'&
A 5B/6B coding scheme in which five data bits are encoded into six-bit
codewords is used in conjunction with offsetting the data on different
channels by three bits in quartet signaling. It provides the level of error
detection necessary, produces a signal balanced within narrow limits, and
restricts strings of consecutive 0s or 1s to a maximum length of 6. It is
also efficient.
& ! $ # %
2',1* 2) '$7$ %()25( 75$160,66,21 +$6 7+5(( 0$,1 385326(6
,567 ,7 &$1 (1685( 7+$7 7+( 75$160,77(' 6,*1$/6 $5( '& %$/@
$1&(' 7+$7 ,6 7+$7 7+(5( $5( (48$/ 180%(56 2) 6$1'6
29(5 $1 (;7(1'(' 3(5,2' 2) 7,0( !+,6 ,6 ,03257$17 ,1 75$16@
)250(5@&283/(' 6<67(06 :+,&+ $5( '(6,*1(' 72 35(9(17
*5281' /2236 (&21' &2',1* &$1 $,' &/2&. 6<1&+521,=$7,21
%< 0,1,0,=,1* 7+( /(1*7+ 2) 675,1*6 2) &216(&87,9( 6 25 6
!+,6 (1685(6 7+$7 7+(5( ,6 $ +,*+ '(16,7< 2) 6,*1$/ 75$16,7,216
!+,5' 72*(7+(5 :,7+ &<&/,& 5('81'$1&< &+(&.6 6 ,7 &$1
(1$%/( (55256 72 %( '(7(&7(' !+(5( ,6 $1 52-(&7 )81&7,21$/ 5(48,5(0(17 7+$7 $1< 7+5(( %,76 ,1 (5525 :,7+,1 $
'$7$ )5$0( 0867 %( '(7(&7(' !+(5( ,6 $/62 $ 1((' 72 '(7(&7
%8567 (55256 $1< &2',1* 6<67(0 0867 127 &203520,6( 7+(
'(7(&7,21 3529,'(' %< 7+( !+( #@1< 7($0 &+26( $ &2',1* 6&+(0( ,1
:+,&+ ),9( '$7$ %,76 $5( (1&2'(' ,172 6,;@%,7 &2'(:25'6 !+,6
,6 86(' ,1 &21-81&7,21 :,7+ 2))6(77,1* 7+( '$7$ 21 ',))(5(17
&+$11(/6 %< 7+5(( %,76 ,1 48$57(7 6,*1$/,1* 7 3529,'(6 7+(
/(9(/ 2) (5525 '(7(&7,21 1(&(66$5< 352'8&(6 $ 6,*1$/ %$/@
$1&(' :,7+,1 1$552: /,0,76 $1' 5(675,&76 675,1*6 2) &216(&8@
7,9( 6 25 6 72 $ 0$;,080 /(1*7+ 2) 7 7+( 6$0( 7,0( ,7
,6 ()),&,(17 $'',1* 21/< 72 7+( '$7$ /2$' &203$5(' :,7+
7+( 352'8&(' %< 7+( $1&+(67(5 &2',1* 86(' ,1
$6(@! !+( &2'( ,6 $/62 ())(&7,9( :+(1 7+( )285
'$7$ &+$11(/6 $5( 08/7,3/(;(' )25 75$160,66,21 21 ! 25
),%(5@237,& &$%/(6
!+,6 $57,&/( '(6&5,%(6 7+( '(6,*1 2) 7+( %/2&. &2'(
86(' ,1 #@1< $1' 025( *(1(5$//< ,1 7+( 352326(' /2&$/ $5($ 1(7:25. 67$1'$5' $1' (;3/$,16
7+( 5($6216 %(+,1' ,76 '(6,*1
& !+( 0(7+2' 86(' 72 75$160,7 ',*,7$/ '$7$ +$6 72 7$.( $&&2817
2) 7+( 3+<6,&$/ &21675$,176 7+$7 7+( 75$160,66,21 0(',80
,0326(6 8321 7+( 75$160,66,21 6<67(0 !+( 3+<6,&$/ 0(',$
86(' %< #@1< ,1&/8'( 816+,(/'(' 7:,67(' 3$,5
"! &$%/( 6+,(/'(' 7:,67(' 3$,5 ! &$%/( $1' 237,&$/
),%(5 !+( '(0$1'6 ,0326(' %< 7+( 86( 2) "! &$%/( 0($1
7+$7 72 75$160,7 $7 %,766 (9(1 :+(1 86,1* 48$57(7
6,*1$/,1* ',*,7$/ '$7$ +$6 72 %( &2'(' %()25( ,7 ,6 75$16@
0,77(' !+( '(6,*1 2) 7+( &2'( +$6 72 7$.( $&&2817 2) 9$5,@
286 &21675$,176 ,0326(' %< ,03/(0(17$7,21 &216,'(5$7,216
