linear algebra spring 2014-2015 (1)

8/9/2019 Linear Algebra Spring 2014-2015 (1)

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Error Control CodingError Control Coding

Saswat ChakrabartiSaswat Chakrabarti

GS Sanyal School of Telecommunications,GS Sanyal School of Telecommunications,

IIT KharagpurIIT Kharagpur


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Some milestones in the history ofSome milestones in the history of

Error Control CodingError Control Coding

1948-49: C.E. Shannons pionring paprs on !"#a$h%a$i&a' Thor( o) Co%%uni&a$ion

1950: *. +. a%%ing rpor$s !a%%ing Co 1954-55: . E'ias in$rou&s &ono'u$iona' &os 1959-0: *. C. os an . K. *a(-Chauhuri an ".

o&3ungh% inpnn$'( inn$ !C Cos 190: I. S. * an . So'o%on s&ri !*-

So'o%on 6*S7 &os

19: . . orn( p'ains &on&a$na$ &oing 19: ". ;. 'ihoo

6#=7 &oing a'gori$h% )or &ono'u$iona' &os


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Some more recentmilestones

19:

199?: C. rrou, ". 'aiu an . Thi$i%aAshi%a)or%u'a$ Turo Cos

1995: . ;. #a&Ka( an *. #. Ba' ris&or =C&os 'ong a)$r *. . a''agr 192D

1998: S. "'a%ou$i s&ris !Spa&-$i% &oing


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Reference

Shu =in an . ;. Cos$''o, ;r., Error Control Coding, 2ndEd.,arson rn$i& a'', 2004

*. . a''agr, Information Theory and ReliableCommunications, ;ohn +i'(, B For>, 198

*. E. 'ahu$, Theory and Practice of Error ControlCodes,"ison +s'(, 198?

T. K. #oon, Error Correction Coding, mathematicalmethods and algorithms, +i'( Inia E., 2005

+. +. $rson an E. ;. +'on, ;r., Error CorrectingCodes, 2n-E., #IT rss Ca%rig, 192


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In)or%a$ion

sour&

Sour&

n&or

Chann'

n&or

#ou'a$or

Chann'

6s$orag%iu%7

Bois

%ou'a$ors$ina$ion Sour&&or

Chann'&or

G r

u

Block diagram of a typical data transmission system


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Coder ModemRF/IF

Stages

RF/IF

StagesModem Decoder

Radio

Channel

Discrete / Coding Channel

PropagationChannel

Modulation Channel

TransmitterReceiver


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A Classification of FEC Codes

FEC Codes

Block"ree

(Con#olutional)

Binary $on%Binary Binary $on%Binary

&ystematic

$onsystematic

&ystematic $onsystematic &ystematic

$onsystematic

&ystematic

$onsystematic


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" inar( 'o&> &o i$h > H 4 an n H

Messages600007

610007

601007611007

600107

610107601107

611107

Codewords600000007

611010007

601101007610111007

611100107

600110107610001107

01011107


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Messages

600017

610017

601017

611017

600117

610117

601117

611117

Codewords

610100017

601110017

611001017

600011017

601000117

610010117

600101117

611111117


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C'assi)i&a$ion o) &oing $&hni3us )or EC Cos:

&oing T&hni3us

#ini%u% is$ &oun is$an&&oing

#aori$( =ogi&&oing J $hr-sho' &oing

S(nro%

&oing Tr''is&oing

S3un$ia'&oing


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S(nro% &oing

=is$ /Ta' 'oo>

up

S(s$%a$i&Sar&h

S$p( s$p

#ggi$$Error

$rapping

"'grai&

@sua''( )or 'inar'o&> &os

C'assi)i&a$ion o) S(nro% &oing $&hni3us


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Memoryless Channels

" %%or('ss &hann' is on in hi&h $hou$pu$ rna$ $h n-$h s(%o' $i% pns on'( on$h inpu$ a$ $i% n.

in $h inpu$ a$ $i% n, $h ou$pu$ a$ $i% n iss$a$is$i&a''( inpnn$ o) $h ou$pu$s a$ o$hr$i%s.

"i$i aussian &hann' an inar(S(%%$ri& Chann' 6SC7 %a( i as%%or('ss &hann's.


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s1 s1

31

3231 32

1 L 31 1- 32

31 321 L p1

1 L p1

p1

p1

00

11

1 L p2

1 L p2

p2

p2

00

11

&ate s' &ate s

A &implified model of a channel with memory


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#orn a'grai& $hor( &'assi)is %an( ari$h%$i&

s(s$%s a&&oring $o $hir %a$h%a$i&a's$rng$h. or a%p',

*roup: " s$ o) %a$h%a$i&a' o&$s 6'%n$s7

$ha$ &an MaN an Msu$ra&$N. Ring: " s$ o) %a$h%a$i&a' o&$s $ha$ &an

MaN, M su$ra&$N an M %u'$ip'iN. Field: " s$ o) %a$h%a$i&a' o&$s $ha$ &an

MaN, Msu$ra&$N, M%u'$ip'iN an MiiN.

Introduction to inear !lgebra


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E+ of a Field ,*F ()-.Si%p's$ on i$h 2

'%n$s 6sa(, 0 J 17.

