)%$6$7,8 (9)!
!%B%9D8;µ%,% .'%(B',C1 3%1%4'C1
–
!"#$ &' ()*
µ 4 # 62
-
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4,$5"%, !.%&)%$6$7,8 (10)
! !%B%9D8;µ%,% .'%(B',C1 3%1%4'C1 –
>2 ()* Z
11
00
)1(
1
! ! ! " !
! ! ! " !
# f
f J, pX(xi)
x1
x2
2, p Y(y j)
y1
y2
( ) ( )[ ] ( ) ( )[ ]( ) ( ) ( )( ) ( ) ( )111
100
1
01
1010
xY
x xY
x xY Y
p f p
fp p p
f f p p p p
!=
+=
"#
$
%&
'
!=
!"#$$ & '()*+, -./01(/( 2*+/345 #67 >@
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4,$5"%, !.%&)%$6$7,8 (11)!
!%B%9D8;µ%,% .'%(B',C1 3%1%4'C1
–
?" µµ 47- &)
()* #
– @-
2-*,% !" µµ #$%&'( -5$ )% 21,µµ>8 *)@ .7$,&, µ-*5B,/#8 [pij]=[PY/X]µ.)1)E$ $, .1)&EK)@$ ,.' /@$?@,/µ)E8 *6$ /*)%=-76$ *#8 &5F- 21,µµ08 &,% *) 7?%) $, /@µB,7$-% &,% 2%, *%8 /*0"-8
•
L$, &,$5"% ">2-*,% µ #%&')* !" µµ #$%&'( (weakly
symmetric)
-5$ )% 21,µµ>8 *)@ .7$,&, µ-*5B,/#8 [pij]=[PY/X]
µ.)1)E$ $, .1)&EK)@$ ,.' /@$?@,/µ)E8 *6$ /*)%=-76$ *#8 &5F- 21,µµ08 &,% *) 5F1)%/µ, *6$ /*)%=-76$ *#8 &5F- /*0"#8 $, -7$,% .5$*, *) 7?%)
[ ] [ ]!!!
"
#
$$$
%
&=
!!!
"
#
$$$
%
&=
6
1
2
1
3
1
2
1
6
1
3
1
3.05.02.0
2.03.05.0
5.02.03.0
ijij p p
=
=
=
3
2
3
2
3
2
+" µµ #$%&'( ,-./0&
1#%&')* +" µµ #$%&'( ,-./0&
!"#$$ & '()*+, -./01(/( 2*+/345 #67 >8
-
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4,$5"%, !.%&)%$6$7,8 (12)
! !%B%9D8;µ%,% .'%(B',C1 3%1%4'C1
– ?" µµ 47-
&' ()*
J, pX(xi)
x1
x3
2, p Y(y j)
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
3213
3212
3211
321321
3.02.05.0
,5.03.02.0
,2.05.03.03.05.02.0
2.03.05.0
5.02.03.0
x p x p x p y p
x p x p x p y p
x p x p x p y p
x p x p x p y p y p y p
x x xY
x x xY
x x xY
x x xY Y Y
++=
++=
++=
!
!!
"
#
$
$$
%
&
=
y1
y2
y3
x2 [ ]!!!
"
#
$$$
%
&
=
3.05.02.0
2.03.05.0
5.02.03.0
ij p
!"#$$ & '()*+, -./01(/( 2*+/345 #67 >9
-
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4,$5"%, !.%&)%$6$7,8 (13)!
!%B%9D8;µ%,% .'%(B',C1 3%1%4'C1
– A4- &B/
?" µµ 47- &' ()*
J, pX(xi)
x1
x2
2, p Y(y j)
( ) ( ) ( )[ ] ( ) ( )[ ]
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )213
212
211
21321
6
1
2
1
,2
1
6
1
,
3
1
3
1
6
1
2
1
3
12
1
6
1
3
1
x p x p y p
x p x p y p
x p x p y p
x p x p y p y p y p
x xY
x xY
x xY
x xY Y Y
+=
+=
+=
!!!
"
#
$$$
%
&
=
y1
y2
y3
[ ]!!!
"
#
$$$
%
&
=
6
1
2
1
3
1
2
1
6
1
3
1
ij p
!"#$$ & '()*+, -./01(/( 2*+/345 #67 >:
-
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4,$5"%, !.%&)%$6$7,8 (14)!
!'K%1F,+,D@ 5'&F97= 3%1%4'7/ D)1 ;10B8M7=µD ,+1 :N797
(3%1F1%@ Bayes)
!
!
1. L/*6 >$, 9=%9'(F &=µµD,B'(F (%1)4' -.%&)%$6$7,8 '.)@ f =0.15
&,% PX={pX(0)=0.9,
pX(1)=0.1}. H.)F>*)$*,8 '*% y=1
*'*- -7$,% ?@$,*'$ $, @.)")27/)@µ- *#$ .%F,$'*#*, *)
/EµB)") .)@ µ-*,?'F#&- $, 0*,$ -.7/#8 1
–
–
!
2. L/*6 >$, (%1)4' D6'(7'1018%@ * '.)@ f =0.15
&,% PX={pX(0)=0.9,
pX(1)=0.1}.H.)F>*)$*,8 '*% y=1
*'*- -7$,% ?@$,*'$ $, @.)")27/)@µ- *#$ .%F,$'*#*, *) /EµB)") .)@ µ-*,?'F#&- $, 0*,$ -.7/#8 1
–
!"
""===
x
x p x y p
x p x y p
y p
x p x y p
y p
y x p y x p
)()/(
)()/(
)(
)()/(
)(
),()/(
!=
===
===
=
=
===
=
=
==
===
1,0
)()/1(
)1()1/1(
)1(
)1()1/1(
)1(
)1,1()1/1(
i
i x pi x y p
x p x y p
y p
x p x y p
y p
y x p y x p
39.09.085.01.085.0
1.085.0)1/1( =
!+!
!
=== y x p
0.1085.01.085.0
1.085.0)1/1( =
!+!
!
=== y x p
!"#$$ & '()*+, -./01(/( 2*+/345 #67 >;
-
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4,$5"%, !.%&)%$6$7,8 (14)
!
!4+B7?7B8% + 7678% µD,%?:BD,%' %6F ,7 (%1)4'.
–
;- *)$ '1) ."#1)3)17, .5$6 ,.' *) &,$5"% -$$))Eµ- *) .)/)/*' *#8 ."#1)3)17,8 .)@ µ-*,3>1-*,% *-"%&5 =6178 "5F)8 µ>/, ,.' *) &,$5"%.
–
"$)@µ- $, µ.)1)E$ $, ,$,&*#F)E$ (.,1,"#3F)E$)
µ- .)"E µ%&10 .%F,$'*#*, "5F)@8.
–
;- 5"", "'2%, F>")@µ- $, µ-*10/)@µ- *#$ ."#1)3)17, # ).)7, /=-*7M-*,% µ- *# : (-7/)?)8 /*) &,$5"%)
&,% # ).)7, -µ.-1%>=-*,% /*#$ H (>A)?)8 /*) &,$5"%)
?#". *#$ ,µ)%B,7, ."#1)3)17, I(:;H)
• I(X;Y)=H(X)-H(X/Y)=H(Y)-H(Y/X)
–
+1,&*%&5 ,@*' .)@ µ,8 -$?%,3>1-% -7$,% $, B1)Eµ- *1'.)@8 >*/% C/*- *, bits
(# ."#1)3)17,)
*, ).)7, /*>"$)$*,% .5$6 ,.' *) &,$5"% $,
µ.)1)E$ $, ,$,&*#F)E$ µ- *>*)%) *1'.) )@*)/C/*- # .%F,$'*#*, "5F)@8 $, -7$,% ,µ-"#*>,.
–
($*7 *)@ /@µB)"%/µ)E I(:;H)
=1#/%µ).)%-7*,% &,% ) /@µB)"%/µ'8 R
&,% *'*- µ%"5µ- 2%, B=KµF µD,)97&+@
!"#$$ & '()*+, -./01(/( 2*+/345 #67 ><
-
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4,$5"%, !.%&)%$6$7,8 (15)
!
N1%/µ>$-8 -1µ#$-7-8 /=-*%&5 µ- *%8 .)/'*#*-8 *#8 ."#1)3)17,8
– G(:/H):
4,"-7*,% &,% %GDG%'F,+,% ,3)E -&315M-% ,&1%BC8 *) .'/) ,B>B,%)% -7 µ,/*- 68 .1)8 *) /EµB)") *#8 -%/'?)@ x
,$ /*#$ >A)?) >=-% "#3F-7 *) y.
1-*,% ,.' *%8 -%/'?)@8 &,% *%8 -A'?)@8 *)@ /@/*0µ,*)8 &,% &,*’
,$*%/*)%=7, *)$ µ>/) ,1%Fµ' bits
&5F- &6?%&08 ">A#8.
– I(:;H):
G (µ)%B,7, ."#1)3)17, /@µB)"7M-% *) ."#1)3)1%&' .-1%-='µ-$) .)@ µ-*,?'F#&- *-"%&5 .5$6 ,.' *) &,$5"%,
?#". G(:)-G(:/H)
!"#$$ & '()*+, -./01(/( 2*+/345 #67 >=
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4,$5"%, !.%&)%$6$7,8 (16)
! Hµ7'G%8% !4+B7?7B8% 3%1%4'C1
–
1. L/*6 >$, 9=%9'(F &=µµD,B'(F (%1)4' -.%&)%$6$7,8 '.)@ f =0.15
– 4#%56$2!7 17: PX={pX(0)=0.9, pX(1)=0.1}.
•
!3,1µ'M)$*,8 *)$ *E.) µ- *)@8 .7$,&-8 -7$,% ?@$,*'$ $, @.)")27/)@µ- PH={pH(0)=0.78,
pH(1)=0.22}.
•
I(:;H)=G(H)-G(H/:)=G(0.22)-G(0.15)=0.76-0.61=0.15 bits
• G(:)=0.47 bits
– 4#%56$2!7 2 7: PX={pX(0)=0.5,
pX(1)=0.5}.
•
!3,1µ'M)$*,8 *)$ *E.) µ- *)@8 .7$,&-8 -7$,% ?@$,*'$ $, @.)")27/)@µ- PH={pH(0)=0.5,
pH(1)=0.5}.
• I(:;H)=G(H)-G(H/:)= G(0.5)-G(0.15)=1-0.61=0.39 bits
• G(:)=1 bit
–
+,1,*#1)Eµ- '*% # ,µ)%B,7, ."#1)3)17, .,71$-% *#$ µ>2%/*# *%µ0 *#8 /*# 2# .-17.*6/# '*,$ µ-2%/*).)%-7*,% *) Px
!"#$$ & '()*+, -./01(/( 2*+/345 #67 $?
-
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4,$5"%, !.%&)%$6$7,8 (17)
! Hµ7'G%8% !4+B7?7B8% 3%1%4'C1
–
2. L/*6 >$, (%1)4' D6'(7'1018%@ * '.)@ f =0.15
– 4#%56$2!7 17: PX={pX(0)=0.9,
pX(1)=0.1}.
•
Nµ)768, PH={pH(0)=0.915, pH(1)=0.085}.
•
I(:;H)=G(H)-G(H/:)= G(0.085)-[0.9*G(0)+0.1*G(0.15)] =0.36
bits
•
G(:)=0.47 bits –
4#%56$2!7 2 7: PX={pX(0)=0.5,
pX(1)=0.5}.
• Nµ)768, PH={pH(0)=0.575, pH(1)=0.425}.
•
I(:;H)=G(H)-G(H/:)=
G(0.575)-[0.5*G(0)+0.5*G(0.15)]=0.98-0.3=0.679 bits
•
G(:)=1 bits –
+,1,*#1)Eµ- '*% &,% /*#$ .-17.*6/# *)@ &,$,"%)E O # ,µ)%B,7, ."#1)3)17, .,71$-% µ-2,"E*-1# *%µ0 *#8 -.7/#8 /*# 2# .-17.*6/# '*,$ µ-2%/*).)%-7*,% *) Px
–
+,1,*#1-7/*- '*% *) O &,$5"% >=-% µ-2,"E*-1# .%/*'*#*, ,.’
'*% *) /@µµ-*1%&' &,$5"%.
!"#$$ & '()*+, -./01(/( 2*+/345 #67 $>
-
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4,$5"%, !.%&)%$6$7,8 (18)
! J% .1>.-% ")%.'$ $, &5$)@µ- 2%, $, µ-*,?C/)@µ- '/) *) ?@$,*'$ .-1%//'*-1# ."#1)3)17,;
H(X/Y) I(X; 2) #( 2 /J)
H(Y)H(J)
H(:,H)H(J, 2)OD;'&,7678+&+
Hµ7'G%8%@
!4+B7?7B8%@ max I(J; 2)
!"#$$ & '()*+, -./01(/( 2*+/345 #67 $$
-
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4,$5"%, !.%&)%$6$7,8 (19)
!
OD;'&,7678+&+ Hµ7'G%8%@ !4+B7?7B8%@
–
G ,µ)%B,7, ."#1)3)17, .,1,*#1)Eµ- '*% -A,1*5*,% ,.' *#$ &,*,$)µ0 PX *#8 *.µ.
: *#8 -%/'?)@ /*) &,$5"%.
–
P%, $, µ-2%/*).)%0/)@µ- *#$ max I(:;H)
F, .1>.-% $, B1)Eµ- *#$ &,"E*-1# ?@$,*0 &,*,$)µ0 PX.
–
L*/% ")%.'$ &,"02)@µ- /*) /@µ.>1,/µ, '*% # =61#*%&'*#*, C,
*)@
&,$,"%)E Q, -7$,%:
•
C(Q)=max I(X;Y)
•
G &,*,$)µ0 P: # ).)7, µ-2%/*).)%-7 *#$ .,1,.5$6 /=>/# ">2-*,% B>"*%/*# &,*,$)µ0.
•
;.)1-7 $, @.51=)@$ .,1,.5$6 ,.' µ7, B>"*%/*-8 &,*,$)µ>8. –
(.' *, .,1,.5$6 -7$,% 3,$-1' '*% ) *1'.)8 µ- *)$ ).)7) )17/,µ- *#$ =61#*%&'*#*, &,$,"%)E C,
,$*%/*)%=-7 /- -&-7$) *) µ>2-F)8 *) ).)7) µ-*15 *#$ .)/'*#*, *#8 ."#1)3)17,8 .)@ -7$,% ?@$,*'$ $, µ-*,3-1F-7 .5$6 ,.' >$, -$F'1@B) ?%,&1%*' &,$5"% =6178 µ$0µ#.