$1' %< 7+( 1((' 72 %( &203$7,%/( :,7+ 27+(5 '$7$ 75$160,6@
6,21 6<67(06 1 3$57,&8/$5 7+( #@1< &2'(
6&+(0( :$6 '(6,*1(' 72 %( &203$7,%/( :,7+ 7+( 3$&.(7
6758&785( $1' (5525 '(7(&7,21 &$3$%,/,7,(6 '(),1(' %< 7+( 67$1'$5' 86(' ,1 7+(51(7 /2&$/ $5($ 1(7:25.6 $1' %<
7+( 67$1'$5' 86(' ,1 72.(1 5,1* /2&$/ $5($ 1(7@
:25.6
" !! ! ## (1(5$//< 63($.,1* $ &2'( ,6 $ 0$33,1* )520 $1 $/3+$%(7
2) 6<0%2/6 72 $ 6(7 2) 6(48(1&(6 2) 6<0%2/6 )520 620(
27+(5 $/3+$%(7 2 )25 (;$03/( :( &28/' '(),1( $ &2'(
)520 7+( $/3+$%(7 >$%&=? 72 7+( 6(7 2) 6(48(1&(6 2) =(526
$1' 21(6 %< $ → % → & → $1' 62 21
25 $127+(5 (;$03/( 7+( 5(35(6(17$7,21 2) '(&,0$/ 180%(56
$6 7+(,5 %,1$5< (48,9$/(176 &$1 %( 5(*$5'(' $6 $ &2'( !+,6
,'($ 2) $ &2'( 6+28/' 127 %( &21)86(' :,7+ 7+( 127,21 2) $
&,3+(5 :+,&+ ,6 $ &2'( 7+$7 ,6 63(&,),&$//< '(6,*1(' 72 +,'(
7+( 0($1,1* 2) 75$160,77(' '$7$ )520 7+26( 127 0($17 72
81'(567$1' ,7 !+( &2'(6 7+$7 :( $5( 86,1* $5( '(6,*1(' 72
%( ($6< 72 '(&2'( $1' $5( &$3$%/( 2) '(7(&7,1* (55256 ,1
75$160,66,21 25 7+( 0$7+(0$7,&$/ %$&.*5281' 2) &2',1*
7+(25< 6(( 5()(5(1&( 25 $1 (1*,1((5,1* 3(563(&7,9( 6((
5()(5(1&( %/2&. &2'( ,6 $ 63(&,),& 7<3( 2) &2'( ,1 :+,&+ (9(5< &2'(@
:25' +$6 7+( 6$0( /(1*7+ 1 $ %/2&. &2'( ($&+
(/(0(17 2) $1 $/3+$%(7 2) ',))(5(17 6<0%2/6 :+,&+ ,76(/)
&$1 %( 5(35(6(17(' %< $ @%,7 180%(5 ,6 (1&2'(' $6 $ @%,7
&2'(:25'
!+( 0$-25 3+<6,&$/ &21675$,17 7+$7 $))(&76 7+( 86( 2) $ &2',1*
6&+(0( ,6 7+( 1((' 72 0$,17$,1 '& %$/$1&( 7+$7 ,6 7+( 63(&@
75$/ &217(17 2) 7+( 75$160,77(' 6,*1$/ 6+28/' +$9( 12 =(52@
)5(48(1&< &20321(17 1 0$7+(0$7,&$/ 7(506 7+,6 75$16/$7(6
,172 5(48,5,1* 7+$7 7+( ',))(5(1&( %(7:((1 7+( 180%(5 2)
21(6 $1' 7+( 180%(5 2) =(526 75$160,77(' $7 $1< 7,0( 0867
%( .(37 $6 &/26( 72 =(52 $6 3266,%/(
/($5/< $ 6,03/( :$< 72 *(7 $ %$/$1&(' &2'( ,6 72 5(35(6(17
$ '$7$ &+$5$&7(5 2) %< 7+( &2'(:25' $1' $ '$7$ %<
7+( &2'(:25' !+( 180%(5 2) 6 75$160,77(' :,// 7+(1
(48$/ 7+( 180%(5 2) 6 75$160,77(' $7 7+( (1' 2) ($&+
&2'(:25' "1)25781$7(/< 7+,6 ,6 $/62 $ 9(5< ,1()),&,(17 :$<
2) $&+,(9,1* '& %$/$1&( %(&$86( ,7 5(48,5(6 7+( 3+<6,&$/
75$160,66,21 2) 7:2 &2'( %,76 )25 (9(5< '$7$ %,7 6(17 !+(
#@1< 7($0 &+26( 72 ,03/(0(17 $ &2'(
%(&$86( ,7 *,9(6 $ 68,7$%/( 75$'(@2)) %(7:((1 ()),&,(1&< $1'
&267 2) ,03/(0(17$7,21
8*867 (:/(77@$&.$5' 2851$/
IEEE
802.3 or 802.5
Frame Bit Stream
11000101
5-Bit Data Words
11000
10111
11000
A1
B1
C1
11110001
00101011
10010
01011000
10110
D1
00110011
10110
A2
B2
00001
C2
10011
D2
If necessary, the
last data word is
padded to the
5-bit boundary.