0 O 0 H 0 0 . 0 H 0

0 O 1 H 1 0 . 1 H 01 O 0 H 1 1 . 0 H 01 O 1 H 0 1 . 1 H 1

6"i$ion in 627 is $h 6#u'$ip'i&a$ion in 627%ou'o-2 ai$ion or is %ou'o-2 %u'$ip'i&a$ion

EPQ* ai$ion7 or "B opra$ion 7



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ur$hr, 1 O 1 H 0 - 1 H 1 Msu$ra&$ionN

J 1. 1 H 1 1R1 H 1 MiisionN

$ote. #os$ o) $h $&hni3us o) 'inar a'gra

6.g. %a$ri opra$ions7 &an us$i)i / or i''ha an ana'ogous opra$ion in a )ini$ )i'.

So, a asi& unrs$aning o) )ini$ )i's is

i%por$an$ )or 'arning EC $&hni3us.


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/efinition (*roup). " group ! is a s$ o)

'%n$s $og$hr i$h an opra$ion on pairs o)

'%n$s in $h s$ 6no$ ( 7 sa$is)(ing $h

)o''oing )our propr$is:

' Closure.or r( !a an ! in $h s$,

& H a is a'so in $h s$.

Associati#ety.or r( !a, !, !& in $h s$a 6&7 H 6 a 7 &


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Introduction to Linear Algebra

0 1dentity.Thr is an '%n$ ! in ! &a'' $h

!in$i$( '%n$ $ha$ sa$is)isa H a H a, )or r( a in $h s$.

2 1n#erses. I) !a is in $h s$, $hn $hr is so%

'%n$ ! in $h s$ &a'' an !inrs o) a su&h$ha$, a H a H , )or a'' a J in .

So% groups sa$is)( $h ai$iona' propr$( o)

&o%%u$a$ii$(, i.. a H a, )or an( a J Su&h a group is &a'' a &o%%u$a$i group or an

a'ian group 6a)$r Bi's "', 1802 L 18297


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I) ! has )ini$ no. o) '%n$s, i$ is a )ini$ group.

&ome con#entional sym!ols for a!elian groups.

as O 6ai$ion7

0 6ro7

inrs o) a - a

a H a H a O 6-a7H6-a7O a H0 D


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6Q*7 . 6%u'$ip'i&a$ion7

1 6on7

inrs o) a a-1

a H a H a . a-1H a-1.a H 1D

"heorem 3' (*roup 4'). In r( group, $h

in$i$( '%n$ is uni3u. "'so, $h inrs o)

a&h group '%n$ is uni3u, an, 6a-17-1H a

Q*, )or !O an !- rprsn$a$ion, - 6-a7 H aD


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5roof.=$, J $o possi' in$i$( '%n$s

H H H . 6pro7B$, '$ J ar $h inrss o) a.

a H 6( in$i$( propr$(7.

Bo, H H 6a 7 H 6a7 6"sso&ia$i7

H H

=as$'(, a-1

a H a a-1

H 1.So, !a is an inrs o) a-1. "s us$ sho $ha$

$h inrss ar uni3u, 6a-17-1H a.



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&ome e+amples of *roups.

1nfinite groups.

a7In$grs unr $h opra$ion o) ai$ion 6O , 0,

- U in$i$( '%n$ : 077osi$i ra$iona' nu%rs unr $h opra$ion o)

%u'$ip'i&a$ion.

&7S$ o) 2 2 ra' a'u %a$ri&s unr %a$ri

ai$ion "'ian group V D



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Finite groups.

a7 W0, 1X unr EPQ* opra$ion 7 W0, 1, .., 8, 9X unr %ou'o-10 ai$ion.

E+ample of a finite non%a!elian group.

$rans)or%a$ion 6ro$a$ion J r)'&$ion7 o) an

3ui'a$ra' $riang':

=$ us )in $h )o''oing si $rans)or%a$ions:1 H 6"C "C7 no &hangDa H 6"C C"7 ,anti%clockwise rotation !y'6-


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H 6"C C"7 &'o&>is ro$a$ion (120D

& H 6"C "C7 *)'&$ion aou$ is&$or o) "D H 6"C C"7 *)'&$ion aou$ is&$or o) D

H 6"C "C7 *)'&$ion aou$ is&$or o) CD

=$, $h group 6, 7 )in as, HW1, a, , &, , X

J ( is a group '%n$ $ha$

no$s $h $rans)or%a$ion ong$s ( pr)or%ing s3un$ia''( )irs$ $h

C

"


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$rans)or%a$ion no$ ( ( an $hn $h

$rans)or%a$ion no$ ( .

a

H 6"C C"7 6"C C"7H 6"C "C7 H .

Bo, ri)( $ha$, a H &

Si%i'ar'(, a& & a


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Ta' )or (:

x\y a ! c d e

a ! c d e

a a ! d e c

! ! a e c d

c c e d ! a

d d c e a !

e e d c ! a

Bo$: r( '%n$ appars on& in a&h &o'u%n J on& ina&h ro. This a'a(s happns in a )ini$ groupD


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Q. /o permutations on n letters form a group7

&u!group. I) is a group J is a sus$ o) ,

$hn ! is &a'' a sugroup .r.$ $h rs$ri&$ion o)

6opra$or7 $o .1t implies that a non empty set is a su!group of

* if it is shown that the closure 8 in#erse properties of

are #alid "he other properties of associati#ity 8

identity are in herited from the group *


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E.In $h s$ 67 o) in$grs unr ai$ion, $h s$

o) n in$grs 6or $h s$ o) %u'$ip's o) ?7 is asugroup .