P X
!"#$$ & '()*+, -./01(/( 2*+/345 #67 $@
-
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4,$5"%, !.%&)%$6$7,8 (20)
!
OD;'&,7678+&+ Hµ7'G%8%@ !4+B7?7B8%@ 3%1%4'C1
–
1. .=%9'(F 3%1)4' J0B8@ EFB=G7
• C(Q)=max I(X;Y)=1 bit, +1)/)=0: I(:,H)=G(:)
–
2.
51KFB=G7 (%1)4' µD µ+ D6'(%4=6,FµD1D@ DNF97=@
• C(Q)=max I(X;Y)=1 bit
–
3. 51KFB=G+ ;B%?7µ+$) '*% >=)@µ- 24?%,3)1-*%&5 /EµB)", -5$ µ-*,?7?,µ- µ'$) &5F- ?-E*-1) /EµB)") ?#".B,?,M,F,...,=,6,
*'*- µ'$) ,@*5 *, 12
/EµB)", ,.' *, 24
F, µ.)1)E/,$ $, µ-*,?)F)E$ &,% /*# /@$>=-%, $, ,.)&6?%&).)%#F)E$ =6178
/35"µ,*,.
;- 5"", "'2%, # =61#*%&'*#*, *)@ &,$,"%)E -7$,% log12
bits•
1,/µ, F, &,*,"02,µ- -5$ =1#/%µ).)%)E/,µ- *)$ )1%/µ'
– max I(X;Y)=max [H(Y)-H(Y/X)]=max H(Y)
-1=log24-1=log12
P X
P X
P X P X P X
!"#$$ & '()*+, -./01(/( 2*+/345 #67 $8
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4,$5"%, !.%&)%$6$7,8 (21)!
OD;'&,7678+&+ Hµ7'G%8%@ !4+B7?7B8%@ 3%1%4'C1
–
.=%9'(F &=µµD,B'(F (%1)4', f . •
I(X;Y)=H(Y)-H(Y/X)
=H(Y)-!p(x)H(Y/X=x)
=H(Y)- !p(x)H(f)
=H(Y)-H(f)
R1-H(f) –
.=%9'(F (%1)4' %6F&GD&+@, %.
• max I(X;Y)=max(H(Y)-H(Y/X))
=max(H(Y)-H(,))
=maxH(Y)-H(,)
•
D, µ.)1)E/- $, -7$,% max H(Y)=log3
,""5 ,@*0 # *%µ0 ?-$ -7$,% -3%&*0 2%, &,µµ7, *%µ0 *#8 p:(xi),
i=1,2
•
($ F>/)@µ- pX(x1)=1-., &,% pX(x2)=.,
*'*- ,.' *, pY(yi), i=1,e,2
?7$)$*,% ,.' *)@8 *E.)@8 (B". ?%,35$-%,)
*'*-
– maxH(Y)=max
H((1-,).,,,(1-,)(1-.))=max[(1-,)*G(.)+G(,)]=(1-,)*maxH(.)+G(,)
• N.'*- .1)&E.*-% '*% max
I(X;Y)=maxH(Y)-H(,)=(1-,)*maxH(.)+G(,)-G(,)=1-,
!"#$$ & '()*+, -./01(/( 2*+/345 #67 $9
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4,$5"%, !.%&)%$6$7,8 (22)
! OD;'&,7678+&+ Hµ7'G%8%@ !4+B7?7B8%@ 3%1%4'C1
–
3%1)4' *, f .
• ($ F>/)@µ- pX(x1=0)=1-.,
&,% pX(x2=1)=., *'*- ,.' *, pY(yi), i=1,2
?7$)$*,% ,.' *)@8 *E.)@8 (B". ?%,35$-%,)
*'*-
– H(Y)=H((1-f).)
–
H(Y/X)=.*G(H/:=1)=.*G(f)•
N.'*- max I(X;Y)=max(H(Y)-H(Y/X))
=max(H((1-f).) - .*G(f))
• P%, f=0.15
µ.)1)Eµ- $, B1)Eµ- '*% µ-2%/*).)%-7*,% '*,$ .=0.445
&,% 51, C=0.685
•
+,1,*#1-7/*- '*% pX(x1=0)=0.555,
&,% pX(x2=1)=0.445,
.)@ -7$,% ?%,3)1-*%&0 ,.' *#$ pX(x1=0)=0.5,
&,% pX(x2=1)=0.5,
# ).)7, µ-2%/*).)%-7 *#$ -$*1).7,,
G(:)
!"#$$ & '()*+, -./01(/( 2*+/345 #67 $:
-
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4,$5"%, !.%&)%$6$7,8 (23)
! CB$ &4/
@ µ )$,( !
'-0,*)@µ- ."-)$5M)@/, ."#1)3)17, ,$5 bit
F,
.1)/F>*)@µ- ,$5 )µ5?-8
–
4C?%&,8 Nµ5?,8 )17M-*,% *) /E/*#µ, -&-7$) &6?%&).)7#/#8 &,*5 *) ).)7) µ#$Eµ,*, .#208 s,
µ0&)@8
4, µ-*,*1>.)$*,% /- µ#$Eµ,*, µ-*5?)/#8 t,
µ0&)@8 S,'.)@ S>4.
– J, ."-)$5M)$*, bits, S-4,
)$)µ5M)$*,% bits %/)*%µ7,8 (parity)
–
!5$ *, ."-)$5M)$*, bits
.1)&E.*)@$ 68 21,µµ%&>8 /@$,1*0/-%8 *6$ bits
*#8 .#208 s,
*'*- ) &C?%&,8 ">µ- '*% -7$,% 21,µµ%&'8.
!"#$$ & '()*+, -./01(/( 2*+/345 #67 $;
-
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4,$5"%, !.%&)%$6$7,8 (24)
! EDCB+µ% Shannon
;'% .'%(B',) 3%1)4'%
-
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4,$5"%, !.%&)%$6$7,8 (25)
! (@*' .)@ µ,8 ">-% *) F-C1#µ, -7$,% '*% – ,/8# 9&-'%&$( '-./0& :-%-'$7%5;#$-& -6( 3.- µ 3!= 7 67
-
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+,1,?-72µ,*, 4,$,"%C$ (1)
!
P&(+&+
– 97$-*,% >$, ?%,&1%*' &,$5"% =6178 µ$0µ#.
-
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+,1,?-72µ,*, 4,$,"%C$ (2)
! H6)1,+&+
–
,)
+1C*, .1>.-% $, @.)")27/)@µ- *%8 .%F,$'*#*-8 "0K#8 *6$ &6?%&C$ /@µB'"6$ y j , j=1,
2, 3, 4. !7$,%
–
p(y 1 )= p(y 1,x 1 )+
p(y 1,x 2 )=p(x 1 )
p(y 1 /x 1 )+ p(x 2 )
p(y 1 /x 2 )=a(0,8)+(1-a)(0)=0,8 - .
–
4,*5 ,$5")2) *1'.) @.)")27M)@µ-
–
p(y 2 )= p(y 2 ,x 1 )+
p(y 2 ,x 2 )=p(x 1 )
p(y 2 /x 1 )+ p(x 2 )
p(y 2 /x 2 )=0,8(1-- ),
– p(y 3 )=
p(y 3,x 1 )+
p(y 3,x 2 )=p(x 1 )
p(y 3 /x 1 )+ p(x 2 )
p(y 3 /x 2 )=- (0,1)+(1-- )(0,1)=0,1,
–
p(y 4 )= p(y 4,x 1 )+
p(y 4,x 2 )=p(x 1 )
p(y 4 /x 1 )+ p(x 2 )
p(y 4 /x 2 )=0,1.
–
(.' *%8 .%F,$'*#*-8 ,@*>8,
*6$ ).)76$ *) 5F1)%/µ, -7$,% 1,@.)")27M)@µ- *# µ>/# ."#1)3)17, *#8 -A'?)@ *)@ &,$,"%)E.
– D ( E )=-p(y 1 )log
p(y 1 )-p(y 2 )logp(y 2 )-p(y 3 )log
p(y 3 )-p(y 4 )log p(y 4 )=
–
=-0,8 - log(0,8 - )-0,8(1-a)log(0,8(1-a))
-2(0,1)log(0,1)
–
=0,66-0,8 - log(0,8 - )-0,8(1-a)log(0,8(1-a)).
!"#$$ & '()*+, -./01(/( 2*+/345 #67 @>
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+,1,?-72µ,*, 4,$,"%C$ (3)
! B)
+1C*, @.)")27M)@µ- *%8 /@$?@,/µ>$-8 .%F,$'*#*-8 *6$ /@µB'"6$
-%/'?)@ &,% -A'?)@ *)@ &,$,"%)E (?-7*- &,% *) -1C*#µ, 1).!
!7$,% p(x 1,y 1 )=
p(y 1,x 1 )= p(x 1 )
p(y 1 /x 1 )=0,8 - ,
! p(x 1,y 2 )=
p(y 2 ,x 1 )= p(x 1 )
p(y 2 /x 1 )=0,
! p(x 1,y 3 )=
p(y 3,x 1 )= p(x 1 )
p(y 3 /x 1 )=0,1a,
! p(x 1,y 4 )=
p(y 4,x 1 )= p(x 1 )
p(y 4 /x 1 )=0,1a,
!
p(x 2 ,y 1 )=
p(y 1,x 2 )= p(x 2 )
p(y 1 /x 2 )=0,
! p(x 2 ,y 2 )=
p(y 2 ,x 2 )= p(x 2 )
p(y 2 /x 2 )=0,8(1-a),
! p(x 2 ,y 3 )=
p(y 3,x 2 )= p(x 2 )
p(y 3 /x 2 )=0,1(1-a),
! p(x 2 ,y 4 )=
p(y 4,x 2 )= p(x 2 )
p(y 4 /x 2 )=0,1(1-a).
! JC1, @.)")27M)@µ- *#$ ,B-B,%'*#*, *)@ &,$,"%)E
!
D ( E / F )=-p(x 1,y 1 )logp(y 1 /x 1 )
- p(x 1,y 2 )log
p(y 2 /x 1 ) -
p(x 1,y 3 )log p(y 3 /x 1 )
- p(x 1,y 4 )log
p(y 4 /x 1 )-p(x 2 ,y 1 )logp(y 1 /x 2 )
-p(x 2 ,y 2 )logp(y 2 /x 2 )
- p(x 2,y 3 )logp(y 3 /x 2 )
-p(x 2 ,y 4 )logp(y 4 /x 2 )=
!
-0,8a.log0,8+0-0,1alog0,1-0,1alog0,1-0-0,8(1-a)log0,8-0,1(1-a)log0,1-0,1(1a)log0,1=-0,8log0,8-0,1log0,1-0,1log0,1=0,2575+0,66.
!"#$$ & '()*+, -./01(/( 2*+/345 #67 @$
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+,1,?-72µ,*, 4,$,"%C$ (4)
! 2)
P%, *)$ .1)/?%)1%/µ' *#8 =61#*%&'*#*,8 *)@ &,$,"%)E F, .1>.-% $, B1)Eµ- *%8 .%F,$'*#*-8 -µ35$%/#8 *6$ /@µB'"6$ *#8 -%/'?)@,
2%, *%8 ).)7-8 µ-2%/*).)%-7*,% # ,µ)%B,7, ."#1)3)17, µ-*,AE *#8 -%/'?)@ &,% *#8 -A'?)@ *)@ &,$,"%)E,?#",?0 *#$ *%µ0 - .
! !7$,%
!
G /@$51*#/# ,@*0 µ-2%/*).)%-7*,% '.68 2$617M)@µ- 2%, *#$ *%µ0 *)@ a
.)@ µ#?-$7M-% *#$ .1C*# *#8 .,15262).
[ ]
[ ].2575,0))1(8,0log()1(8,0)8,0log(8,0max
)66,02575,0))1(8,0log()1(8,0)8,0log(8,066,0max
)]/()([max);(max
)(
)(
)()(
!!!!!=
!!!!!!=
!==
aaaa
aaaa
X Y H Y H Y X I C
x p
x p
x p x p
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-
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+,1,?-72µ,*, 4,$,"%C$ (5)
! !.)µ>$68,
!
D>*)$*,8 ,$6*>16 *#$ *%µ0 ,@*0 *)@ a,
",µB5$)@µ- *# =61#*%&'*#*, *)@ &,$,"%)E
! C=0,8 bits/symbol.
[ ]
[ ][ ]
[ ]( )
.2
112
10
1log
018,0
8,0log8,0))1(8,0log()8,0log(8,0
log8,0))1(8,0log(8,0log8,0)8,0log(8,0
log11)1)(1())1(8,0log()1(8,0log)8,0log(8,0
))1)(log(1())1(8,0log()1(8,0
))8,0(log()8,0log()(8,0
2575,0)1(8,0log()1(8,0)8,0log(8,0);(
0=
!==
"#
$
%&
'
(!=
(
!=(
(=(((=
+(+((=
)*+,-
.(
((+((()*+,-
. +(=
/((+(/((
/+/(=
/(((((=
a
a
a
a
a
a
aaa
eaea
ea
aaaeaa
aaaa
aaaa
aaaada
Y X dI
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+,1,?-72µ,*, 4,$,"%C$ (6)! L/*6 >$, ?@,?%&' /@µµ-*1%&' &,$5"% =6178 µ$0µ#.
J) &6?%&'
,"35B#*) /@µB'"6$ /*#$ -7/)?) *)@ &,$,"%)E ?7?-*,% ,.' *#$ *@=,7, µ-*,B"#*0 :={0,1}
µ- .%F,$'*#*-8 -µ35$%/#8 P(X=0)=1/2,P(X=1)=1/2,
-$C # *@=,7, µ-*,B"#*0 H={0,1}
/@µB)"7M-% *%8 *%µ>8 -A'?)@.
N .7$,&,8 µ-*5B,/#8 *)@ &,$,"%)E -7$,%
!
(i) S, @.)")2%/*-7 # ,B-B,%'*#*, *#8 : ?-?)µ>$)@ '*% 2$617M)@µ- *#$ H.
!
(ii) S, @.)")2%/*-7 # ,µ)%B,7, ."#1)3)17, *)@ &,$,"%)E.!
(iii) P%, .)%, *%µ0 *)@ ‘-’,
# =61#*%&'*#*, *)@ &,$,"%)E C
.,71$-%
*#$ µ>2%/*# *%µ0 *#8;
! (iv) P%, .)%, *%µ0 *)@ ‘-’,
# =61#*%&'*#*, *)@ &,$,"%)E C
.,71$-% *#$ -"5=%/*# *%µ0 *#8;
1
1
! !