Data Words on 4 Channels
0
1
1
0
1
0
0
0
1
1
A2
0
1
1
0
1
A1
1
1
1
0
1
Channel A
B2
1
0
0
0
0
B1
0
0
0
1
1
Channel B
C2
1
1
0
0
1
D2
C1
0
1
0
0
1
D1
Channel C
,675,%87,21 2) $ '$7$
675($0 21 )285 &+$11(/6 )25 48$5=
7(7 6,*1$/,1*
Channel D
!+( 0$-25 &203$7,%,/,7< &21675$,17 ,0326(' 8321 7+( '(6,*1
2) 7+( "=1< &2'( :$6 7+$7 ,7 +$' 72 %(
&203$7,%/( :,7+ 7+( )5$0( )250$76 '(),1(' %< 7+( $1' 67$1'$5'6 $1' ,1 3$57,&8/$5 :,7+ 7+( (5525=
'(7(&7,21 &$3$%,/,7,(6 2))(5(' %< 7+( &<&/,& 5('81'$1&< &+(&.
%8,/7 ,172 7+26( )250$76 (( ##2/<120,$/ 5,7+0(7,&
$1' <&/,& ('81'$1&< +(&.6 21 3$*( =
7+(
'(),1(' %< %27+ $1' '(7(&76 83 72
7+5(( 6,1*/(=%,7 (55256 2&&855,1* $1<:+(5( ,1 $ )5$0( 25 $
6,1*/( %8567 (5525 2) 83 72 %,76 ,1 7+( )5$0( !+( 35272&2/
:,// 7+(1 ',6&$5' 7+$7 3$&.(7 $6 )/$:(' '$7$ 3$&.(7 &21=
6,676 2) 7+( )5$0( '(/,0,7(' %< $ 35($0%/( $1' 67$57
$1' (1' '(/,0,7(56 !+( "=1< 7($0 +$' 72 '(6,*1
7+( &2'( 62 7+$7 6,0,/$5 (5525='(7(&7,21 &+$5$&7(5,67,&6
:,// %( 0$,17$,1(' :+(1 25 )5$0(6 $5( &2'('
$1' 75$160,77(' ,1 3$5$//(/ 86,1* 48$57(7 6,*1$/,1* 1 3$57,&8=
/$5 7+( 5(48,5(0(17 72 '(7(&7 83 72 7+5(( 6,1*/(=%,7 (55256
$1<:+(5( ,1 $ )5$0( ,6 $ &2038/625< 5(48,5(0(17 2) 52-(&7 62 7+( 1(: 67$1'$5' +$' 72 0((7
7+,6 '(0$1'
1 7+( 1(;7 7:2 6(&7,216 :( '(6&5,%( 7+( 352&(66 2) &2',1*
7+( '$7$ )5$0( $1' 7+( (5525 '(7(&7,21 &$3$%,/,7,(6 2) "=
1<
$7$ )520 7+( 25 )5$0( ,6 ',9,'(' ,172 =%,7
'$7$ %/2&.6 :,7+ 3$'',1* $''(' $7 7+( (1' ,) 1(&(66$5<
$1' ,6 ',675,%87(' %(7:((1 7+( )285 '$7$ 675($06 75$160,77('
86,1* 7+( 48$57(7 6,*1$/,1* 6&+(0( 6(( ,* Cipher A
Cipher B
Cipher C
Cipher D
0
0
0
0
1
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
A2
B2
C2
D2
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
1
1
A1
B1
C1
D1
A
8*867 (:/(77=$&.$5' 2851$/
B
C
D
!+( '$7$ :25'6 $5( XOR('
:,7+ 36(8'25$1'20 675($0 &,=
3+(56