"o form a su!group of a finite group *.Ta> an( '%n$ !h )ro% J )or% a

s3un& o) '%n$s

h, h h, hhh, hhhh, YYYYY h h2 h? h4


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En$ua''(, i$ %a( )oun $ha$, h&H 1U hn !& is

&a'' $h orr o) $h '%n$ !h.Cyclic *roup. " group $ha$ &onsis$s o) a'' $h

!pors o) on o) i$s '%n$s is &a'' a &(&'i&

group.

Coset /ecomposition of a finite group 9*:.

=$ ! a sugroup o) ! J i$s '%n$s h1, h2, h?, Y.. i$h h1as $h in$i$( '%n$.


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Bo, &ons$ru&$ an arra( as)o''os:

h1H 1 h2 h? h4YY hng2 h1H g2 g2 h2 g2 h? g2 h4YY g2 hn.

g? h1 H g? g? h2 g? h? g?h4YY g? hn

..............................................................................................g% h1 H g% g% h2 g% h? g% h4Y.. g% hn

g2: an '%n$ o) no$ &onsir ar'ir.S$op hn a'' $h group '%n$s appar so%hr in $h

arra(U i$ has $o s$op as ! is )ini$. Th &os$ &o%posi$ion

is a'a(s r&$angu'ar.


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"heorem 3.Er( '%n$ o) ! appars on&

J on'( on& in a &os$ &o%posi$ion o) .5roof. Er( '%n$ appars a$ 'as$ on&

6o$hris $h pro&ss os no$ s$op7.

Bo, suppos, $o '%n$s in $h sa% ro g i

hJ gi h>ar 3ua'. Thn %u'$ip'(ing a&h i$h

giL1

gis hi H h>, hi&h his &on$rai&$or(. So, an'%n$ &an no$ o&&ur $i& in $h sa% ro.


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ur$hr, suppos $o '%n$s in i))rn$ ros, gi

han g> h', ar 3ua' an $ha$ >i.

Bo %u'$ip'(ing on $h righ$ ( hL1. + g$,

gi H g> h' hL1.

Thn, !gi is in $h > L $h &o-s$ &aus h'hL1is in

$h sugroup. This is &on$rai&$or( again. So, an'%n$ &anno$ appar in $o i))rn$ ros.

Corollary. i) is a sugroup o) , $hn $h no. o)'%n$s in iis $h no. o) '%n$s in .


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Tha$ is,

6orr o) 7.6Bo.o) &os$s o) .r.$ 7H6orr o) 7"heorem 30. Th orr o) a )ini$ group is iisi'( $h orr o) an( o) i$s '%n$s.

R1$*&.

Definition: " ring * is a s$ o) '%n$s i$h $oopra$ions )in : $h )irs$ is &a'' ai$ion 6O7 J

$h s&on is &a'' %u'$ip'i&a$ion 6no$ (u$aposi$ion7. ur$hr, $h )o''oing aio%s arsa$is)i:


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6i7 * is an a'ian group unr ai$ion 6O7.

6ii7 Closure: or an( a, in *, $h prou&$ a is in*.

6iii7 Associatie la!: a 6&7 H 6a7&

6i7 Distributie la!: a6 O &7 H a O a& 6 O &7 a H a O &a

Bo$: !O opra$ion is a'a(s &o%%u$a$i in a

*ing, u$ !. %a( no$ . I) %u'$ip'i&a$ionopra$ion is a'so &o%%u$a$i i$ is a

&o%%u$a$i ring.


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"heorem 32.or an( $o '%n$s a, in a ring *,

6i7 a 0 H 0 a H 0

6ii7 a 6-7 H 6-a7 H -6a7

roo) : 6i7 a 0 H a60 O 07 H a 0 O a 0a0 L a0 H a.0 O a 0 L a 0

or 0 H a 0.

roo) o) 6ii7 is ')$ as an r&is. I) a *ing has an in$i$( '%n$ or %u'$ip'i&a$ion, i$ is aring !ith identity, i.. a H a.1 H 1.a, )or a'' !a


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In !*, r( '%n$ has an inrs or !O, u$

%a( no$ ha inrss or 6.7. or, in !*

i$h in$i$(, i) a H 1, $hn ! is &a'' as $h

"right inerse#o) a.

"heorem 3;. In a ring i$h in$i$(,

6i7 Th in$i$( is uni3u

6ii7 I) an '%n$ !a has o$h a righ$ inrs an a

')$ inrs !&, $hn H &.


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In $his &as $h '%n$ !a is said to hae an

inerse6no$ ( a-17. Th inrs in uni3u.

roo): Sa% as Thor% Z1, ')$ as an r&is.

"n '%n$ i$h an inrs is &a'' a uni$. Th

s$ o) a'' uni$s is &'os unr %u'$ip'i&a$ion 6i) a

J ar uni$s, & H a has inrs &L1

H -1

a-1

7


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"heorem 3 ,Finite field o#er polynomial

Ring-.