! !
"# $% &"' (
!"#$$ & '()*+, -./01(/( 2*+/345 #67 @9
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+,1,?-72µ,*, 4,$,"%C$ (7)
! i)
P%, $, B1)Eµ- *#$ ,B-B,%'*#*, *#8 : ?-?)µ>$)@ '*% 2$617M)@µ- *#$ H,
?#",?0, G(:/H)
F, &5$)@µ- =10/# *)@ *E.)@
– G(:,H) = G(H) + G(:/H)
!
P%, ,@*' .1>.-% .1C*, $, B1)Eµ- *%8 .%F,$'*#*-8 U(H=0)
&,% U(H=1).
(@*>8 B17/&)$*,% ,.' *#$ /=>/#
–
– G(H) = 1
! N% .%F,$'*#*-8 U(:,H)
?7$)$*,% ,.' *)$ *E.)
U(:,H)=U(H/:)U(:)'.)@ )% U(H/:)
-7$,% )% .%F,$'*#*-8 *)@ .7$,&, µ-*5B,/#8.
( ) ( ) ( ) ( )
( ) ( )( ) ( )( ) ( ) ( )( )
( )
( )
10 1 0 1
1
0 0 1 1
1 0 1 1
10
2
11
2
P Y P Y P X P X
P Y P X P X
P Y P X P X
P Y
P Y
! !
! !
! !
! !
"# $# $ # $= = = = = %& '( ) ( ) "( )
= = = " + =%
= = = + = "
= =
= =
1
!"#$$ & '()*+, -./01(/( 2*+/345 #67 @:
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+,1,?-72µ,*, 4,$,"%C$ (8)
– U(:=0,H=0) = U(H=0/:=0)U(:=0) = (1--)(1/2)
–
U(:=0,H=1) = U(H=1/:=0)U(:=0) = -/2
– U(:=1,H=0) = U(H=0/:=1)U(:=1) = -/2
– U(:=1,H=1) = U(H=1/:=1)U(:=1) = (1--)(1/2)
– H(X,Y) = -(1--) log (1--) – - log
- +1
!
(.' *%8 1 &,% 2 .1)&E.*-%
!
G(:/H) = G(:,H) - G(H) = -(1--) log (1--) – - log
-
!
ii)
(.' *)$ *E.) *#8 ,µ)%B,7,8 ."#1)3)17,8 &,% .,1,*#1C$*,8 '*% G(:)=1>=)@µ-
– I(:;H) = G(:) - G(:/H) = 1 + (1--) log (1--)
+- log -
!
iii)
G =61#*%&'*#*, *)@ &,$,"%)E .,71$-% *#$ µ>2%/*# *%µ0 *#8 '*,$ @.51=-%
,.)@/7, F)1EB)@ &,% # .#20 .,71$-% *#$ µ>2%/*# *%µ0 *#8.
$# .-17.*6/# # G(:)
>=-% *#$ µ>2%/*# *%µ0 *#8 &,% *) &,$5"% -7$,% =6178 F'1@B) '*,$ -=0
0 -=1
!
i$)
G =61#*%&'*#*, *)@ &,$,"%)E .,71$-% *#$ -"5=%/*# *%µ0 *#8 '*,$ # I(:;H)
.,71$-% *#$ -"5=%/*# *%µ0 *#8.
+,71$)$*,8 *#$ .1C*# .,15262) 68 .1)8 - &,% -A%/C$)$*,8 µ- µ#?>$ B17/&)@µ- '*% # -"5=%/*# *%µ0 -7$,% '*,$
-=0.5
&,% 51, # =61#*%&'*#*, *)@ &,$,"%)E -7$,% 0.
2
-
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+,1,?-72µ,*, 4,$,"%C$ (9)!
97$-*,% >$, ?%,&1%*' &,$5"% =6178 µ$0µ#.
-
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+,1,?-72µ,*, 4,$,"%C$ (10)
!
i) +1C*, .1>.-% $, @.)")27/)@µ- *%8 .%F,$'*#*-8 "0K#8 *6$ &6?%&C$
/@µB'"6$ y j , j=1, 2, 3.
!7$,%
! p(y 1 )= p(y 1,x 1 )+
p(y 1,x 2 )=p(x 1 )
p(y 1 /x 1 )+ p(x 2 )
p(y 1 /x 2 )=a(1/2)+(1-a)(1/2)=1/2.
! 4,*5 ,$5")2) *1'.) @.)")27M)@µ-
! p(y 2 )=
p(y 2 ,x 1 )+
p(y 2 ,x 2 )=p(x 1 )
p(y 2 /x 1 )+ p(x 2 )
p(y 2 /x 2 )=(1+- )/4
&,% ! p(y 3 )=
p(y 3,x 1 )+
p(y 3,x 2 )=p(x 1 )
p(y 3 /x 1 )+ p(x 2 )
p(y 3 /x 2 )=(1-- )/4.
! (.' *%8 .%F,$'*#*-8 ,@*>8,
*6$ ).)76$ *) 5F1)%/µ, -7$,% 1,@.)")27M)@µ- *# µ>/# ."#1)3)17, *#8 -A'?)@ *)@ &,$,"%)E.
! D ( E )=-p(y 1 )log
p(y 1 )-p(y 2 )logp(y 2 )-p(y 3 )log
p(y 3 )=
!
=-(1/2)log(1/2)-((1+a)/4)log((1+a)/4)
-((1-a)/4)log((1-a)/4)
! =(1/2)-[((1+a)/4)(log(1+a) -2)]
–[((1-a)/4)(log((1-a)-2),
-@=A &!:A#& log(x/ y)=logx-logy.
! =(3/2)-(1/4)(1+a)log(1+a)-(1/4)(1-a)log(1-a).
!"#$$ & '()*+, -./01(/( 2*+/345 #67 @=
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+,1,?-72µ,*, 4,$,"%C$ (11)
!
ii) +1C*, @.)")27M)@µ- *%8 /@$?@,/µ>$-8 .%F,$'*#*-8 *6$ /@µB'"6$
-%/'?)@ &,% -A'?)@ *)@ &,$,"%)E (?-7*- &,% *) -1C*#µ, 1).
! !7$,% p(x 1,y 1 )=
p(y 1,x 1 )= p(x 1 )
p(y 1 /x 1 )=- (1/2),
! p(x 1,y 2 )=
p(y 2 ,x 1 )= p(x 1 )
p(y 2 /x 1 )=- (1/2),
! p(x 1,y 3 )=
p(y 3,x 1 )= p(x 1 )
p(y 3 /x 1 )=0,
!
p(x 2 ,y 1 )=
p(y 1,x 2 )= p(x 2 )
p(y 1 /x 2 )=(1-a)(1/2), ! p(x 2 ,y 2 )=
p(y 2 ,x 2 )= p(x 2 )
p(y 2 /x 2 )=(1-a)(1/4),
! p(x 2 ,y 3 )=
p(y 3,x 2 )= p(x 2 )
p(y 3 /x 2 )=(1-a)(1/4).
!
JC1, @.)")27M)@µ- *#$ ,B-B,%'*#*, *)@ &,$,"%)E
!
D ( E / F )=-
p(x 1,y 1 )logp(y 1 /x 1 ) -
p(x 1,y 2 log p(y 2 /x 1 )
- p(x 1,y 3 )log p(y 3 /x 1 )
- p(x 2 ,y 1 ) log
p(y 1 /x 2 ) -
p(x 2 ,y 2 )logp(y 2 /x 2 )
-
p(x 2 ,y 3 )logp(y 3 /x 2 )=
! =(a/2)+(a/2)+0+[(1/2)-(a/2)]+ [(1/2)-(a/2)]+
[(1/2)-(a/2)]=
! =(3/2)-(a/2).
!"#$$ & '()*+, -./01(/( 2*+/345 #67 8?
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+,1,?-72µ,*, 4,$,"%C$ (12)
!
iii) P%, *)$ .1)/?%)1%/µ' *#8 =61#*%&'*#*,8 *)@ &,$,"%)E F, .1>.-% $,
B1)Eµ- *%8 .%F,$'*#*-8 -µ35$%/#8 *6$ /@µB'"6$ *#8 -%/'?)@,
2%, *%8 ).)7-8 µ-2%/*).)%-7*,% # ,µ)%B,7, ."#1)3)17, µ-*,AE *#8 -%/'?)@ &,% *#8 -A'?)@ *)@ &,$,"%)E,
?#",?0 *#$ *%µ0 - .
! !7$,%
!
G /@$51*#/# ,@*0 µ-2%/*).)%-7*,% '.68 2$617M)@µ- 2%, *#$ *%µ0 *)@ a.)@ µ#?-$7M-% *#$ .1C*# *#8 .,15262).
!.)µ>$68,
!"
#$%
&'''++'=
!"
#$%
&'''''++'=
'==
)1log()1(
4
1)1log()1(
4
1
2
max
)22
3()1log()1(
4
1)1log()1(
4
1
2
3max
)]/()([max);(max
)(
)(
)()(
aaaaa
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X Y H Y H Y X I C
x p
x p
x p x p
!"#$$ & '()*+, -./01(/( 2*+/345 #67 8>
+ ? 7 4 " C (13)
-
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+,1,?-72µ,*, 4,$,"%C$ (13)!
D>*)$*,8 ,$6*>16 *#$ *%µ0 ,@*0 *)@ a,
",µB5$)@µ- *# =61#*%&'*#*, *)@ &,$,"%)E
! C=0,162 bits.
!
4-%-$7%#5!$# #65!7* ($&
-
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1-*,% .5$6
,.' *) &,$5"% µ- ?-?)µ>$-8 *%8 .%F,$'*#*-8 /@µB'"6$ -%/'?)@
! :61#*%&'*#*, 4,$,"%)E =
;>2%/*) +)/' ."#1)3)17,8 .)@
-7$,% ?@$,*'$ $, µ-*,3-1F-7 .5$6 ,.' *) &,$5"%.
–
;-2%/*).)7#/# (µ)%B,7,8 +"#1)3)17,8 68 .1)8 *%8 .%F,$'*#*-8 -%/'?)@
!
J, .,1,.5$6 .1)V.)F>*)@$ E.,1A# F)1EB)@ /*) &,$5"%.
!
$-8 *%8 .%F,$'*#*-8 /@µB'"6$ -%/'?)@ -7$,% # G(:)
–
G =61#*%&'*#*, *)@ &,$,"%)E -7$,% ,."5 # maxH(X)
.)@ -.%*@2=5$-*,% /- .-17.*6/# %/).7F,$6$ /@µB'"6$
! I/=E)@$ ?#",?0
! I(:;H) R C R H(X) R maxH(X)=logn
!"#$$ & '()*+, -./01(/( 2*+/345 #67 8@
-
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!16*0/-%8
!"#$$ & '()*+, -./01(/( 2*+/345 #67 88
-
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46?%&).)7#/# !">2=)@ "-%)% 4C?%&-8
!
9@W&)7 4C?%&-8
-
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!%/,2620 /*#$ 46?%&).)7#/#
+#20
46?%&).)%#*08 (.)&6?%&).)%#*08
9>&*#8
ŝ s
t r 4,$5"%
!.%&)%$6$7,8
46?%&).)%#*08 +#208/") ;)$*>")
;)$*>")
s!
;)$*>") ;)$*>")
s!ˆ
;)$*>")
D'1@B)8
-
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!%/,2620 /*#$ 46?%&).)7#/# !
G ED0B8% 309'(7678+&+@ -7$,% # µ-">*# µ-F'?6$ 2%, *#$
,.)*-"-/X,*%&0 &,% )1F0 X-*,3)15 *#8 ."#1)3)17,8 ,.' *#$ .#20 /*)$
.1))1%/X'.! Y,/%&'8 /*'=)8 =
I.5:.#"!7 + G&(%82!7 /3,"µ5*6$.
P%, *)$ /&).' ,@*'
.1)/*7F-*,% '/)8 ."-)$,/µ'8 -7$,% ,.,1,7*#*)8 2%, -.%F@µ#*0 µ-76/# *#8 .%F,$'*#*,8 /35"µ,*)8 µ-*5?)/#8
!
N% &6?%&>8 ">A-%8 -7$,% ,&)")@F7-8 ?@,?%&C$ K#376$ ..=.
C ={00, 10, 01,11}.
J) ."0F)8 *6$ &6?%&C$ ">A-6$ -$'8 &C?%&, C
/@XB)"7M-*,% X- |C|.
!
L$,8 &C?%&,8 )$)X5M-*,%
8 ">A-%8 >=)@$ *) 7?%) X0&)8.
!
$%&'(:@ !%B%97&*#8 >=-% *#$ ?@$,*'*#*, $, "5B-% '"-8 *%8 ">A-%8 .)@ µ-*,?'F#&,$,
X-
0 =6178 /35"X,*,. +.=.
-5$ /- -7/)?) 01001
*'*- /- >A)?) ",$F,/µ>$# 7/68 ,""5 %/)µ0 ,&)")@F7, K#376$ ?#".
00101 ,""5 '=% 0011
! T, /35"X,*,, ?#",?0 ) F'1@B)8,
-X3,$7M)$*,% ?%,/&)1.%/X>$, &,*5 *@=,7) *1'.) &,% '=% /- /@/*5?-8 (bursts).
G .%F,$'*#*, ?#" $, ,"")%6F-7 >$, bit
&,*5 *# X-*5?)/# -7$,% # 7?%, X- ,@*0 ).)%)@?0.)*- 5"")@ bit
&,% ?-$ -.#1-5M-*,% ,.' /35"X,*, /- 2-%*)$%&5 ?@,?%&5 K#37,.
-
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!%/,2620 /*#$ 46?%&).)7#/# ! I8
26 /6 *)@ &,$,"%)E.
–
!.-%?0 &5F- &,$5"% ,A%).%/*7,8 0RpR1/2
µ.)1-7 $, µ-*,*1,.-7 /- >$, &,$5"% 1/2RpR1,
F, F-61)Eµ- '*% 1/2RpR1.
!
N B=KQF@ 64+B7?7B8%@ -$'8 &C?%&, -7$,% *) .)/)/*' *#8 ,1=%&08 ."#1)3)17,8 (?#".
*, K#37, ."#1)3)17,8 .1%$ *#$ &6?%&).)7#/#)
.)@ µ-*,3>1-*,% ,.' *#$ &6?%&0 ">A#.