#%* &#5# 453'#. +4 '9%-64+7';OR'& 8+5* # 453'#. 0( 14'6&0; • "'+)*5;580 #/& 8'+)*5;(063 %0&'803&4 %0/5+/6' 50 #-5'3;
3#/&0. $+54 130&6%'& $: 453'#. %+1*'34 4'' +) *'
/#5' 8*'/'7'3 #/ 6/$#-#/%'& %0&'803& 0%%634
%+1*'3'& &#5# $-0%,4 0/ 5*' (063 453'#.4 #3' 5*'/ %0&'&
*' 0$4'37#/5 3'#&'3 8+-- /05+%' 5*#5 +/ #-- $65 0/' 0( 5*'
#%%03&+/) 50 5*' (0--08+/) %0&' 5#$-' #4 4*08/ +/
%0&' 1#+34 5*' #-5'3/#5+7' 6/$#-#/%'& %0&'803&4 #3' -0)+%#+) /')#5+7'4 0( '#%* 05*'3 *+4 +4 /05 536' 0( 5*' 4'%0/& 1#+3
Data
Code
Data
Code Pair
(03 3'#40/4 *#7+/) 50 &0 8+5* 5*' '3303 1301'35+'4 0( 5*'
%0&'
5 5*' '/& 0( 5*' 453'#. 0/' 0( 580 '/& &'-+.+5'34 +4 64'&
+/&'1'/&'/5-: +/ '#%* 453'#. 4'' +) "*'/ 5*' '/& 0(
5*' 453'#. +4 3'#%*'& +( 5*' /'95 6/$#-#/%'& %0&'803& +4
&6' 50 *#7' 8'+)*5 #%%03&+/) 50 5*' 36-'4 #$07' '/&
&'-+.+5'3 +4 64'& ( +5 +4 &6' 50 *#7' 8'+)*5 +4
64'& *' +/7#-+& 1#%,'5 .#3,'3 +4 64'& $: 3'1'#5'34
50 .#3, '3303'& 1#%,'54 (03 &+4104#- 03 (635*'3 130%'44+/)
(5'3 %0&+/) 5*' &#5# 453'#.4 #3' 0((4'5 $: $+54 8+5* 3';
41'%5 50 '#%* 05*'3 $'(03' 53#/4.+44+0/ 4'' +) )#+/
5*' 3'#40/ (03 5*+4 *#4 50 &0 8+5* '3303;&'5'%5+0/ 1301'35+'4
#/& 8+-- $' '91-#+/'& $'-08
*' #%56#- 53#/4.+55'& $+5 4'26'/%' &08/ '#%* %*#//' -00,4
#4 4*08/ +/ +) *' 45#35 #/& '/& 0( 1#%,'5 .#3,;
'34
#3'
&'4+)/'& 50 .#+/5#+/ '3303 &'5'%5+0/ %#1#$+-+5+'4 *'
64'
0(
#-5'3/#5+7'
'/& &'-+.+5'34 '(('%5+7'-: #/ '953# 1#3+5:
%*'%, +4 #/ '44'/5+#- 1#35 0( 5*' '3303 &'5'%5+0/ 4%*'.' 0(
!;/:
/ 5*' 4'%0/& %0-6./ #-- 5*' ;$+5 %0&'803&4 #3' $#-#/%'&
5*#5 +4 5*'3' #3' '26#- /6.$'34 0( 4 #/& 4 *'3' #3' /05
'/06)* $#-#/%'& ;$+5 %0&'803&4 50 %0&' #-- 1044+$-' ;$+5
&#5# 4:.$0-4 40 58'-7' &#5# 4:.$0-4 #3' %0&'& $: # %*0+%'
0( 580 &+(('3'/5 ;$+5 %0&'803&4 0/' 0( 8'+)*5 580 580 4
#/& (063 4 #/& 0/' 0( 8'+)*5 (063 4 #/& 580 4 *'
580 %0&'803&4 #3' 64'& +/ 5*' (0--08+/) (#4*+0/ +/&'1'/;
&'/5-: +/ '#%* 453'#.