Th ring o) po'(no%ia's %ou'o a %oni&

po'(no%ia' p67 is a )i' i) an on'( i) p67 is apri% po'(no%ia'.

roo):=$, p67 a pri% po'(no%ia'. To pro $ha$

$h ring is a )i', ha $o s$a'ish $ha$ r( nonAro '%n$ has a %u'$ip'i&a$i inrs.


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=et@ s(+) !e a non ero element of the ring

"hen@ deg s(+) J deg p(+)Further@ *C/,s(+)@ p(+)- ' and hence@

' a(+)p(+) D !(+)s(+)@ for some polynomials a(+) and

!(+)' Rp(+),'- Rp(+),a(+) p(+) D !(+)s(+)-

Rp(+),Rp(+),!(+)- Rp(+),s(+)--

Rp(+),Rp(+),!(+)-s(+)-

(Rp(+),!(+)-)s(+) ,mod p(+)-

+ s $ha$ !*p67 67D is a %u'$ip'i&a$i inrs o) s67in $h ring o) po'(no%ia's %ou'o p67.


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Bo, suppos $ha$ p67 hos gr is a$ 'as$

2, as 'or g. po'(no%ia's ar pri%D is no$ pri%.Thn,p67 H r67.s67, sa( )or so% r67 an s67, a&h

o) g. a$ 'as$ !1. J )in or $h ring o) po'(no%ia'.

I) $h ring is a )i', $hn r67 has an inrs po'(no%ia' r-167.

n&,

s(+) Rp(+),s(+)- Rp(+),r%'(+) r(+) s(+)- Rp(+),r

%'(+) p(+)- 6

u$ s67 0 J hn& a &on$rai&$ion. So, $h po'(no%ia'ring is no$ a )i' i) p67 is no$ a pri% po'(no%ia'.


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&an &ons$ru&$ a )ini$ )i' 63n7D, hos

'%n$s ar rprsn$ ( po'(no%ia's or637 o) gr 'ss $han !n.

+ha$ is $h no o) '%n$s o) $h )ini$ )i' so &ons$ru&$V

"ns: 63n7D.E: Cons$ru&$ 62?7 )ro% 627 using $h pri%

po'(no%ia' p67 H ?O O 1. 6in$. In$i)( $h '%n$s J&ons$ru&$ $h $a's )or ai$ion J %u'$ip'i&a$ion7.

5rimiti#e Field Element


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)ini$ion :" pri%i$i )i' '%n$ o) 637 is an

'%n$ !^ su&h $ha$ r( )i' '%n$6&p$ Aro7 &an prss as a por o)

!^ .E: In 657, 21H 2, 22H 4, 2?H ?, 24H 1.

ri%i$i '%n$sar r( us)u' )or

&ons$ru&$ing )i's &aus, i) on pri%i$i'%n$ is )oun, $h %u'$ip'i&a$ion $a' &an &ons$ru&$ asi'(.


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"heorem 3' ,?n Field Elements-

=et@

'@

@ @

K 4' denote the non ero field

elements of *F(K) "hen@

+K%'4' (+ % ') (+ % ) (+ % K%')

roo):Th s$ o) non Aro '%n$s o) 637 is a

)ini$ group unr $h opra$ion o) %u'$ip'i&a$ion.

Bo, '$ an( non Aro '%n$ o) 637 an!h is i$s orr unr %u'$ip'i&a$ion, i..

hH 1.

Bo r%%ring $ha$ $h no o) '%n$s in a


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Bo, r%%ring $ha$ $h no o) '%n$s in a

)ini$ group 6i.. $h orr o) $h group7 is iisi'

( $h orr o) an( '%n$ o) $h group, &an

sa( $ha$ !h iis 63 L 17.

_3-1H 6_h763-17/h H1

So, ! is a Aro o) $h po'(no%ia' 63-1L 17an

hn&, &onsiring a'' iLs,

3-1L1 H 6 - 17 6 - 27 Y. 6 - 3-17


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"heorem 3' ,Cyclic 4 group property of non%

ero elements of *F(K)-.

Th group o) non-Aro '%n$s o) 637

unr %u'$ip'i&a$ion is a &(&'i& group 6*%%r: "&(&'i& group &on$ains a'' $h pors o) on o) i$s '%n$s7.

roo):Q%i$$. or, no$ $ha$, i) 63 L 17 is a pri%, $h proo) is

oious as r( '%n$, &p$ 1, has orr 63L17. So, r( '%n$is pri%i$i, i) 63 L17 is a pri%.


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"heorem 36 , E+istence of a primiti#e element in

*F(K)-.

E#ery *alois field has a primiti#e element5roof. Th prious $hor% s$a$ $ha$ $h non-Aro

'%n$s o) 637 )or% a &(&'i& group 6i.. a'' pors o) an

'%n$ o) $h group is$7.

I) is $ha$ '%n$, $hn, $h non Aro '%n$s o)

637 &an prss as, , 2, ?, 4, Y.., 3-1, as $hr

ar 63 L 17 '%n$sD.

So, $hr is on '%n$ hos orr is 3 L1, i.. 3-1

H 1. Thus is $h pri%i$i '%n$ o) 637.