N 1@FX'8 ."#1)3)17,8 -$'8 ?@,?%&)E &C?%&, C
µ0&)@8 n -7$,% 7/)8 X- (1/n)log
2
|C| , &-7*,% ?- µ-*,AE 0 (|C|=1 )
&,% 1(|C| = 2n)
– ..=. -5$ C1 = {00, 10, 01, 11},
*'*- ) 1@FX'8 ."#1)3)17,8 -7$,%: (1/n)log2|C1|
= (1/2)log24= 1).
– !5$ C2 = {000, 101, 011, 110},
*'*- ) 1@FX'8 ."#1)3)17,8 -7$,%: (1/n)log2|C2|
=(1/3)log24 = 2/3).
–
L/*6 '*% µ-*,?7?-*,% *) µ0$@µ, 0010
&,% .,1,",µB5$-*,% *) 0011.
9-$ @.51=-% ?@$,*'*#*, ,$7=$-@/#8 /*#$ .-17.*6/# *)@ C1.
–
($ 'µ68 =1#/%µ).)%0/)@µ- *)$ C2
'.)@ *) *-"-@*,7) K#37) &5F- &6?%&08 ">A#8 -7$,% *) K#37) -">2=)@ %/)*%µ7,8 *'*- '*,$ /*-7")@µ- *#$ &6?%&0 ">A# 000
&,% .,1,"5B)@µ- *#$ ">A# 001,
,@*0 ?-$ ,.)*-"-7 &6?%&0 ">A# &,% >*/% &,*,",B,7$)@µ- '*% .,1,"5B,µ- "5F)8.
–
9-$ -7$,% .5$*)*- ?@$,*0 # ,$7=$-@/# &,% ,&'µ, ?@/&)"'*-1# # ?%'1F6/#
!
Y,/%&>8 +15A-%8 .)@ )17M)$*,% µ-*,AE *6$ ?@,?%&C$ ">A-6$:
– (XOR) .1'/F-/# : 0 + 0 = 0, 0 + 1 = 1, 1 + 0
= 1 &,% 1 + 1 = 0
–
(AND) .)"//µ'8: 0 . 0 = 0, 0 . 1 = 0, 1 . 0 = 0 &,% 1 .
1 = 1.
-
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!%/,2620 /*#$ 46?%&).)7#/#
! RB'&QF@:
J)-2/ (Weight)
Hamming 0 ,."5 B51)8, wt(x), X%,8
">A#8 x X0&)@8 n
K#376$ )$)X5M-*,% *) ."0F)8 *6$ K#376$ *#8 ">A#8,
*, ).)7, -7$,% 7/, X- *) «1».
J) B51)8 .,71$-% *%X>8 ,.' 0
>68 n
–
!%B)9D';µ%: P%, x1 = 000000, x2 = 000010, x3 = 110001
wt(x1)=0,wt(x2)=1, wt(x3)=3
! RB'&QF@:
I6'A-%8 -X3,$7M)@$ ,/@X36$7, *)@ ?@,?%&)E K#37)@.
G ,.'/*,/# .,71$-% *%X>8 ,.' 0
>68 n.
!
!E&)", µ.)1-7 $, .,1,*#1#F-7 '*% d(x,y)=wt(x+y)
– !%B)9D';µ%: d(x1, x2)= wt(x1+x2) = wt(000000
+000010)=wt(000010)=1 d(x1, x3)=wt (x1+x3)=wt(110001)=3, d(x2, x3)
=wt(110011)=4
-
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!%/,2620 /*#$ 46?%&).)7#/# !
AD1'() ;'% (09'(7678+&+:
!
+1C*, -.%">2-*,% >$,8 F-*%&'8 ,&>1,%)8 k,
*) µ0&)8 &5F- ?@,?%&08 ">A#8 .)@ ,$*%/*)%=-7 /- >$, X0$@X,.
J) X0&)8 *6$ ">A-6$ -.%">2-*,% 2kT|M|,'.)@ |M|
-7$,% *) ."0F)8 *6$ ?@$,*C$ X#$@X5*6$.
! !.%.">)$,
-.%">2-*,% *) ."0F)8 *6$ n-k
?@,?%&C$ K#376$ .)@ F, .1)/*-F)E$ /- &5F- ">A# (."-)$,/X'8)
>*/% C/*- $, X.)1-7 $, ,$%=$-E-*,% 0 &,% $, ?%)1FC$-*,% *) -.%F@X#*' ."0F)8 /3,"X5*6$.
L*/% .1)&E.*)@$ )% &6?%&>8 ">A-%8 .)@ ,$*%/*)%=)E$ /*, ?@$,*5 X#$EX,*,,X0&)@8 n
K#376$,
– J, ."-)$5M)$*, bits, n-k,
)$)µ5M)$*,% bits %/)*%µ7,8 (parity)
– !5$ *, ."-)$5M)$*, bits
.1)&E.*)@$ 68 21,µµ%&>8 /@$,1*0/-%8 *6$ bits
*#8 .#208 s,
*'*- ) &C?%&,8 -7$,% 21,µµ%&'8.
!
!5$ *, µ# &6?%&).)%#µ>$, k
?@,?%&5 K#37, (K#37, ."#1)3)17,8)
µ-*,?7?)$*,% µ,M7 µ- *, (n-k)
."-)$5M)$*, .)@ .,152)$*,% &,*5 *#$ &6?%&).)7#/#,
*'*- ) &C?%&,8 )$)µ5M-*,% /@/*#µ,*%&'8-/- ,$*7F-*# .-17.*6/# -7$,% µ#-/@/*#µ,*%&'8
".=.
– !5$ u1, u2, u3 K#37, ."#1)3)17,8
&,% # &6?%&0 ">A# -7$,% u1,u2,u3,x4
µ- x4=bit %/)*%µ7,8=u1+u2+u3
) &C?%&,8 -7$,% /@/*#µ,*%&'8
–
!5$ # &6?%&).)%#µ>$# ">A# -7$,% # x1,
x2, x3 µ- x1= u1+u2, x2=u1+u3,x3=u2+u3
*'*- ) &C?%&,8 -7$,% µ# /@/*#µ,*%&'8
-
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!%/,2620 /*#$ 46?%&).)7#/# !
!SRT2!R >UH"OHTR> -
H!R>TH># 3V.I3H
!
Z8 K-'7"62 A# y
.)@ -"03F# /*)$ ,.)?>&*#,?#",?0 - =
x + y.
! [>X- '*% ) &C?%&,8 C
,$%=$-E-% *) .1'*@.) /35"X,*)8 -,
,$ &,% X'$) ,$ x + - = y
?-$ -7$,% &6?%&0 ">A#,
(2%, &5F-) " x # C.
!
Z8 I6'$)@ '*% d(x, y) = wt(x + y),
# ,.'/*,/# *)@ &C?%&, C
-7$,% 7/# X- *#$ -"5=%/*# *%X0 *)@ B51)@8 wt(x
+ y), µ- x, y # C &,% x \ y.
! !%B)9D';µ% –
L/*6 ) C = {0000, 1010, 1111}
!5$ # 1010
µ-*,?'F#&- *1-%8 3)1>8 &,% ) ?>&*#8 >",B- X- *# /-%15 *%8 ">A-%8 0100,
1001 &,% 1010
*'*- *, ,$*7/*)%=, .1'*@., /35"µ,*)8 -7$,% 1010
+ 0100 = 1110, 1010 + 1001= 0011&,% 1010 + 1010 = 0000.
G ,.'/*,/# ?- *)@ &C?%&, -7$,% 7/# X- d(0000,
1010) =d(1010, 1111) =wt (1010 +1111) =wt (0101) = 2.
-
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!%/,2620 /*#$ 46?%&).)7#/# !
AD1'() ;'% %67(09'(7678+&+:
!
L/*6 x,y # ,.)/*,"-7/,/"#3F-7/, ">A#.
!5$ y \x &,% *) y
?-$ -7$,% &6?%&0 ">A#, y$ C
(,$7=$-@/# "5F)@8),
*'*- 2%, "0K# ,.'3,/#8 =1#/%µ).)%-7*,% # H67(09'(7678+&+ OD;8&,+@ !'K%1F,+,%@ (HO!) :
!
($ @.51=-% X'$) X7, &6?%&0 ">A# x,
# ).)7, -X3,$7M-% *#$ X%&1'*-1# ,.'/*,/# ,.' *# ">A# y,
/- /E2&1%/# X- '"-8 *%8 5""-8 &6?%&>8 ">A-%8,
*'*- # y
,.)&6?%&).)%-7*,% 68 x(µ #8 *%8 &6?%&>8 ">A-%8 =>
!4-B+@ H67(09'(7678+&+
O:;'&,+@ !'K%1F,+,%@ (!HO!)
–
!7*- -.,$,",µB5$-*,% # µ-*5?)/# =>
H,D4-@ H67(09'(7678+&+
O:;'&,+@ !'K%1F,+,%@ (HHO!). !.,$5"#K# X-*5?)/#8 X.)1-7 $, M#*#F-7 &,% '*,$,@.51=-% X%, X'$) &6?%&0 ">A# -22E*-1# /*# ">A# y,
µ- .)"E µ-25"# ,.'/*,/#
!
!%B)9D';µ%:!5$ |M|=2, k = 1, n = 3 &,% C={000,
111}, B,/%M'µ-$)% /*#$
((;+: –
N ,.)?>&*#8 ,.)3,/7M-% 2%, «111»,
-3'/)$ -7*- y= 111 0 y= «011», «101
0 «110»,
&,FC8 >=)@$ X%&1'*-1# ,.'/*,/# .1)8 *#$ «111»
,.' ',*% .1)8 «000».
– //-1%8 ?@,?%&>8 ,&)")@F7-8.
G .-17.*6/# *#8 7/#8 ,.'/*,/#8 *#8 "#3F-7/,8 ">A#8 ?-$ -7$,% ?@$,*0 X- *)$ &C?%&, ,@*'
-
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!%/,2620 /*#$ 46?%&).)7#/#
!
Y,/%&0 @.'F-/# '*% >=)µ- /=-?'$ ,.)&"-%/*%&5 ?@,?%&' /@µµ-*1%&' &,$5"% (Binary
Symmetric Channel)
• L/*6 .(x, y)
# .%F,$'*#*, $, X-*,?'F#&- # ">A# x
&,% $, -"03F# ,.' *)$ .,1,"0.*# # ">A# y.
• !5$ )% ">A-%8 x,y µ0&)@8 n
?%,3>1)@$ &,*5 d K#37, *'*- n-d
K#37, µ-*,?'F#&,$ /6/*5 &,% d
-/3,"µ>$, &,% %/=E-%: .(x,y)=pn-d(1 – p)d.
! EDCB+µ%
! D-61)EX- >$, BSC
&,$5"% X- Q < p < 1,
?E) &6?%&>8 ">A-%8 x1 &,% x2
&,% X7, ">A# y, '"-8 X0&)@8 n
K#376$, &,FC8 &,% '*% )% x1
&,% y ?%,3>1)@$ /- d1
K#37, &,% )% x2 &,% y /- d2 K#37,.
J'*- .(x1, y) R .(x2, y)
,$ &,% X'$) ,$ d1 T d2.
!
(.'?-%A# P%, $, %/=E-% # ,$%/'*#*, .(x1,
y) R .(x2, y) ,1&-7 $, %/=E-%
! pn – d1(1 – p)d1 R pn – d2(1– p)d2
0 (p/(1 – p))d2 – d1 R 1.
!3'/)$ %/=E-% (p/(1 – p))>1 *'*- d2
R d1.
11
00
1
!" !
!" !
#
p
p
p
-
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!%/,2620 /*#$ 46?%&).)7#/#
! H18?%4µ),01:
!
L4B-5 µ # I(
+(4"A# y =x + - ?-$ ,$0&-% /*)$ C
(-3'/)$ ,.'/*,/# C =d)
&,% /@$-.C8 ,$%=$-E/%µ#. !5$ -=x+y
µ- wt(-)=d
*'*- @.51=-% *)@"5=%/*)$ >$, *>*)%) - µ- y=x+- # C
&,% 51, y µ# ,$%=$-E/%µ) /35"µ,
! !%B)9D';µ%: N &C?%&,8 C = {000,
111} >=-% ,.'/*,/# d=
wt(000 +111) =wt(111)=3.
]1, ,$%=$-E-*,% ,.' *)$ &C?%&, &5F- .1'*@.) /35"X,*)8 - B51)@8 1
0 2 ('.68 *, 011 , 100 &"..).
9-$ ,$%=$-E-% 'µ68 *) .1'*@.) /35"X,*)8 «111»
B51)@8 3(".=. y= 000+-= 000+111=111 # C)
-
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!%/,2620 /*#$ 46?%&).)7#/#
!
.'FBK0&+ >?%4µ),01
!
RB'&µF@: L$,8 &C?%&,8 C
9'7BKC1D' *) .1'*@.) /35"X,*)8 - ,$ " x
# C, *'*- # r = x +
- -7$,% -22E*-1, /*# x
,.' ',*% /- ).)%,?0.)*- 5""# ">A# z
# C. L$,8 &C?%&,8 C
&,"-7*,% B – ?%'1F6/#8,
,$ ?%)1FC$-% '", *, - µ- wt(-)
R B &,% ?-$ ?%)1FC$-% *)@"5=%/*)$ >$, - B51)@8 B +
1.
!
L4B-5 µ # !
'-0,.-% $, >=-% ,.'/*,/# dW2t+1
–
H6F9D'N+: L/*6 - µ- ->0
&,% wt(-) R d-1 &,% x,y # C , x\y.
P%, *) 1) /&>")8 ,1&-7 $, ?-%=F-7 '*% d(x,
x + -) < d(y, x + -).
P-$%&5 µ-*,AE *6$ ,.)/*5/-6$ *1%C$ ">A-6$ 7/)@ µ0&)@8,
%/=E-%: d(y, x + -) + d(x + -, x) Td(y, x) Td.
[,XB5$)$*,8
@.'K# '*% d(x + -, x) = wt(x + - + x) = wt(-)
&,% wt(-) R (d – 1)/2 0 2wt(-) + 1
R d,%/=E-% d(y, x + -) + wt(-) Td T 2wt(-) + 1
&,% 51, d(y, x + -) Twt(-) + 1 =d(x, x + -)+ 1.
– !.)X>$68, d(x, x + -) < d(y, x + -)
&,% .1)&E.*-% '*% ) &C?%&,8 ?%)1FC$-% *) .1'*@.) /35"X,*)8 -.