• 03 5*' (+345 &#5# 4:.$0- 5*#5 %0&'4 50 #/ 6/$#-#/%'& %0&';
803& 5*' 8'+)*5;580 %0&'803& +4 %*04'/
• "*'/ 5*' /'95 6/$#-#/%'& %0&'803& 0%%634 5*' 8'+)*5;
(063 %0&'803& +4 64'&
/05*'3 &'4+)/ %0/453#+/5 0( 5*' %0&+/) 4%*'.' 8#4
5*#5 +5 4*06-& $'*#7' 8'-- 8*'/ 5*' (063 &#5# 453'#.4 #3'
.6-5+1-'9'& 0/50 ('8'3 %*#//'-4 (03 /'5803,4 64+/) 03
(+$'3;015+% %#$-'4 4 /05'& +/ 5*' #35+%-' 0/ 1#)' #/& (+$'3;015+% 4 1#44 (063 1#3#--'- 453'#.4 0( &#5#
5*306)* # .6-5+1-'9'3 8*+%* %0.$+/'4 5*' (063 %0&'803&
453'#.4 +/50 0/' 453'#. %0&'803& $: %0&'803& +) "*'/ %0.$+/'& +/ 5*+4 8#: 5*' 1*:4+%#- #/& '3303 1305'%;
5+0/ %#1#$+-+5+'4 0( 5*' %0&' #3' .#+/5#+/'& #4 '9;
1-#+/'& +/ 5*' /'95 4'%5+0/
0
1
1
0
0
0
1
0
0
1
0
0
0
0
0
1
1
1
0
1
0
1
1
1
0
0
0
1
1
0
1
1
0
0
0
1
1
1
1
0
1
0
0
0
0
1
1
0
1
0
1
0
1
0
0
0
1
1
0
1
1
0
0
0
Channel A
Channel B
Channel C
Channel D
Stream of Ciphered Data Symbols
Channel A
0
0
1
1
0
0
1
0
0
0
0
1
1
0
0
0
1
1
0
1
1
0
0
1
Channel C
Channel D
Channel B
6-Bit Codewords
0/7'34+0/ 0( ;$+5 &#5#
4:.$0-4 50 ;$+5 %0&'803&4 *'
5*3'';$+5 0((4'5 0( %*#//'-4 #/&
+4 (03 '3303 &'5'%5+0/
6)645 '8-'55;#%,#3& 063/#-
The IEEE 802.3 frame (Fig. 1) consists of a 48-bit destination address, a 48-bit
source address, a 16-bit length field, and then a data field ranging between 368
and 12,000 bits in length and consisting of data organized into octets. This is
followed by a 32-bit cyclic redundancy check (CRC).
End of
Packet
ED2 = Weight 2 Expected
ED4 = Weight 4 Expected
IPM = Invalid Packet Marker
111111
000000
110000
000011 000001
111100 111110
011111 110000
SDH = High-Priority Packet
SDN = Normal-Priority Packet
100000
111100
111110
000011
6-Bit
Codewords
32-Bit CRC
Data Field
Start of
Packet
...010101
Preamble
16-Bit Length Field
48-Bit Source Address
!5$160,77(' %,7 6(48(1&( 21 ($&+ &+$11(/
48-Bit Destination Address
'(6,*1 ())257 :$6 '(',&$7(' 72 /,0,7,1* 7+( 180%(5 2) '$7$
%,76 $))(&7(' %< $ 6,1*/( &2'( %,7 ,1 (5525
Fig. 1. IEEE 802.3 frame structure.
The IEEE 802.5 token-ring frame (Fig. 2) consists of an 8-bit access control field
(not used in the 802.12 standard), an 8-bit frame control field, a 48-bit destination
address, a 48-bit source address, between 0 and 240 bits of routing information,
and then a data field ranging between 0 and 36,016 bits in length and consisting of
data organized into octets. This data field is followed by a 32-bit CRC.
32-Bit CRC
Data Field
0 to 240 Bits Routing Information
48-Bit Source Address
48-Bit Destination Address
8-Bit Frame Control Field
8-Bit Access Control Field
Fig. 2. IEEE 802.5 frame structure.
The CRC provides two types of protection for an IEEE 802.3 or 802.5 frame:
• Any three single-bit errors occurring anywhere in the frame are detected.
• Any burst of errors for which the distance between the first corrupted bit and the
last corrupted bit is less than or equal to 32 bits is detected.