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Th orr o) r( non Aro '%n$ o) 637iis 63 L17.

+hn &ons$ru&$ing an $nsion )i' as

a s$ o) po'(no%ia's, i$ is usua''( &onnin$ i)$h po'(no%ia' p67 &orrspons $o a pri%i$i'%n$ o) $h )i' )or H ^.This is on ( &hoosing a sp&ia' po'(no%ia'

L $h pri%i$i po'(no%ia'.

)i i$i i i$i ' i 'D


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)ini$ion ri%i$i po'(no%ia'D:

" pri%i$i po'(no%ia' p67 or 637 is a

pri% po'(no%ia' or 637 i$h $h propr$( $ha$

in $h $nsion )i' &ons$ru&$ %ou'o p67, $h

)i' '%n$ rprsn$ ( ! is pri%i$i.ri%i$i po'(no%ia' o) r( gr is$s

or r( .

" pri%i$i po'(no%ia' is a pri% po'(no%ia'haing $h pri%i$i '%n$ as i$s Aro.


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"E &"RLC"LRE ?F F1$1"E F1E=/&.

)ini$ion 6Chara&$ris$i&7:Th nu%r o) '%n$sin $h s%a''s$ su)i' o) 637 is &a'' $h

&hara&$ris$i& o) 637.

"heorem 3' ,on the characteristic of a *alois

Field-.

Ea&h a'ois )i' &on$ains a uni3u s%a''s$su)i', hi&h has a pri% nu%r o) '%n$s.


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roo):6Q%i$7Th )i' &on$ains 0 J 1. To )in $h

su )i', &onsir $h sus$ H W0, 1, 1O1, 1 O1O1, Y..X no$ing $hs ( W0, 1, 2, ?, YX. This is

a &(&'i& su group unr ai$ion an i$ %us$

&on$ain a )ini$, sa( !p no. o) '%n$s.

+ ha $o sho $ha$ !p is a pri% J H

6p7. In !, !O is %ou'o p 6as i$ is a &(&'i& gr.

@nr O7. ur$hr, ^._ H 6 1 O 1 O Y.O 17 _ H _ O

Y.. O _, hr $hr ar !^ &opis o) _ in $h su%.


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n&, %u'$ip'i&a$ion is a'so %ou'o !p.

#oror, a&h '%n$ _ has an inrsunr !. &aus, $h s3un& _, 2 _, ? _, Y., is

a &(&'i& sugroup o) . I$ &on$ains !1 so $ha$ ^_ H 1

)or so% !^ in .

Thus, $h sus$ ! &on$ains $h in$i$(

'%n$, is &'os unr ai$ion J %u'$ip'i&a$ionan &on$ains a'' inrs unr !O J !..


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n&, i$ is a su)i' J has %o. p ari$h%$i&.

ro% our $hor% on $h 3uo$in$ ring sa( $ha$!p has $o a pri%.

)ini$ion 6%ini%a' po'(no%ia'7:I) 6[7 is an $nsion )i' o) 637 an _

is an '%n$ o) 6[7, $hn $h pri% po'(no%ia'

)67 o) $h s%a''s$ gr or 637 i$h )6_7 H 0

is &a'' $h %ini%a' po'(no%ia' o) _ or 637.


100/257


"heorem 3 ,?n minimal polynomial-.

Er( '%n$ !_ o) 6[7 has a uni3u%ini%a' po'(no%ia' or 637.

ur$hr, i) !_ has $h %ini%a' po'(no%ia' )67

an a po'(no%ia' g67 has !_ as a Aro, $hn )67

iis g67.

roo): 6Q%i$7 + no$ $ha$, !_ is a'a(s a Aro o)

6[L 7, hi&h is a po'(no%ia' or 637.

Bo, i$h $h h'p o) uni3u )a&$oriAa$ion $hor%,


101/257


p 3

[L H )167. )267Y.. )>67.

Bo, i) !_ is a Aro o) $h ')$ si, $hn i$ %us$ aAro o) so% $r% on $h righ$ si 6o) on'( on $r%,

&aus, or $h $nsion )i' 6[7, $h pri% $r%s

&an )ur$hr )a&$or in$o 'inar an &ons$an$ $r%7.To pro $h s&on par$ o) $h $hor%, '$ us ri$,

g67 H )67.h67 O s67, hr g. s67 g. )67 an hn&

s67 &an no$ ha !_ as a Aro.u$, 0 H g6_7 H )6_7.h6_7 O s6_7 H s6_7.

s67 %us$ Aro an $h $hor% is pro.


102/257


"heorem 30 ,on the e+pression of a field

element-.=$ !^ a pri%i$i '%n$ in 6[7, an

$nsion )i' o) 637 an !% $h gr o)

)67, $h %ini%a' po'(no%ia' o) !^ or 637. Thn,

$h nu%r o) '%n$s in $h )i' 6[7 is, [ H 3%

an a&h '%n$ !_ &an ri$$n as,

_ H a%-1^%-1O a%-2^

%-2O YY O a1 ^ O a0, hr, a%-

1, a%-2, Y., a1, a0ar '%n$s o) 637.