Nµ)768 ,.)?-%&$E-*,% &,% *) 2) /&>")8
(*)⌊=⌋=
µ- *) µ-2,"E*-1) ,&>1,%) $R=
d(y, x) + d(x, z) Td(y, z)
d(y, z) + d(z, x) Td(y, z)
!%/,2620 /*#$ 46?%&).)7#/#
-
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!%/,2620 /*#$ 46?%&).)7#/# !
P1,3%&0 (.'?-%A# *)@ D-610µ,*)8 9%'1F6/#8 A-%8 &,% ,&*7$, t
-7$,% µ# -.%&,"@.*'µ-$-8 ).'*- *, /35"µ,*, ('.68 *) r)
.)@
&-7$*,% /*#$ .-1%)=0 *)@8 -7$,% ?%)1FC/%µ, &,*5 (;+ /- ,$*7F-/# µ- (B)
.-17.*6/# '.)@ )% /3,71-8 -.%&,"E.*)$*,% &,% 51, &5.)%, /35"µ,*, .)@ ,$0&)@$ /*#$ *)µ0 *6$ .-1%)=C$ %/,.>=)@$ ,.' &6?%&>8 ">A-%8 &,% /@$-.C8 -7$,% µ# ?%)1FC/%µ,.
# ) I6'
-
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!%/,2620 /*#$ 46?%&).)7#/# !
!%B)9D';µ%: +)%, .1'*@., /35"µ,*)8 ?%)1FC$)$*,% ,.' C
= {000,
111};
!
A# «111»,
-3'/)$ ("5B-% ,@*0$ 0 "5B-% X7, ,.' *%8 «011»,
«101 0 «110», ?#",?0 ?%)1FC$-%:
–
*, .1'*@., /35"X,*)8 -1= 111 + 011 = 100, -2=111
+ 101 = 010 &,% -3=111 + 110 = 001
/*#$ .-17.*6/# X-*5?)/#8 *#8 ">A#8 «111».
!
Nµ)768 ) ,.)?>&*#8 /@X.-1,7$-% *#$ «000»,
-3'/)$ "5B-% ,@*0$ 0 "5B-% X7, ,.' *%8 «001»,
«010» 0 «100», ?#",?0 ?%)1FC$-%:
–
J, .1'*@., /35"X,*)8 1000 + 100 = 100, 000 + 010
= 010 &,% 000 + 001
= 001.! $, /- «000».
-
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PU(;;I4NI 4Z9I4!
-
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P1,µµ%&)7 4C?%&-8 !
5'&%;0;- - RB'&µ78 - I9'F,+,D@
!
L$,8 &C?%&,8 C )$)X5M-*,% O-#HH
&'/
,$ " x,y#C, *'*- &,% (x +
y)#C.+.=. ) C = {00, 11} -7$,% 21,XX%&'8,
-$C ) C = {00, 10, 11} ?-$ -7$,% 21,XX%&'8,
,3)E # 11 + 10 = 01
?-$ -7$,% &6?%&0 ">A#.
!
45F- ">A# -$'8 21,µµ%&)E &C?%&, ,.)*-"-7*,% ,.' *, bits
*)@ ,1=%&)E µ#$Eµ,*)8 &,% ,.' >$, ,1%Fµ' ."-o$,M'$*6$ bits
%/)*%µ7,8
.)@ .1)&E.*)@$ 68 21,µµ%&>8 /@$,1*0/-%8 *6$ bits
*)@ ,1=%&)E µ#$Eµ,*)8
!
45F- 21,XX%&'8 &C?%&,8 .-1%>=-% *# X#?-$%&0 ">A#,
(.1)&E.*)@/, ,.' *) 5F1)%/X, X%,8 &6?%&08 ">A#8 X- *)$ -,@*' *#8).
!
+"-)$-&*0µ,*, 21,µµ%&C$ &6?7&6$ /- /=>/# µ- µ# 21,µµ%&)E8.
–
G ,.'/*,/# -$'8 21,µµ%&)E &C?%&, -7$,% 7/# X- *) -"5=%/*) *6$ B,1C$ *6$ X# µ#?-$%&C$ &6?%&C$ ">A-6$.
–
G &6?%&).)7#/# -$'8 21,XX%&)E &C?%&, -7$,% .%) ,."0 &,% >=-% µ%&1'*-1-8 ,.)F#&-@*%&>8 ,.,%*0/-%8.
–
H.51=-% X7, .%) ,."0 ?%,?%&,/7, ,.)&6?%&).)7#/#8 µ>2%/*#8 .%F,$'*#*,8 (+(;+ 0 ((;+)
,.' '*% >=-% ,$,3-1F-7 >68 *C1,
–
G .-1%21,30 *6$ /@$'"6$ *6$ .1)*E.6$ /35"X,*)8 .)@ ,$%=$-E-% 0 ?%)1FC$-% >$,8 21,µµ%&'8 &C?%&,8 -7$,% .%) -E&)"# ,.' ',*% /*# 2-$%&0 .-17.*6/#.
-
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P1,µµ%&)7 4C?%&-8 -
;,F#µ,*%&' H.'B,F1)
! H.51=-% /*-$0 ,$*%/*)%=7, -$$)%C$ *#8 21,XX%&08 5"2-B1,8 &,% -$$)%C$ *#8 F-617,8 &6?%&).)7#/#8
–
(.,1,7*#*# # -µB5F@$/# /*%8 .,1,&5*6 >$$)%-8 *#8 21,µµ%&08 5"2-B1,8 2%, /,3>/*-1# &,*,$'#/# *6$ 21,µµ%&C$ &6?7&6$
X117'D@ ED0B8%@ 309'(7678+&+@
" P1,XX%&'8 4C?%&,8
" [>A#
" P1,XX%&'8 &C?%&,8 C =
" 46?%&0 ">A# *)@ C =
" _@W&'8 &C?%&,8
" Y5/# 21,XX%&)E &C?%&,
"
_%5/*,/# 21,XX%&)E &C?%&,
X117'D@ AB%µµ'(-@ P4;DGB%@
" _%,$@/X,*%&'8 @.)=C1)8
" _%5$@/X,
"
P1,XX%&' ,$5.*@2X, @.)/@$'")@ S*)@ ?%,$@/X,*%&)E =C1)@,
(_%,$@/X,*%&'8 @.)=C1)8)
" _%5$@/X, *)@
" N1F)2C$%) /@X."016X, @.)/@$'")@ S
" Y5/# ?%,$@/X,*%&)E @.)=C1)@
"
_%5/*,/# ?%,$@/X,*%&)E @.)=C1)@
-
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P1,µµ%&)7 4C?%&-8-;,F#µ,*%&' H.'B,F1)
! N ?%,$@/X,*%&'8 =C1)8 4n (K =
{0, 1}), ,.,1*7M-*,% ,.' *%8 B,FX6*>8
.)/'*#*-8 (scalars) 0 &,% 1
&,% *) /E$)") *6$ ?%,$@/X5*6$ (0 ">A-6$)X0&)@8 n,
&,FC8 &,% ,.' *%8 .15A-%8 *#8 .1'/F-/#8 ?%,$@/X5*6$ (">A-6$)
&,% *)@ .)"//X)E B,FX6*C$ .)/)*0*6$ µ- ?%,$E/µ,*, .)@ %&,$).)%)E$ µ7, /-%15 %?%)*0*6$
! E$ '7574/
7,( 6-)84,( 75/ 6-'A-%8 *)@ %?7)@ X0&)@8 n),
, &,% B )% B,FX6*>8 .)/'*#*-8 *)@ ?%,$@/X,*%&)E =C1)@ 4n &,% 0
# X#?-$%&0 ">A#.
1. (y + z) # 4n ,
2. , ⋅y # 4n ,
3. x + y = y + x,
4. (x + y) + z = x + (y + z),
5.
(, ⋅ B) ⋅ y = , ⋅(B ⋅ y),
6. (, + B) ⋅ x = , ⋅ x +
B ⋅ y,
7.
a ⋅(x + y) = a ⋅ x + a ⋅ y,
8. (x + 0) = x,
9. 1 ⋅ x = x,
10.
H.51=-% ">A# x' # 4n,
*>*)%, C/*- x + x' = x' + x = 0.
-
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P1,µµ%&)7 4C?%&-8-;,F#µ,*%&' H.'B,F1) !
L$, X# &-$' @.)/E$)") S
*)@ ?%,$@/X,*%&)E =C1)@ 4n -7$,% >$,8 @.)=C1)8
,@*)E,
,$ 2%, &5F- M-E2)8 ?%,$@/X5*6$ (">A-6$)
y, z # S, %/=E-% (y + z) # S&,% ,⋅y
# S 2%, &5F- B,FX6*0 .)/'*#*, ,.
]1, C
-7$,% >$,8 21,XX%&'8 &C?%&,8 ,$ &,% X'$) ,$ C
-7$,% @.)=C1)8 *)@ 4n.
!
J) /E$)") '"6$ *6$ 21,XX%&C$ /@$?@,/XC$ *6$ ?%,$@/X5*6$ (0 ">A-6$)-$'8 /@$'")@ S
= {z1, z2,`, zk} )$)X5M-*,% *) O-#HH
&' #()67"OH# 72" S
A# y
-7$,% 21,XX%&'8 /@$?@,/X'8 *6$ ">A-6$ z1,`,
zk ,$ % B,FX6*>8 .)/'*#*-8 ,1,`, ,k
*>*)%-8 C/*-: y = a1⋅z1+ `+ ,k⋅zk.
! J) 21,XX%&' ,$5.*@2X,
&5F- @.)/@$'")@ S
*)@ 4n -7$,% @.)=C1)8 *)@ 4n. >2
*4 7# O-#HH &'/
&B$ /
.1)&E.*6$ -& *)@ S.
! J) 21,XX%&' ,$5.*@2X,
*)@ @.)/@$'")@ S
*)@ 4n .-1%>=-% *# X#?-$%&0 ">A#,
'"-8 *%8 ">A-%8 *)@ S
&,% *, -.7 µ>1)@8 ,F1)7/X,*5 *)@8
! !%B)9D';µ%: J) 21,XX%&' ,$5.*@2X, *)@ @.)/@$'")@ S
= {00011,
11100},.)*-"-7*,% ,.' *%8 ,&'")@F-8 ">A-%8:
00000, 00011, 11100, 00011 + 11100 =11111. !.)X>$68, ={00000,
00011, 11100, 11111}.
! To 1#0H,7' O ('H4(2 $M2 $ #("
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P1,µµ%&)7 4C?%&-8-;,F#µ,*%&' H.'B,F1) !
!M2 $ #(M
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P1,µµ%&)7 4C?%&-8-;,F#µ,*%&' H.'B,F1)
! L$, /E$)") ?%,$@/X5*6$ S = {z1,
z2,`, zk} -7$,% 9 -# HH & B/
I(48)-7572 ,$ ∌ B,FX6*>8 .)/'*#*-8 ,1,
,2,`, ,k,
X- *)@"5=%/*)$ X7, -A ,@*C$ \ 0,
>*/% C/*-: ,1 ⋅z1 + ` + ,k ⋅zk. = 0.
9%,3)1-*%&5, S =21,XX%&C8 -A,1*#X>$).
– L$, /E$)"), S
.)@ .-1%>=-% *# X#?-$%&0 ">A# =
21,XX%&C8 -A,1*#X>$).
– ($ S\{0},
.-1%>=-% >$, X>2%/*#8 ?%5/*,/#8 21,X.
,$-A51*#*) @.)/E$)")
! !%B)9D';µ%: J) S = {110, 011, 101, 111}
=21,XX%&C8 -A,1*#X>$)
–
I/=E-% ,1 ⋅110 + ,2 ⋅011 + ,3 ⋅101 + ,4 ⋅111 = 000
' ,1 ,1 0 + 0 ,2 ,2+,3 0 ,3 + ,4 ,4 ,4 = 000, ' ,1 + ,3 +
,4 = 0, ,1 + ,2 + ,4 = 0 &,% ,2 +,3+ ,4 = 0.
+1)/F>*)$*58 *-8 ' ,4 = 0. L*/%, ,1 + ,3 =
0, ,1 + ,2 = 0 &,% ,2 + ,3 = 0, ' ,1 = ,2 = ,3.
;.)1)EX- $, -.%">A)@X- ")%.'$ ,1 = ,2 = ,3=
1
&,% -.)X>$68 *) S=21,XX%&C8 -A,1*#X>$)
!
P%, *#$ -E1-/# -$'8 ,.' *, X>2%/*#8 ?%5/*,/#8,
21,X. ,$-A51*#*)@,
@.)/@$'")@ *)@ S, ,3,%1)EX- ,.' *) S
*) .1C*) ?%5$@/X, .)@ B17/&)@X- $, -7$,% 21,XX.
/@$?@,/X'8 *6$ 5""6$ ?#". *) 110 (110=011+101)
&,% -A-*5M)X- ,$ *) S' = {011,101, 111} =
21,X. ,$-A51*#*).
–
(.' *#$ 0 ,1 ,1 +,2 0 ,2 + ,3 ,3 ,3 = 000, ',2
+ ,3 = 0 (1), ,1 + ,3 = 0(2), ,1 + ,2 + ,3 =0 (3).
(.' *%8 (2), (3) ' ,1 = 0 51, ,3 = 0
&,% ,2 = 0. ' ∌ B,FX6*>8 .)/'*#*-8,
\ 0, .)@ %&,$).)%)E$ *#$ (1) ' S’ =
>$,, X>2%/*#8 ?%5/*,/#8, 21,XX.
,$-A51*#*) @.)/E$)") *)@ S.
-
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P1,µµ%&)7 4C?%&-8-;,F#µ,*%&' H.'B,F1) !
L$, X# &-$' @.)/E$)") ?%,$@/X5*6$ Y -$'8 ?%,$@/X,*%&)E @.)=C1)@ V
-7$,%
X7, J)$)) -7$,% *) B ={011, 101, 111} ' Y=
B5/# *)@ = {000,011,101,111,110,100,010, 001}.
!
L$,8 ?%,$@/X,*%&'8 (@.))=C1)8 >=-% .)"">8 B5/-%8 µ- 7?%) ."0F)8 ?%,$@/µ5*6$ k
µ- k=*4 7# $ )
-
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!E1-/# Y5/#8 P1,µµ%&)E 4C?%&, C!
TDA-%8 *)@ C), µ- C =
! L$,8 .7$,&,8 B17/&-*,% /- X)130 C*
H#&,7;/
!
)7#85/ 9-#HHB( ( C!9 ; Row
Echelon Form), ,$ '"-8 )% X#?-$%&>8
21,XX>8 -7$,% /*) &5*6 X>1)8 &,% *) .1C*) K#37) «1»
(«)?#2'8»)X%,8 21,XX08 -7$,% /- /*0"# .%) ?-A%5 /- /=>/# X- *) .1C*) (*)$ )?#2')
«1» *6$ .1)#2)EX-$6$ 21,XXC$.