IEEE Project 802 requires that any transmission scheme developed under its
aegis can perform the first type of error detection, that is, that at least three singlebit errors can always be detected.
!+( &+2,&( 2) &2'( 72*(7+(5 :,7+ 7+( $/7(51$7,21 2)
81%$/$1&(' &2'(:25'6 $1' 7+( 86( 2) 7+( $/7(51$7,9( (1'
'(/,0,7(56 (1685(6 7+$7 6 5(48,5(0(17 ,6 0(7 83 72
7+5(( 6,1*/(>%,7 (55256 2&&855,1* $1<:+(5( 21 7+( )285 &+$1>
1(/6 :,7+,1 $ '$7$ )5$0( $5( '(7(&7(' !+,6 ,6 127 $ 75,9,$/
0$77(5 72 &21),50 %(&$86( 7+( 86( 2) $ &2'( 0($16 7+$7 $
6,1*/(>%,7 (5525 ,1 7+( 75$160,77(' %,7 675($0 0$< &$86( 0$1<
025( 7+$1 21( 6,1*/( %,7 72 %( ,1 (5525 ,1 7+( '$7$ 675($0
$)7(5 '(&2',1* 68%67$17,$/ 3257,21 2) 7+( &2'(
8*867 (:/(77>$&.$5' 2851$/
7 ,6 127 ',)),&8/7 72 6(( +2: 7+( 86( 2) 7+( $/7(51$7,21 58/(
)25 &+226,1* 81%$/$1&(' &2'(:25'6 72*(7+(5 :,7+ 7+( 86( 2)
7+( $/7(51$7,9( (1' '(/,0,7(56 /($'6 72 7+( '(7(&7,21 2) $//
75,3/(6 2) 6,1*/(>%,7 (55256 :+(1 7+26( (55256 2&&85 ,1 ',67,1&7
&2'(:25'6 21 $ 6,1*/( &+$11(/ ,* 6+2:6 6(9(5$/ &$6(6
,1 :+,&+ 7+5(( 6,1*/( (55256 $5( '(7(&7(' %< 7+( $/7(51$7,21
58/(6 $1' 7+( 86( 2) (1' '(/,0,7(56
6(&21' 0$-25 '(6,*1 7+5867 :$6 72 0$.( 685( 7+$7 %8567
(5525>'(7(&7,21 3523(57,(6 ,1+(5,7(' )520 >
$5( 127
&203520,6(' ,1 ">1< (&$86( :( $5( 12: 7$/.>
,1* $%287 $ 6<67(0 7+$7 &$1 75$160,7 29(5 )285 3$5$//(/
675($06 $6 :(// $6 $ 6,1*/( 675($0 :( +$9( 72 %( &$5()8/ 72
'(),1( :+$7 :( 0($1 %< $ %8567 (5525 #( '(&,'(' 7+$7 7+(
&/($5(67 &$6( &28/' %( 0$'( %< &216,'(5,1* $ %8567 (5525 72
%( $1< (5525 &$86(' %< $5%,75$5< &255837,21 $&5266 $// 7+(
675($06 )25 $ &(57$,1 180%(5 2) %,7 3(5,2'6 6(( ,* 2
)25 (;$03/( )25 $ %8567 (5525 2) 6(9(1 &2'( %,7 3(5,2'6
7+(5( ,6 $ %/2&. 6(9(1 &2'( %,76 /21* %< )285 &2'( %,76 :,'(
&255(6321',1* 72 7+( )285 &+$11(/6 :,7+,1 :+,&+ $1<
*,9(1 &2'( %,7 0$< 25 0$< 127 %( ,19(57('
!+( ),567 7+,1* :( 5($/,=(' :$6 7+$7 ,) $// 7+( &2'(:25'6
:(5( 75$160,77(' 6<1&+521286/< 21 $// )285 &+$11(/6 $ %8567
(5525 2) /(1*7+ &255837,1* (,*+7 &2'( %,76 &28/' &$86(
7528%/( ,) ,7 2&&855(' $7 $ &2'(:25' %281'$5< 6(( ,* !+( (5525 &28/' &255837 (,*+7 &2'( :25'6 $1' 7+86 (,*+7
'$7$ %/2&.6 &255(6321',1* 72 )257< '$7$ %,76 !+,6 ,6 :(//
$%29( 7+( 3527(&7(' %< 7+( 7 :28/' 7+(5()25( 5(>
48,5( $ 5(0$5.$%/< *22' &2',1* 6&+(0( 72 3527(&7 $*$,167
(9(1 68&+ $ 6+257 (5525 %8567
D2
D1
C2
C1
B2
B1
A2
A1
D2
C2
B2
A2
D1
C1
8/7,3/(;,1* )285 675($06 ,172 21(
B1
A1
Correct Codeword Weights:
3
3
3
3
3
3
3
3
ED2
Errored Codeword Weights:
3
2
3
4
3
3
2
3
ED2
Error! The last apparently unbalanced codeword has weight 2, so the next
would have weight 4, yet the ED2 delimiter indicates that the next should
have weight 2.