103/257


roo): + no$ $ha$ an( '%n$ !_ %a( ri$$n

in $h )or%_ H a%-1%-1O a%-2%-2 O Y. O a1 O a0, is an

'%n$ o) 6[7 as ! is $h pri%i$i '%n$D

ur$hr, $his prssion is uni3uU &aus i)

_ H %-1%-1O %-2%-2O Y. O 1 O 0, $hn,

0 H 6a%-1

L %-1

7 %-1O 6a%-2

L %-2

7 %-2O YY

O 6a1L 17 O 6a0L 07 J hn&, ! is a Aro o) a

po'(no%ia' o) gr 6% L 17


104/257


This is &on$rar( $o $h )ini$ion o) !n. "s $hr ar

!3% su&h _, [ is a$ 'as$ as 'arg as 3%.Qn $h o$hr han, >no $ha$ r( non

Aro )i' '%n$ &an prss as a por o)

!. or i) )67 is $h %ini%a' po'(no%ia' o) !,

)67 H 0. hn&,

%

O )%-1%-1

O Y. O )1 O )0H 0Qr, %H - )%-1%-1- Y. L )1 - )0.


105/257


@sing $his r'a$ionship, an( por o) ! &an

ru& $o a 'inar &o%ina$ion o) 6%-1, %-2, Y..,, 07.

or a%p',

%O1H %.

H -)%-1. %-)%-2%-1- YY - )1 2L )0

H -)%-16-)%-1%-1

- )%-2%-2

- ..Y - )1- )0

H - )%-2%-1- )%-?%-2- Y. L )12L )0


106/257


n&, r( '%n$ o) 6[7 &an prss

as a is$ri&$ 'inar &o%ina$ion o) %-1

,%-2

, Y.,0

.So, [ is no$ 'argr $han 3%J $h $hor% is pro.

Coro''ar( 6on $h no. o) '%n$s7:

Er( a'ois )i' has !p% '%n$s )or so%posi$i in$gr !% J pri% !p.

roo):Er( has a su)i' i$h !p '%n$s $o

hi&h $h prious $hor% app'is.


107/257


Th prious $hor% h'ps us $o asso&ia$ a

po'(no%ia' o) g 6%-17 i$h a&h )i' '%n$si%p'( ( rp'a&ing ! ( !.

Ths po'(no%ia's %a( rgar as )i'

'%n$sU %a( a J %u'$ip'i %ou'o $h%ini%a' po'(no%ia' )67.

This is us$ $h )i' ou' o$ain )ro% $h

$hor% on )ini$ )i's or po'(no%ia' ring using

)67 as $h pri% po'(no%ia'.


108/257


So, a&h &an &ons$ru&$ (

po'(no%ia' ari$h%$i& %ou'o a pri% po'(no%ia'an $h no o) '%n$s is a pri% por.

Th )o''oing $hor% s$a'ishs $ha$ )or r(

pri% !p an posi$i in$gr !%, $hr is a o)

po'(no%ia's.


109/257


"heorem 32 (Alge!raic property of the field

elements in term f the characteristic).

=$, 637 ha &hara&$ris$i& !p. Thn )or

an( posi$i in$gr !% an )or an( '%n$s !^

an ! in 637,

6^ 7p%H ^p%p%

roo): Suppos $h $hor% is $ru )or % H 1. Thn6^ 7pH ^pp


110/257


This &an rais $o $h p $h por.

66^ 7p7pH 6^pp7pH ^p2p2

*pa$ing $his 6% L17 $i%s, g$,

6^ 7p%H ^p%p%

So, i$ is n&ssar( $o pro $h $hor% )or % H 1.

Bo, ( ino%ia' $hor%,

( ) ( )"

pp p ip ii

i

C

= =


111/257


Bo, or 1 i 6p L17 an !p

is a pri% nu%r.

So, $h no%ina$or iis )or a&h!i

61i

6p L177 . hn& pC

iis a %u'$ip' o) !p.

pCi H 0 6%o p7 )or 1 i 6p L17.

In 637, hn $h in$gr ari$h%$i& is

%ou'o p,

( )

( )

& p

i

p ppC

i p i i p i

= =

p


112/257


Bo, i) p H 2, H H - . J i) !p is an o

pri%,

J hn&, in gnra',

( ) ( )"

pp p ip i

i

i

C

=

=

( ) ( )" ""& " " &&&& &

p p = + + + +

( )pp = +

( )$

$ $

( )p p =

( )p p p =

( )m m mp p p =


113/257


"heorem 3; (?n the no of elements of the smallest

splitting field).

=$ !% a posi$i in$gr an !p a pri%.

Thn $h s%a''s$ sp'i$$ing )i' o) $h po'(no%ia'

g67 H p% - rgar as a po'(no%ia' or

6p7 an has p%'%n$s.


114/257


Coro''ar( 6Eis$n& o) a )or r( % J p7:

or r( pri% !p J posi$i in$gr %, $hr

is a a'ois )i' i$h p%'%n$s.

ina''(, n i) !3 is no$ a pri% u$ a pri% por

6sa(, pn7 $hn 63%7 &an &ons$ru&$ as an

$nsion )i' o) 637.