;%, /*0"# .)@ .-1%>=-% >$,$ «)?#2'»
1 &,"-7*,% /*0"# «)?#2'8»
! L$,8 .7$,&,8 B17/&-*,% /- X)130 !DB'7B'&µ:1+@ 34'Q%(0,-@ .'),%N+@ AB%QQC1 (!3.A - Reduced
Row Echelon Form), -5$
-7$,% /- µ)130 49P &,% -.%.">)$ %/=E-% '*% /- &5F- /*0"# )?#2' *)@ .7$,&, (?#",?0 /*0"# .)@ .-1%>=-*,% ) )?#2'8 «1»
µ%,8 21,XX08)
?-$ .-1%>=)$*,% 5"", K#37, «1»
,""5 X'$) K#37, «0»
–
45F- .7$,&,8 X- 0 &,% 1
X.)1-7 $, *-F-7 *'/) /- X)130 49P '/) &,% +49P.
–
L$,8 .7$,&,8 (X- 0 &,% 1)
X.)1-7 $, >=-% .)"">8 X)13>8 49P ,""5
µ'$)$ µ7, X)130 +49P
-
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!E1-/# Y5/#8 P1,µµ%&)E 4C?%&, C
! L$,8 .7$,&,8 *7F-*,% /- µ)130 49P 0 +49P µ- *)$ µ-*//µ' *)@ /- ?%,?)=%&)E8 %/)?E$,µ)@8 .7$,&-8 µ>/6 *6$ ,&)")EF6$ ?E) *E.6$ /*)%=-%6?C$ .15A-6$ .)@ -3,1µ'M)$*,% /*%8 21,µµ>8 *)@:
– ($*,"",20 ?E) 21,µµC$ &,%
–
($*%&,*5/*,/# µ7,8 21,µµ08 ,.' *) 5F1)%/µ, *)@ -,@*)E *#8 µ- &5.)%, 5""#
! !%B)9D';µ%
– L/*6 ) &C?%&,8 C =
{0000, 1110, 0111, 1001},
*)@ o.)7)@ )% ">A-%8 ,.)*-")E$ *%8 21,XX>8 *)@ .7$,&, P.
J)$ µ-*/M)µ- /- X)130 4_P =1#/%µ).)%C$*,8 *%8 68 5$6 .15A-%8 68 -A08:
-
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!E1-/# Y5/#8 P1,µµ%&)E 4C?%&, C
! !%B)9D';µ% (>=1:
-
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!E1-/# Y5/#8 P1,µµ%&)E 4C?%&, C!
!%B)9D';µ% (>=1:=-% *)$ )?#2' «1»
X%,8 21,XX08 $, X#$ .-1%>=-% 5"", K#37, «1».
! (.' *)$ ,1=%&' .7$,&, P
)?#2)EX,/*- /*) 2) .7$,&, µ- ,$*,"",20 *#8 1#8
µ- *#$ 4# 21,µµ0 &,% *#8 2#8 µ- *#$ 3#.
! (.' *) 2) .7$,&, /*)$ 3) X- *#$ ,$*%&,*5/*,/# *#8 3#8 21,XX08 X- *)
5F1)%/X, *)@ -,@*)E *#8 &,% *#8 1#8 21,XX08.!
J>")8,
,$*%&,F%/*C$*,8 *#$ 3# 21,XX0 *)@ 3)@ .7$,&, X- *) 5F1)%/X, *)@
-,@*)E *#8 &,% *#8 2#8 21,XX08 &,*,"02)@X- /*)$ 4) .7$,&,
! +152X,*%,
) 4)8 .7$,&,8 -7$,% /- X)130 +4_P,
,3)E )% ?E) .1C*-8 *)@ /*0"-8 .)@ .-1%>=)@$ *)@8 )?#2)E8 «1»
*6$ ?E) X# µ#?-$%&C$ 21,XXC$
.-1%>=)@$ µ'$) K#37, «0»
/*%8 @.'")%.-8 F>/-%8
-
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!E1-/# Y5/#8 P1,µµ%&)E 4C?%&, C!
H1%(D?%4%'0,'() >2%/*#8 ?%5/*,/#8 21,XX%&C8 ,$-A51*#*, @.)/E$)", *)@ ?%,$@/X,*%&)E (@.))
=C1)@ C (0 *)@ /@$'")@ S, '.)@ C= )
-7$,% B5/# *)@.
! !.7/#8, '.68 .1),$,3>1F#&-,
)% X# µ#?-$%&>8 21,XX>8 -$'8 .7$,&, /- X)130 4_P 0 +4_P .)@ /=#X,*7M-*,% ,.' *%8 ">A-%8 *)@ /@$'")@ S
(0 ,.' *%8 ">A-%8 *)@ C, '.)@ C =
),
-7$,% )% ">A-%8 -$'8 X>2%/*#8 ?%5/*,/#8 21,XX%&C8 ,$-A51*#*)@ @.)/@$'")@ *)@ C
&,%, -.)X>$68, X7, B5/# *)@.
!
1)@X- /- X)130 4_P 0 +4_P *)$ .7$,&, .)@ /=#X,*7M-*,% ,.' *%8 ">A-%8 *)@ /@$'")@ S
(0 &,% ,.' '"-8 *%8 ">A-%8 *)@ C,
C = ).
! ..=. /*) .1)#2)Eµ-$) +,15?-%2X,,
*) @.)/E$)") {1110, 0111} -7$,% X%,
B5/# *)@ &C?%&, C = {0000, 1110, 0111,
1001}, ,.' *#$ ).)7, .1)&E.*)@$,
X- 21,XX%&)E8 /@$?@,/X)E8,
'"-8 )% ">A-%8 *)@ C.
!
Nµ)768 µ.)1-7 $, B1-F-7 &,% X7, B5/# *)@ ?@W&)E &C?%&, C&,=1#/%µ).)%C$*,8 *%8 ">A-%8 .)@ *)$ ,.,1*7M)@$ 68 ,$6*>16.
–
!$,"",&*%&5 µ.)1-7 $, -3,1µ)/*-7 >$,8 .%) ,."'8 ,"2'1%Fµ)8 B,/%/µ>$)8 /*#$ 2$C/# *6$ ">A-6$ *)@ /@$'")@ S
0 *)@ C, C =
) ).)7)8 .-1%2153-*,% /*# /@$>=-%,.
!E1-/# Y5/#8 9@W&)E &C?%&, C&
-
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1 # #8
!
G ?%,?%&,/7, -E1-/#8 µ%,8 B5/#8 *)@ ?@W&)E &C?%&, C&(C
= ) µ- k=?%5/*,/# C, n=µ0&)8 ">A-6$ C
?%,&17$-*,% /*, ,&'")@F, B0X,*,:
1.
8 *)@ ;.
–
8 *)@ ,$5/*1)3)@ .7$,&, GT)
&,% -7$,% ?%5/*,/#8 n-k.
!E1-/# Y5/#8 9@W&)E &C?%&, C&
-
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1 # #8
! !%B)9D';µ%:
O#*-7*,% X7, B5/# *)@ C& µ- S =
{11101, 10110, 01011,
11010} &,% C= !
8 *#8 ">A#8 *)@ S
) ).)7)8 µ-*/
M-*,% /- +49P =1#/%µ).)%C$*,8 ,$*,"",20 0 &,% ,$*%&,*5/*,/# 21,µµ08 ,.' *) 5F1)%/µ5 *#8 µ- 5""#
G
M
!E1-/# Y5/#8 9@W&)E &C?%&, C&
-
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1 # #8
! !%B)9D';µ% (&=1:
-
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P-$$0*)1,8 +7$,&,8 P1,µµ%&)E
4C?%&, C
! RB'&µF@: 45F- .7$,&,8,
*)@ ).)7)@ )% 21,XX>8 ,.)*-")E$ X7,
B5/# *)@ &C?%&, C,
)$)X5M-*,% 2-$$0*)1,8 .7$,&,8 G
2%, *)$ C.
–
J) ."0F)8 *6$ 21,XXC$ k *)@ G =
X- *# ?%5/*,/# *)@ C,?#",?0 =
µ- *) ."0F)8 *6$ ">A-6$ .)@ ,.,1*7M)@$ X7, B5/# *)@.
–
($ # ?%5/*,/# -$'8 &C?%&, C
= k,
*) µ0&)8 *6$ &6?%&C$ ">A-6$ =
n &,% # ,.'/*,/0 *)@ d, *'*- ) C
/@µB)"7M-*,% 68 (n, k,
d)21,XX%&'8 &C?%&,8.
! ($ G
-7$,% >$,8 2-$$0*)1,8 .7$,&,8 2%, >$,$ &C?%&, C,
*'*- &5F- .7$,&,8 %/)?E$,X)8 *)@ G,
68 .1)8 *%8 21,XX>8,
-7$,% 2-$$0*)1,8 .7$,&,8 2%, *)$ C.
!
P%, *#$ -E1-/# *)@ G
-$'8 &C?%&, C,
,1&-7 $, B1)EX- X%, B5/# *)@ ' *%8 X# X#?-$%&>8 ">A-%8 *)@ .7$,&, /- X)130 +4_P,
.)@ /=#X,*7M-*,% ,.' *%8 ">A-%8 *)@ C
(0 *)@ S, '.)@ C =
), '.68 -?-7=F# .1)#2)@µ>$68
-
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P-$$0*)1,8 +7$,&,8 P1,µµ%&)E
4C?%&, C
!
309'(7678+&+: L$,8 2-$$0*)1,8 .7$,&,8 G
2%, >$, 21,XX%&' &C?%&,
(n, k, d), C,
µ.)1-7 $, ,A%).)%#F-7 2%, *#$ &6?%&).)7#/# ">A-6$ (µ#$@X5*6$)
u X0&)@8 k K#376$ 68 -A08:
! C=u.G= [a1,a2,`ak] [g1,g2,...gk]T = a1.g1+
a2.g2+`+ak.gk =[b1,b2,..., bn],
– u= [a1,a2,`ak]
# µ# &6?%&0 ">A# *)@ µ#$Eµ,*)8 &,% c=[b1,b2,...,
bn] # &6?%&0 ">A# µ0&)@8 n
K#376$ &,% [g1,g2,...gk]T *, ?%,$E/µ,*, ">A-%8 (."0F)@8 n)
.)@ ,.)*-")E$ *#$ B5/# *)@ C
&,% ,.,1*7M)@$ *)$ G
–
J) ,.)*>"-/µ, c
-7$,% &6?%&0 ">A# *)@ C
(-3'/)$ u.G = 21,µ.
/@$?@,/µ'8 *6$ 21,µµC$ *)@ G,
.)@ /=#µ,*7M-*,% ,.' *%8 ">A-%8 gi
.)@ ,.,1*7M)@$ µ7, B5/# *)@ &C?%&, C).
! L$,8 2-$$0*)1,8 .7$,&,8 G,
(kxn), *)@ ).)7)@ )% .1C*-8 k
/*0"-8 /=#X,*7M)@$ *)$ .7$,&, *,@*'*#*,8 Ik,
(kxk), ?#",?0 G = [Ik ,
M],">2-*,% '*% B17/&-*,% /- ,=6'(- Q7B?- (standard
form).
–
..= N 2-$$0*)1,8 .7$,&,8 G,
) ).)7)8 .1)&E.*-% /*) B0X, 2
*)@ /-
.1)#2)Eµ-$) .,15?-%2µ, -7$,% /- *@.%&0 X)130.
– N &C?%&,8 C,
) ).)7)8 >=-% 2-$$0*)1, .7$,&, G
/- *@.%&0 X)130,=,1,&*#17M-*,% &=&,+Q%,'(F@ (C9'(%@
-
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P-$$0*)1,8 +7$,&,8 P1,µµ%&)E
4C?%&, C!
$, *>*)%) /@/*#µ,*%&' &C?%&,
*) µ0$@X, u
,.)*-"-7*,% ,.' *, .1C*, kK#37, *)@ c,
,3)E c = u.G = u.[I, M] = [uI, uM] = [u, uM].
J, .1C*, k K#37,
*6$ &6?%&C$ ">A-6$ ">2)$*,% K#37, ."#1)3)17,8 &,% *, @.'")%., n
–
k,."-)$,/X'8 0 K#37, -">2=)@ %/)*%X7,8.
!
9-$ >=)@$ '")% )% 21,XX%&)7 &C?%&-8 2-$$0*)1-8 .7$,&-8 /- *@.%&0 X)130 [I,;]
–
A-6$ ,.)*-")E$*,% ,.' *, ,1=%&5 K#37, *)@ µ#$Eµ,*)8 ?-$ -7$,% 'µ68 /@$-='µ-$, ,""5 -$*).7M)$*,% /- k
.1)&,F)1%/µ>$-8 F>/-%8 *#8 &6?%&08 ">A#8 .)@ )17M)$*,% ,.' *%8 ,$*7/*)%=-8
F>/-%8 *6$ /*#"C$-)?#2C$ *)@ G –
,-& !# -"$? $7. 6#%56$2!7 = G
µ 6=%#5 .- µ #$-!:7 µ -$&!8#5 !# $"6&'? µ =%@? µ #
'-$/00707 #.-00-
-
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+7$,&,8 I/)*%µ7,8 P1,µµ%&)E
4C?%&, C
! K (#/
3*=O+2" E$*)8 *)@ C
$, µ.)1)EX- .5$*, $, .1)/?%)17/)@X- *)$ .7$,&, H
! ($ # ?%5/*,/# *)@ C
-7$,% k, *'*- # ?%5/*,/# *)@ C&,
-7$,% n – k = µ- ."0F)8
*6$ /*#"C$ *)@ H,
-$C *) ."0F)8 *6$ 21,XXC$ *)@ -7$,% n=7/) X- *) X0&)8 *6$ &6?%&C$ ">A-6$ *)@ &C?%&, C.
–
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+7$,&,8
I/)*%µ7,8 P1,µµ%&)E 4C?%&, C!
EDCB+µ%: G ,.'/*,/# d
-$'8 21,µµ%&)E &C?%&, =
µ- *)$ -"5=%/*)
,1%Fµ' 21,µµC$ *)@ +7$,&, I/)*%µ7,8 H
*6$ ).)76$ *) 5F1)%/µ,
%/)E*,% µ- 0 (0 %/)?E$,µ, o
-"5=%/*)8 ,1%Fµ'8 21,µµC$ .)@ -7$,% 21,µµ%&5 -A,1*#µ>$-8).