Correct Codeword Weights:
3
3
3
3
3
3
3
3
ED2
Errored Codeword Weights:
3
4
3
3
2
4
3
3
ED2
The calculation of cyclic redundancy checks (CRCs) depends upon the arithmetic
of modulo 2 polynomials. A modulo 2 polynomial is an expression of the form a0 +
a1x + a2x2 + ...+ anxn, where the coefficients a0, a1, ... ,an are integers modulo 2,
that is, they can take values of 0 or 1. The coefficients obey regular modulo 2
arithmetic as follows:
Addition
Error! The first apparently unbalanced codeword has weight 4, yet the
rules stipulate that the first must have weight 2.
Correct Codeword Weights:
3
2
3
4
3
2
4
3
ED2
Errored Codeword Weights:
3
2
3
3
3
2
3
4
ED2
Error! The alternation rule is violated—apparently two unbalanced codewords in succession have weight 2.
"0" 0&+* +# "..+./ 0%.+1$% 0%" (0".*0&+* .1(" *! "*!
!"(&)&0"./
%" /+(10&+* 0+ 0%&/ ,.+(") &/ 0+ +##/"0 0%" +!"3+.!/ +*
03+ +# 0%" %**"(/ ."(0&2" 0+ 0%" +0%". 03+ &$ (".(5
/%+3/ 0%0 3&0% /1 % * +##/"0 "2"* * "..+. 1./0 +# ("*$0%
#+1. +..1,0&*$ +!" &0/ * +*(5 +..1,0 0%" +!" &0/
+# 0 )+/0 +!"3+.!/ %&/ +.."/,+*!/ 0+ !0 (+ '/
!0 &0/ *! /+ 3&(( " !"0" 0"! 10+)0& ((5 5 0%"
1./0 "..+. ,.+,".0&"/ +# 0%" %1/ "2"* 3&0% * .&7
0..5 +!" *5 1./0 "..+. +# ("*$0% #+1. &/ !"0" 0"!
1/0*0&( 0%"+."0& ( 3+.' ("! 1/ 0+ !"/&$* +!" +* 0+,
+# 0%&/ +##/"0 / %")" 0%0 "40"*!/ 0%" 1./0 "..+.7!"0" 0&+*
,&(&05 0+ /"2"* +!" &0 ,".&+!/ .&0..&(5 +..1,0&*$ +!" &0/ 3%&(" /0&(( )""0&*$ 0%" ."-1&.")"*0 0+ !"0" 0
0%."" /",.0" /&*$("7&0 "..+./ /"" &$ %" 0.&1),% +#
0%"+.5 +2". 0.&( *! "..+. /%+1(! *+0 " 1*!"."/0&)0"!
%"."80%"." ." ,,.+4&)0"(5 %+& "/ +# +!"
%"* )1(0&,("4"! +*0+ /&*$(" %**"( #+. +. #&".7
+,0& /+(10&+*/ 0%" "..+. !"0" 0&+* ,&(&0&"/ +# 0%" +!" ." "*%* "! %" /&*$("7&0 "..+. ,.+0" 0&+* .")&*/
0 0%."" 3%&(" 0%" 1./0 "..+.7!"0" 0&+* ,&(&0&"/ ." &*7
."/"! 0+ +!" &0/
Multiplication
+
0
1
× 0
1
0
0
1
0
0
0
1
1
0
1
0
1
Electrical engineers will recognize the addition operation as the XOR operation
defined on binary states.
The following are examples:
• (1+x2+x4) + (1+x+x4) = x+x2
• (1+x2+x4) (1+x+x4) = 1+x+x2+x3+x5+x6+x8.
Division of modulo 2 polynomials is done in exactly the same way as it is for ordinary polynomials, remembering that the coefficients obey modulo 2 arithmetic as
defined above.