115/257


"he Chinese Remainder "heorems ,Fast

Algorithm-:

I$ is possi' $o uni3u'( $r%in a non-

nga$i in$gr gin on'( i$s r%ainrs %ou'

rsius i$h rsp&$ $o %ou's o) a&h o) sra'

in$grs, proi $ha$ $h in$gr is >non $o

s%a''r $han $h prou&$ o) $h %ou'a$ion LChins r%ainr $hor%.


116/257


Ep'ana$ion: =$ $h nou'a$ion , %0H ?, %1H 4

J %2H 5 an '$ .

in an in$gr C, '$, CiH *%i&DTh Chins r%ainr $hor% sa(s $ha$ $hr is

a on-$o-on %ap $n $h $o possi' a'us

o) !C an $h si$( a'us $ha$ $h &$or o)

rsius 6C0, C1, C27 &an $a> on.

" $

"

/"k

i

i

M m m m m=

= = =

0 C ? 0 C 4


117/257


0 C1 ?, 0 C24

0 C?5Suppos, C0H 2, C1H 1 J C2H 2

or C0H2, $h a'us o) C $ha$ shou' &onsir

ar,W2, 5, 8, 11, 14, 1, 20, 2?, 2, ...X

Si%i'ar'( )or C1H 1, $h &onsira' a'us o) C

ar W 1, 5, 9, 1?, 1, 21,25,29,Y..XJ )or C2H 2,

W2,,12,1,22,2,?2, Y..X

+ osr $ha$ $h uni3u so'i. or !C is 1.


118/257


First theorem for integer ring ('st Chinese

Remainder "heorem) .

in a s$ o) in$grs %0, %1, Y.., %>$ha$

ar pair is r'a$i'( pri% an a s$ o) in$grsC0, C1, Y., C>i$h Ci%i, $hn $h s(s$% o)

3ua$ions

CiH &6%o %i7, i H 0, Y, > has a'%os$ on

so'u$ion )or !C in $h in$ra'


119/257


roo): Suppos $ha$ !& J !C ar so'u$ions in $his

in$ra'. Thn

C H [i%iO Ci 0 i >J CH [i%iOCi 0 i >

So, 6C - C7 is a %u'$ip' o) !%i )or a&h !i. Thn, 6C

- C7 is a'so a %u'$ip' o) as %i-s ar r'a$i'(

pri%.

"

"k

i

i

C m

=

,CiH aii6%o %i7

Chinese Remainder "heorems o#er Ring of


126/257


Chinese Remainder "heorems o#er Ring of

polynomials.irs$ $hor% Th. 2.8.?D: in a s$ o) po'(no%ia's

%60767, %61767,Y., %6>767 $ha$ ar pair is r'a$i'(

pri% an a s$ o) po'(no%ia's C60767, C61767, Y., C6>7

67 i$h g C6i767 g. #6i767, $hn $h s(s$% o)

3ua$ions

C6i767 H C67 6%o %6i7677, i H 0, Y, >. has a$

%os$ on so'u$ion )or C67 sa$is)(ing


127/257


roo): Si%i'ar $o $h ana'ogous $hor% or

in$gri$(.

=$ C67 J C67 $o so'u$ion.C67 H [6i767 %6i767 O C6i767

J C67 H [6i767 %6i767 O C6i767

So, C67 - C67 is a %u'$ip' o) . u$, gC67 - C67 is 'ss $hn $h gr

( ) ( )"

deg , - degk

i

i

C x m x

=

<

( ) ( )"

ki

i

m x=

k


128/257


o) . So, C67 - C67 H 0 n& proD

&econd "heorem.

=$, a prou&$ o) r'a$i'(

pri% po'(no%ia'U '$, an B6i767 sa$is)(

B6i767 #6i767 O n6i767 %6i767 H 1. Thn $h s(s$% o)

&ongrun&s C6i767 H C67 6%o %6i7677 i H 0, Y., >is uni3u'( so' (

( ) ( )"

ki

i

m x

=

( ) ( ) ( )"

kr

r

M x m x=

=


129/257


roo): Th uni3unss is a'ra( pro ( $h

prious $hor%.

So, ha $o sho $ha$ C67 sa$is)is r(&ongrun&.

or $his, osr in a si%i'ar a( as )or,

C67 H C6i767 B6i767 #6i767 6%o %6i7677.&aus, #6r767 has %6i767 as a )a&$or )or r 1.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )"

modk

i i i

i

C x C x N x M x M x

=

=


130/257

02/01/15 SCK, IIT Kharagpur 1?0

ur$hr, B6i767 #6i767 O n6i767 %6i767 H 1

So, B6i767 #6i767 H 1 6%o %6i7677.C67 H C6i767 6%o %6i7677 )or 0 i >

n& proD.

BC C?/E&

" 'arg &'ass o) #u'$ip' or &onn&$ing 'inar,

*'o&>, &(&'i& &os.


131/257


"$$ra&$i &aus, 6i7 goo &os is$ )or

%ora$ 'o&> 'ng$h 6$hough, as(%p$o$i&a''(, $/n ispoor,7 6ii7 r'a$i'( si%p' n&oing J &oing, 6iii7

&on$ains a r( i%por$an$ su&'ass o) *-S &os,

6i7 '' s$ru&$ur J hn&, i$s s$u( )or%s a asis

)or goo unrs$i%a$ing o) a'grai& &os.