! H6F9D'N+: " x # C ' x ⋅ H= 0
' [x1,x2,`,xn] ⋅ [h1,h2,`,hn]T = 0 ' [x1 h1+x2
h2+`+xn hn] = [0,0,`0], hi= i 21,µµ0 *)@ H.
[,µB5$)$*,8 @.'K# '*% xi =1 0 0
&,% wt(x) =a 1’s ' x H =
µ- *) 5F1)%/µ, wt(x)21,µµC$ *)@ H.
]1, -5$ xmin=# x µ- *) -"5=%/*) wt(x)
= d =,.'/*,/#
*)@ &C?%&, C, *'*- xmin ⋅ H =
µ- *)$ -"5=%/*) ,1%Fµ' 21,µµC$ d
*)@ H.)@ %/)E$*,% µ- 0.
! !%B)9D';µ%: L/*6 ) G
.1)#2)Eµ-$)@ .,1,?-72µ,*)8 µ-
!
J'*- µ%, &,% %/=E-% '*% 0 2# +
3# 21,µµ0 = (1 1) + (1 1)
(0 %/)?E$,µ, # 1# +4# = (0 1) + (0 1) )
*)@ G = (0 0)
' # ,.'/*,/# *)@ &C?%&, -7$,% d=2
< 5? P E 4C? C
-
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=17µ)9% (coset) =
,.,1,7*#*# >$$)%, 2%, %67(09'(7678+&+ D1F@ (C9'(%
C! RB'&µF@: ;%, /@$)X5?, *)@ C
6-2A-6$ *#8 X)1308 c + x, '.)@ c # C
(*) x
-7$,% /*,F-1' &,% *) c&@X,7$-*,% /- '") *) -E1)8 *)@ C).
_#",?0, C + x = {c + x| c# C}.
!
!%B)9D';µ%: O#*)E$*,% '"-8 )% /@$)X5?-8 *)@ &C?%&, C
= {000, 111}, k=1,
n=3, n-k=2. – P%, x = 001 >=)@X- C + 001
= {000 + 001 = 001, 111 + 001 = 110}.
– C + 010 = {010, 101},
– C + 100 = {100, 011},
– C + 011 = {011, 100},
–
C + 101 = {101, 010}, – C + 110 = {110,
001},
–
C + 111 = C + 000 = {000, 111}.
–
+,1,*#1)Eµ- '*% ) ,1%Fµ'8 *6$ '"6$ *6$ /@$)µ5?6$ -7$,% 2n-k=22=4
&,% &5F- /@$)µ5?, ,.)*-"-7*,% ,.' 2k=21=2
">A-%8 '/-8 ,&1%BC8 &,% *) ."0F)8 *6$ &6?%&C$ ">A-6$ *)@ &C?%&, C
–
!.7/#8 .,1,*#1-7/*- '*% -5$ x,y # C+x
(0 %/)?E$,µ, /*) C+y) *'*- x+y # C
-
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8 /@$)X5?-8 '/-8 &,% )% ">A-%8 X0&)@8 n
*)@ Kn &,FC8 -7$,% ?@$,*' ?E) /@$)X5?-8
C + x = C + y $, *,@*7M)$*,%,
,&'X, &,% ,$ x \ y
'.68 ..= %/=E-% /*) .1)#2)Eµ-$) .,15?-%2µ,.
! E$ '7574/
?"(2 µ )$,( :
($ C
-7$,% >$,8 /@/*#X,*%&'8 21,XX%&'8 &C?%&,8 (n,k,d)
&,% x &,% y
?E) ">A-%8 -.7/#8 X0&)@8 n K#376$,
*'*- %/=E)@$ ,$,3)1%&5 X- *%8 /@$)X5?-8 *, ,&'")@F,:
1. 45F- ">A# x
.-1%>=-*,% /*# /@$)X5?, C + x
("'26 E.,1A#8 µ#?-$%&08 &6?%&08 ">A#8 /- 21,µµ%&)E8 &C?%&-8)
2.
J) ."0F)8 *6$ ">A-6$ /- X%, /@$)X5?, -7$,% 7/) X- *) ."0F)8 *6$ ">A-6$ *)@ &C?%&, C,
?#",?0 |C + x| = |C| = 2k.
3.
($ µ%, ">A# x
.-1%>=-*,% /*# /@$)X5?, C + y,
*'*- %/=E-% C + x = C+ y,
?#",?0 &5F- ">A# *#8 /@$)X5?,8 *#$ .1)/?%)17M-%.
4.
J) ."0F)8 *6$ ?%,3)1-*%&C$ /@$)X5?6$ *)@ &C?%&, C,
?%5/*,/#8 k, %/)E*,% X- 2n – k .
( (.' %?%'*#*, 5
' ."0F)8 /*)%=-76$ '"6$ *6$ /@$)µ5?6$=
2n ' -3'/)$ ,.' %?%'*#*, 2,
|C + x| = 2k *'*- ."0F)8 /@$)µ5?6$= 2n / 2k=
2n – k)
-
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A# x# 4n .-1%>=-*,% X'$) /- X7, /@$)X5?, *)@ C,
?#",?0 ?E) /@$)X5?-8 C + x &,% C + y
-7*- *,@*7M)$*,% -7*- ?-$ >=)@$ &,$>$, /*)%=-7) &)%$'.
•
(.'?-%A#:
L/*6 '*% @.51=)@$ ?E) ">A-%8 7?%-8 /- '", *, /E$)", *6$ /@$)µ5?6$ ' ?E) .-1%.*C/-%8:
,)
S, B17/&)$*,% /*#$ 7?%, /@$)µ5?, >/*6 *)@ C+x.
J'*- .1>.-% x +ci = x+cj (ci, cj # C, ci\cj )
' ci= cj 5*).).
•
B)
S, B17/&)$*,% /- ?%,3)1-*%&>8 /@$)µ5?-8 >/*6 *6$ C+
x, C+y. J'*- F, 7/=@-: x + ci = y + cj ' y= x+ (ci +
cj), µ- (ci+cj) # C ' N% x, y
.1>.-% $, ,$0&)@$ /*#$ 7?%, /@$)µ5?,,
5*).) µ%58 &,% # @.'F-/0 µ,8 -7$,% '*% x,
y ,$0&)@$ /- ?%,3)1-*%&>8 /@$)µ5?-8,
*6$ C+ x &,% C+y.
6. ($ *) 5F1)%/X, x + y # C,
*'*- )% x,y
.-1%>=)$*,% /*#$ 7?%, /@$)X5?,.
($ *) 5F1)%/X, x + y ∉ C,
*'*- )% x,y
.-1%>=)$*,% /- ?%,3)1-*%&>8 /@$)X5?-8.
•
(.'?-%A#: (’ /&>")8.
L/*6 '*% x, y
# /*#$ 7?%, /@$)µ5?, >/*6 *)@ z+C&,% ,8 @.)*-F-7 '*% x+y
∉ C J'*- %/=E-% x=z+c1, y=z+c2 ' x+y= (z+z)
+(c1+c2) = 0 + c3 ' x+y=c3, c3 # C ' x+y # C,
5*).).
•
Nµ)768 2%, Y’ /&>")8
>/*6 '*% x+y ∉ C &,%
,8 @.)*-F-7 '*% x, y # /*#$
7?%, /@$)µ5?, >/*6 *)@ z+C.
' &,*,"02)µ- .5"% /*#$ x+y=c3, c3
# C' x+y # C, 5*).)
(.)&6?%&).)7#/# µ-
-
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! L/*6 ) 21,XX%&'8 &C?%&,8 C
&,% '*% >=)µ- µ-*5?)/# *#8 &6?%&08 ">A#8 x,
.)@ )?#2-7 /*# "0K# *#8 ">A#8 y.
J'*- >=)@X- *) .1'*@.) /35"X,*)8 - =
x + y. ]1, %/=E-% &,% x = - + y
&,% /@$-.C8 (I?%'*#*, 6) 7' 4:ND'@ D (6BF,=67 &?)4Q%,7@)
(%' y
(4+?KD8&% 4:N+)6DB':$)@ '*% ,.' y
-$*).7/F#&- # &,*5""#"# /@$)µ5?,,
",µB5$-*,% @.'K# '*% .1'*@., /35"X,*)8 µ%&1)E B51)@8 ",XB5$)@$ =C1, X- *# µ-2,"E*-1# .%F,$'*#*, -
,.)&6?%&).)7#/# X>2%/*#8 .%F,$'*#*,8 ( (;+).!
I*O'-
0 µ 2/ #
62 & ,$ & 26 2 5 < 5/
H4 75 12;04
# 7,( ?"(2H)$,(
1. [0K# *#8 ">A#8 y
&,% @.)")2%/X'8 *#8 /@$)X5?,8 C + y.
2. !.%")20 *#8 ">A#8 -,
-"5=%/*)@ B51)@8,
.)@ .-1%>=-*,% /*# /@$)X5?, C + y.
#
($ @.51=)@$ .-1%//'*-1-8 ">A-%8 *#8 /@$)X5?,8 C
+ y, µ- -"5=%/*) B51)8,*'*- /*#$ (+(;+)
.-17.*6/# ,.)&6?%&).)7#/#8 µ>2%/*#8 .%F,$'*#*,8 -.%">2-*,% ,@F,71-*, X7, -A ,@*C$,
-$C /*#$ .-17.*6/# *#8 ( ((;+),.)&6?%&).)7#/#8 M#*-7*,% -.,$5"#K# *#8 µ-*5?)/#8.
3. H.)")2%/X'8 *#8 &6?%&08 ">A#8 x
.)@ µ-*,?'F#&- .1)/F>*)$*,8 *# "#3F-7/, ">A# y
X- *) .1'*@.) /35"X,*)8 -"5=%/*)@ B51)@8 -,
?#",?0, x = y + -.
7 5
-
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(.)&6?%&).)7#/# µ- "6
&; ! )7#85 I62&,$
&262 5A-%8 *)@ C (n,k,d) µ0&)@8 n,
M=2k &,% c1=µ#?-$%&0 ">A#.
G ?%5*,A# ?- ,@*0 -7$,% 2n-k x
2k µ- *%8 2n-k 21,µµ>8 $,
,.)*-")E$ *%8 /@$)µ5?-8 *)@ C.
J, B0µ,*, 2%, *)$ /=#µ,*%/µ' *#8 -7$,%:1. N% 2k &6?%&>8 ">A-%8 *)@ C
*).)F-*)E$*,%
/*#$ 1# 21,µµ0 µ- *) .1C*)
/*)%=-7) *#8 21,µµ08 (,.' ,1%/*-15)
$, ,.)*-"-7 # c1.
2. Z8 .1C*) /*)%=-7) *#8 2#8 21,µµ08 -.%">2-*,% *) .1'*@.) /35"µ,*)8 -2
µ- *) -"5=%/*) B51)8 &,% *) ).)7) ?-$ @.51=-% 0?# 68 ">A# /*%8 6%=7A-%8 *#8 1#8 21,µµ08).
3. J) B0µ, 2
-.,$,",µB5$-*,% µ- *)$ 7?%) *1'.) >68 '*)@ -A,$*"#F)E$ '"-8 )% ?%,3)1-*%&>8 ">A-%8 µ0&)@8 n
,
&5*% .)@ .1,&*%&5 /@µB,7$-% '*,$ /=#µ,*%/F-7 &,% # 2n-k 21,µµ0 *#8 ?%5*,A#8
( ? 7 < 5?
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(.)&6?%&).)7#/# µ-
-
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(.)&6?%&).)7#/# µ- //-1%8 /@$)X5?-8 *)@ C=
{0000, 1010, 1101, 0111}, {0001, 1011,1100, 0110}, {0010, 1000,
1111, 0101} &,% {0100, 1110, 1001, 0011}.
!
!5$ *C1, ) ?>&*#8 "5B-% *%8 ">A-%8 1110 &,% 1111,
)% ).)7-8 ?-$ -7$,% &6?%&>8 ">A-%8 *'*- /Eµ36$, µ- *%8 +(;+ &,% ((;+ >=)µ-:
! ($ "#3F-7 # ">A# 1110,
*'*- ,3)E -$*).%/*-7 # /@$)X5?, /*#$ ).)7,
.-1%>=-*,%, ?#",?0 # *>*,1*#,
-.%">2-*,% *) .1'*@.) B51)@8 X- *) µ%&1'*-1) B51)8,
?#",?0 *) 0100. !.)X>$68,
/EX36$, X- *%8 +(;+ &,% ((;+ /@X.-1,7$-*,% '*% µ-*,?'F#&- # ">A# 1110
+ 0100 = 1010 # C.
! A#8 1111,
,3)E ?%,.%/*6F-7 '*% .-1%>=-*,% /*#$ 3# /@$)X5?,,
.,1,*#1-7*,% '*% /- ,@*0 @.51=)@$ ?E) .1'*@., /35"X,*)8 X- *) -"5=%/*) B51)8,
*, 0010 &,% 1000.
–
($ >=)@X- -3,1X)20 *#8 ((;+,
M#*-7*,% -.,$,X-*5?)/# *)@ X#$EX,*)8.
–
($ -3,1X'M-*,% 'X68 # +(;+,
) ?>&*#8 -.%">2-% ,@F,71-*, >$, ,.' *, .1'*@., /35"X,*)8 X- *) -"5=%/*) B51)8,
>/*6 *) 1000
&,% /@X.-1,7$-% -.)X>$68 '*% µ-*,?'F#&- # ">A# 1111
+ 1000 = 0111.
( ? 7
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(.)&6?%&).)7#/# µ- A#.
! E$ '7574/
?"($-' µ 2"
! ($ x, y &,% - # Kn,
%/=E)@$ :
1.
($ &,% X'$) ,$ x # C,
*'*- x ⋅ H = 0.
•
(.'?-%A#: " x # C ' x ⋅ H= (u
⋅ G) ⋅ H= u (G H)=u⋅ 0 =02.
($ &,% X'$) ,$ x &,% y
# /*#$ 7?%, /@$)X5?,, *'*- x ⋅ H = y
⋅ H.
R+2"( $5*#$; 72
$
2 /-%8 /35"µ,*)8 /*) - &,F)17M)@$ &,% *%8 ,$*7/*)%=-8 21,µµ>8 *)@ - .)@ -µ.">&)$*,% /*) 5F1)%/µ, *#8 (1)
( ? 7
-
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(.)&6?%&).)7#/# µ- =19BFµ01
–
N ,"2'1%Fµ)8 µ- -&*-*,µ>$# J@.%&0 9%5*,A# /@$)µ5?6$ >=-% ?E) ?@/&)"7-8,*#$ ,$,M0*#/# *#8 /@$)X5?,8 /*#$ ).)7, .-1%>=-*,% # "#3F-7/, ">A# &,% *#$ ,$,M0*#/# /*# /@$>=-%, *)@ .1'*@.)@ /35"X,*)8 X- *) -"5=%/*) B51)8.