A very important operation in the calculation of CRCs is the calculation of the
remainder R(x) when one polynomial M(x) is divided by another G(x). R(x) is
uniquely defined by M(x) = G(x)K(x) + R(x), where the degree of R(x) is less than
the degree of G(x). The degree of a polynomial is defined as the largest value of n
for which the coefficient an is nonzero, so, for example, 1+x2+x5 has degree 5.
Although this sort of polynomial division may look formidable, there are very efficient means of calculation based on shift registers.
CRCs are calculated on a stream of data by assuming that the data represents the
coefficients of some modulo 2 polynomial. So, given a stream of data n bits long,
the first bit can be considered as the coefficient an1 of xn1, the second bit as
the coefficient an2 of xn2, the (n1)th bit as the coefficient a1 of x and the nth
bit as the constant term a0.
Roughly speaking, the 32 bits of the CRC are defined to be the polynomial remainder R(x) when the polynomial defined by the data M(x) is divided by a standard
polynomial:
G(x)= 1+x+x2+x4+x5+x7+x8+x10+x11+x12+x16+x22+x23+x26+x32.
Slight modifications are made for implementation reasons, but as far as the error
properties are concerned, this is what is calculated.
Channel
C
Channel
A
Channel
D
The CRC bits are appended onto the data. When the data is received, the CRC is
calculated and compared with the received CRC. If there is a difference, the data
is known to have been corrupted.
Channel
B
Bits
Possibly
Inverted
by Error
Burst
Codeword Boundary
((1/0.0&+* +# 1./0 "..+. 3&0% ("*$0% +# /"2"* +!" &0
,".&+!/
When a corruption occurs in transmission, a number of bits are inverted. Let the
errored bits define the coefficients of a polynomial E(x); for example, if errors occur
at bit positions p and q, the polynomial will be xnp + xnq. Since the operation of
calculating CRCs is linear, the error is detected if and only if E(x) is not exactly
divisible by G(x). CRC-generating polynomials such as G(x) are chosen precisely
to detect as many polynomials like E(x) as possible.
* !!&0&+* 0+ 0%" "..+. !"0" 0&+* 0%" +!" %/ +0%".
,.+,".0&"/ 0%0 ." %&$%(5 ,".0&*"*0 0+ ,%5/& ( 0.*/)&//&+*
%" #&./0 +# 0%"/" &/ .1* ("*$0%80%" )4&)1) *1)". +#
+*/" 10&2" 6".+/ +. +*/" 10&2" +*"/ 0.*/)&00"! +* *5
+!" /0.") %&/ &/ &),+.0*0 &* 0.*/)&//&+* /5/0")/ 3%"."
/+)" #+.) +# (+ ' &*#+.)0&+* &/ ." +2"."! #.+) 0%" !0
/0.") /&* " (+ ' ." +2".5 1/1((5 !","*!/ +* ." "&2&*$ 1$1/0 "3("007 '.! +1.*(
A
B
C
C
D
A
D
B
Codeword Boundary
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&281()( &%6)/-1) :%1()5 25 7,) *285>3%-5 6',)0) 7,)
! -6 &281()( &)7:))1 * %1( ) 25 7,) 6-1+/)>675)%0
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6<00)75-'%/ &)'%86) 2* 7,) %/7)51%7-21 58/)?% :)-+,7>
'2():25( -6 %/:%<6 6)17 *-567 21 % ',%11)/ :,)1 7,)5) -6 %
',2-') 2* 81&%/%1')( '2():25(6
C
A
D
B
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%1( %1 29) %7 %1( 72 %9-( 811-1+,%0 /-67%-5
2/)6 -5%1(% 2:&5%< %9-( 2.)5 %1( -',%)/ !35%77
%7 %&25%725-)6 :,2 %// 3529-()( 9%/8%&/) %(9-') -1
7,)-5 2:1 63)'-%/ :%<6
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#'"! 9%1 -17 !'%"('"! '" "! "%* " !35-1+)5
#)5/%+ /%,87 "%* ! %' " %%"% "%%'! "& ((->
621 $)6/)< (!'"! $(% !'& %"' 2'%/ %1( )75232/->
7%1 5)% )7:25.6 !7%1(%5(6 200-77)) 5%*7 )7)'7-21 2* % &8567 2* /)1+7, &8567 2* /)1+7, :-//
%/:%<6 &) ()7)'7)( &)'%86) 2* 7,) 2**6)7 &)7:))1 ',%11)/6
8+867 ):/)77>%'.%5( 2851%/