–
($52 2%, ,.")E/*-1) *1'.) ,.)&6?%&).)7#/#8 µ- µ%&1'*-1# ,$52 2%, ,.)F#&-@*%&' =C1)
! # I9'F,+,% 1
>=19BFµ01 µ67BD8 1% A-%8 X%,8 /@$)X5?,8 >=)@$ *) 7?%) /E$?1)X),
*) /E$?1)X) ,@*' X.)1-7 $, =1#/%X).)%#F-7 ,$*7 *#8 /@$)X5?,8 2%, *)$ .1)/?%)1%/X' *#8.
L*/%,
,1&-7 $, >=)@X- ,.' &5F- /@$)X5?, X'$) *) /E$?1)X' *#8 &,% *# ">A# X-
*) -"5=%/*) B51)8 ()?#2' *#8 /@$)X5?,8)
2%, ,.)&6?%&).)7#/# µ%,8 "#3F-7/,8 ">A#8 ]1,:
–
G -&*-*,µ>$# J@.%&0 9%5*,A# (.)&6?%&).)7#/#8 (µ- /@$)µ5?-8)
µ.)1-7 $, ,.").)%#F-7 /-
.7$,&, .)@ .-1%>=-% µ'$)$ *, /E$?1)X, &,% *)@8 )?#2)E8 '"6$ *6$ /@$)X5?6$).
(.)&6?%&).)7#/# µ-
-
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(.)&6?%&).)7#/# µ- 8 ,@F,71-*,
&,% *).)F-*-7*,% 68 ,$*7/*)%=)8 )?#2'8 /@$)µ5?,8 /*)$ .7$,&, ,$ -3,1X'M-*,% +(;+,
-7*- ?-$ -.%">2-*,% &,X7, ,$ -3,1X'M-*,% ((;+ &,% µ>$# &-$0 # ,$*7/*)%=# F>/# /*)$ +7$,&, *)$*,8 *# "#3F-7/, ">A# y
X- *)$ )?#2' *#8 /@$)X5?,8 (*) .1'*@.) /35"X,*)8)
-, ?#",?0 x = y + -.
( ? 7
-
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(.)&6?%&).)7#/# µ- $-8 J9(
(+7$,&-8
-
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(.)&6?%&).)7#/# µ- 1F#&-,
-
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(.)&6?%&).)7#/# µ- $)@8 *)@8 +7$,&-8 &*# )% ">A-%8 y=
0110 &,% 1111,
# ,.)&6?%&).)7#/0 *)@8 &,*5 +(;+ &,% ((;+ >=-% 68 -A08:.
!
H.)")27M)$*,% *, /E$?1)X, *6$ ">A-6$ .)@ -"03F#/,$
&,% -">2=)$*,% µ- B5/#
*)@8 ,$*7/*)%=)@8 )?#2)E8 *6$ /@$)X5?6$ *6$ +%$5&6$.
!
$,,
*, /E$?1)X, *6$ ">A-6$ .)@ -"03F#/,$ -7$,% 0110⋅G =01
&,% 1111⋅G = 10.
! A# 1111
?-$ )?#2)Eµ-F, /- &5.)%) /@X.>1,/X,,
-3'/)$ ?-$ @.51=-% ,$*7/*)%=)8 )?#2'8 &,% M#*-7*,% ,$,X-*5?)/#.
]/ #/# P1, C 4 ?7
-
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]//# P1,µµ%&C$ 46?7&6$
]//# P1,µµ%&C$ 46?7&6$
-
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]//# P1,µµ%&C$ 46?7&6$
]//# P1,µµ%&C$ 46?7&6$
-
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]//# P1,µµ%&C$ 46?7&6$
]//# P1,µµ%&C$ 46?7&6$
-
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]//# P1,µµ%&C$ 46?7&6$
]//# P1,µµ%&C$ 46?7&6$
-
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]//# P1,µµ%&C$ 46?7&6$
]//# P1,µµ%&C$ 46?7&6$
-
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]//# P1,µµ%&C$ 46?7&6$
]//# P1,µµ%&C$ 46?7&6$
-
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]//# P1,µµ%&C$ 46?7&6$
]//# P1,µµ%&C$ 46?7&6$
-
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]//# P1,µµ%&C$ 46?7&6$
]//# P1,µµ%&C$ 46?7&6$
-
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]//# P1,µµ%&C$ 46?7&6$
]//# P1,µµ%&C$ 46?7&6$
-
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]//# P1,µµ%&C$ 46?7&6$
($,&-3,",76/# /*)@8 P1,µµ%&)E8 4C?%&-8
-
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, -3, , 6/# / )@8 1,µµ )E8 C? -8
(n,k,d) = n:
µ0&)8 &6?%&08 ">A#8 *)@ Ck:
?%5/*,/# C
d:
-"5=%/*# ,.'/*,/# µ-*,AE &6?%&C$ ">A-6$ 9%5/*,/# k
*)@ C = P1,µµ>8 2-$$0*)1, G
9%5/*,/# k *)@ C =>
2k &6?%&>8 ">A-%8
;0&)8 n =
-
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($,&-3,",76/# /*)@8 P1,µµ%&)E8 4C?%&-8
I9'F,+,D@ >=17µ)901: ($ C
-7$,% >$,8 /@/*#X,*%&'8 21,XX%&'8 &C?%&,8 (n,k,d)
&,% x &,% y
?E) ">A-%8 -.7/#8 X0&)@8 n K#376$,
*'*- %/=E)@$ ,$,3)1%&5 X- *%8 /@$)X5?-8 *,
,&'")@F,:1) 45F- ">A# x
.-1%>=-*,% /*# /@$)X5?, C + x
("'26 E.,1A#8 µ#?-$%&08 &6?%&08 ">A#8 /-
21,µµ%&)E8 &C?%&-8)
2)
J) ."0F)8 *6$ ">A-6$ /- X%, /@$)X5?, -7$,% 7/) X- *) ."0F)8 *6$ ">A-6$ *)@ &C?%&, C,?#",?0 |C
+ x| = |C| = 2k.
3) ($ X%, ">A# x
.-1%>=-*,% /*# /@$)X5?, C + y,
*'*- %/=E-% C + x = C + y,
?#",?0 &5F- ">A# *#8 /@$)X5?,8 *#$ .1)/?%)17M-%.
4)
N% ">A-%8 '"6$ *6$ /@$)µ5?6$ -7$,% ?%,3)1-*%&>8,
0 %/)?E$,µ,,
&5F- ">A# x∈ 4n .-1%>=-*,% X'$) /- X7, /@$)X5?, *)@ C,
?#",?0 ?E) /@$)X5?-8 C + x &,% C+ y
-7*- *,@*7M)$*,% -7*- ?-$ >=)@$ &,$>$, /*)%=-7) &)%$'.
5)
J) ."0F)8 *6$ ?%,3)1-*%&C$ /@$)X5?6$ *)@ &C?%&, C,
?%5/*,/#8 k, %/)E*,% X- 2n – k .
( (.' %?%'*#*, 4
⇒ ."0F)8 /*)%=-76$ '"6$ *6$ /@$)µ5?6$= 2n
⇒ -3'/)$ ,.' %?%'*#*, 2, |C + x| =
2k *'*- ."0F)8 /@$)µ5?6$= 2n / 2k= 2n – k)
6) ($ *) 5F1)%/X, x + y C,
*'*- )% x,y
.-1%>=)$*,% /*#$ 7?%, /@$)X5?,.
($ *) 5F1)%/X, x + y C, *'*- )% x,y
.-1%>=)$*,% /- ?%,3)1-*%&>8 /@$)X5?-8.
($,&-3,",76/# /*)@8 P1,µµ%&)E8 4C?%&-8
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($,&-3,",76/# /*)@8 P1,µµ%&)E8 4C?%&-8
!
/-%8 /*%8 ).)7-8 .1)&"0F#&,$ /35"X,*, &,*5 *# X-*5?)/#.
!
P%, *#$ ,.)&6?%&).)7#/# ,1&-7 *) /E$?1)µ) &,% ) )?#2'8 *#8 /@$)µ5?,8
–
==y+- (*) - ,$*%/*)%=-7 /*) /E$?1)µ) .)@ @.)")27/,µ-)
!
N )?#2'8 *#8 /@$)µ5?,8 -7$,% # ">A# µ- *) µ%&1'*-1) B51)8 !
N )?#2'8 *#8 /@$)µ5?,8 -7$,% *) .1'*@.) /35"µ,*)8 51, .-1%>=-*,% /*#$ /@$)µ5?,
($,&-3,",76/# /*)@8 P1,µµ%&)E8 4C?%&-8
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($,&-3,",76/# /*)@8 P1,µµ%&)E8 4C?%&-8
! +UN
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($,&-3,",76/# /*)@8 P1,µµ%&)E8 4C?%&-8
!
!3'/)$ *) .1'*@.) /35"µ,*)8 ()?#2'8 /@$)µ5?,8)
.-1%>=-*,% /*#$ /@$)µ5?, &,% -3'/)$ ) ,1%Fµ'8 '"6$ *6$ ">A-6$ '"6$ *6$ /@$)µ5?6$ -7$,% 2n (2n-k
* 2k) µ.)1)Eµ- $,
.,1,*#10/)@µ- *, -A08 ,$ -.%">A)@µ- µ%, &6?%&0 ">A# µ0&)@8 n
µ- *) -"5=%/*) B51)8 (=1#/%µ).)%)Eµ- 2%, .,15?-%2µ, *) .,15?-%2µ, 4.20
*)@ B%B"7)@)
! G=
! 0001*G = 01
! 0010*G = 10
! 0100*G = 11
! 1000*G = 10
!
(.' *, .,1,.5$6 B">.)@µ- '*% # &6?%&0 ">A# 0001
>=-% /E$?1)µ) 01, )% 0010
&,% 1000>=)@$ /E$?1)µ) 10
&,% ,$0&)@$ /*#$ 7?%, /@$)µ5?, &,% # 0100
>=-% /E$?1)µ) 11. 9#",?0 B17/&)@µ- ,&1%BC8 *, 7?%, µ- *) .,15?-%2µ, 4.20
=6178 $, =1-%,/*-7 $, @.)")27/)@µ- *%8 /@$)µ5?-8
!
>D 6DB86,0&+ 67= 7' (09'(:@ 4:ND'@ µD ,% D4)
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($,&-3,",76/# /*)@8 P1,µµ%&)E8 4C?%&-8
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PU(;;I4NI 4Z9I4!
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P1,µµ%&)7 4C?%&-8 (4C?%&-8 Hamming)
!
RB'&QF@: L$,8 &C?%&,8 X0&)@8 n
= 2r – 1 (r T2)
&,% .7$,&, -">2=)@ %/)*%X7,8 G,
*)@ ).)7)@ (G)
)% 21,µµ>8 ,.,1*7M)$*,% ,.' '"-8 *%8 ?@$,*>8 X# X#?-$%&>8 ">A-%8 µ0&)@8 r,
)$)X5M-*,% &B$ /
Hamming
H;&2"/ 2 r –
1.
– G ?%5/*,/# k
*)@ &C?%&, = n-r = 2r – 1 – r
&,% µ.)1-7 $, ?-%=F-7 '*% d=3.]1, µ.)1)E$ $, ?%)1FC/)@$ t=((d-1)/2)=
1 "5F)8 &,% $, ,$%=$-E/)@$ d-1=2
"5F#
– r= ,1%Fµ'8 (."-)$,M'$*6$) bits
%/)*%µ7,8
&,% ,1%Fµ'8 &6?%&C$ ">A-6$ (&,%
µ#$@µ5*6$) |C|= 2& = 2 2r
– r – 1 –
N 1@FX'8 ."#1)3)17,8 -$'8 ?@,?%&)E &C?%&, C
X0&)@8 n -7$,% 7/)8 X- (1/
n)log2|C| = [log2 (22r – r –
1 )]/(2r – 1)= (2r – r – 1
)/(2r – 1) !E&)", 3,7$-*,%
,.' *# /=>/# '*% ) 1@Fµ'8 ."#1)3)17,8 ."#/%5M-% /*) 1
2%, @K#">8 *%µ>8 *)@ r
! ..=. N% .,1,&5*6 4C?%&-8 (n,k,r)
-7$,% &C?%&-8 Hamming
!
(7,4,3), (15,11,4), (31,26,5) ` !
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P1,µµ%&)7 4C?%&-8 (4C?%&-8 Hamming)!
!%B)9D';µ%: 4C?%&,8 Hamming (7,4,3)
!
=-% /,$ 21,µµ>8 '"-8 *%8 µ# µ#?-$%&>8 ">A-%8 µ0&)@8 r=3,
?#":
!
(.' *)$ +7$,&, H
µ.)1-7 -E&)", $, ?-%=F-7 '*% d=3
µ%, &,% ,@*'8 -7$,% ) -"5=%/*)8 ,1%Fµ'8 21,µµC$ *)@ H
*) 5F1)%/µ, *6$ ).)76$ %/)E*,% µ- µ#?>$.
– ..= 1# +2# +3# 21,µµ0 =
[101]+[011]+[111] =[111]+[111]=[000]
! !.%.">)$ %/=E-% k= n-r=4 &,% |C|=
2&
= 16
P1,µµ%&)7 4C?%&-8 (4C?%&-8 Hamming)
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P1,µµ%&)7 4C?%&-8 (4C?%&-8 Hamming)!
!%B)9D';µ% (&=1:A-%8 .)@ .1)&E.*)@$ ,.' *)$ G
µ-*5 *)$ .)"//µ' *)@ µ- &5F- >$, ,.' *, 24 =16
µ#$Eµ,*, (">A-%8 µ0&)@8 4 bits)
0000
0001
0010
0011
0000000
0001011
0010111
0011100
0100
0101
0110
0111
0100110
0101101
0110001
0111010
1000
1001
1010
1011
1000101
1001110
1010010
1011001
1100
1101
1110
1111
1100011
1101000
1110100
1111111
s t s t s t s t
P1,µµ%&)7 4C?%&-8 (4C?%&-8 Hamming)
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P1,µµ%&)7 4C?%&-8 (4C?%&-8 Hamming)!
I62&,$ &262 5
-
8/1