)ini$ion o) C Cos: =$ !3 an !% ginan '$ ! an( '%n$ o) 63%7 o)

orr !n 6i n H 17 Thn )or an( posi$i in$gr


132/257


orr n 6i.. H 17. Thn )or an( posi$i in$gr

!$ an an( in$gr !0, $h &orrsponing C &ois $h &(&'i& &o o) 'o&> 'ng$h !n i$h $h

gnra$or po'(no%ia' g67 H =C# W )067, )0 O167, Y

)0O2$ L 167X

+hn )67 is $h %ini%a' po'(no%ia' o) or

637D. @sua''(, on &hooss 0H 1, hi&h 'as $o $h

s%a''s$ gr o) g67.

ur$hr a pra&$i&a' &o r3uirs a 'arg 'o&>


133/257

02/01/15 SCK, IIT Kharagpur 1??

ur$hr, a pra&$i&a' &o r3uirs a 'arg 'o&>

'ng$h !n 6r%%r ShannonV7 J hn&, ! is&hosn as $h pri%i$i '%n$, hos orr is $h

'args$.

or su&h a s'&$ion o) !, $h C &o is &a''a !pri%i$i C &o J in $his &as, n H 3% L 1

6$h orr o) $h pri%i$i '%n$7.

=$,r67 H *&i po'(no%ia' H

"

n

i i

i

r x

=

&67 H &o or po'(no%ia' H &


134/257


&67 H &o or po'(no%ia' H &ii

67 H rror po'(no%ia' H

r67 H &67 O 67.

Bo, C67 is a %u'$ip' o) g67So, r67 H &67 O 67

H 67, i) is a Aro o) &67, i.. a Aro o) g67

r67 H 67 H , H 1, 2, YY, r )or a''

s $ha$ ar Aro o) g67.

"

n

ii

i

e x

=

"

n ii j

i

e

=

Thi i $ ) ! $i i ' i $h $$


135/257


This is a s$ o) !r 3ua$ion ino'ing $h rror pa$$rn

J no$ $h &oors.To so' $hs 3ua$ion, $h s(nro%s ar

)in as,

SH r67, H 1, 2, Y, r 2$ L 1 or 0O1, 0O2,

Y.., r 0 O 2$ L1.

J gr o) g67 H n L > %$)or a inar( C &osign is$an& H 2$ O 1 J %in62$ O 17 or .

*prsn$a$ions o) 617 as an $nsion )i' o)


136/257



627: Th pri%i$i po'(no%ia': p6A7 H A4

O A O1Eponn$ia'

Bo$a$iono'(no%ia'Bo$a$ion

inar(

Bo$a$ion&i%a'

Bo$a$ion#ini%a'

o'(no%ia'

0

0

1

2

?

0

1

A

A2

A?

00000001

0010

01001000

01

2

48

- O1

4O O1

4O O 14O ?O 2O O 1

Class "est%1 26 minutes '0666>


137/257


Q 3' a) Construct a group 9*: of integers with ten elements !) Nerify whether the group is 9cyclic:

c) 1dentify a su!group 9&: with 0 or more elements and

o!tain coset decomposition of 9*:

Q 3 5ro#e that the identity element is uniKue for a ring withidentity

Q 30 a) 5ro#e that the Kuotient ring OI(K) is a field if and only if 9K:

is a prime integer

!) Construct OI(>) !y identifying its elements and operations$ow@ identify all the groups and su!groups that can also !e

defined with the a!o#e elements and operations


138/257



139/257


"heorem (?n a cyclic code).


140/257


"heorem (?n a cyclic code).

1n the ring of polynomials *F(K) ,+-I(+n4')@ a su!set P is

a cyclic code if and only if it satisfies the following

two properties.

' P is a su!group of *F(K),+-I(+n4 ') under addition

1f c(+) P and a(+) *F(K),+-I(+n4 ')@ then

( ) ( )

n

x

R a x c x

a(+). "akes care of end%around shifting 8 addition


141/257


nra$or po'(no%ia' g67:

Th uni3u non-Aro %oni& po'(no%ia' 6in `7 o)

s%a''s$ gr is &a'' $h gnra$or po'(no%ia'

o) ` an is no$ ( g67

g67 is o) gr6n L >7

"h 30 (? th t f th t


142/257


"heorem 30 (?n the property of the generator

polynomial).

" &(&'i& &o &onsis$s o) a'' %u'$ip's o) $h

gnra$or po'(no%ia' g67 ( po'(no%ia's o) gr

6 > L 17 or 'ss.

>: nu%r o) in)or%a$ion s(%o's / 'o&>D

5 f All h l i l t ! i th d ! th


143/257


5roof. All such polynomials must !e in the code !y the

earlier theorem which defines a cyclic code (as g(+) is inthe code)

$ow@ if any polynomial c(+) is in the code@ then@

c(+) Q(+) g(+) D s(+) where deg s(+) J deg g(+) n 4 k 8 s(+)

c(+) 4 Q(+)g(+) which happens to !e a codeword !ecause@ !oth the

terms on the R& are codeword polynomials 8 the code is linear

But deg of s(+) J (n 4 k)@ which is the smallest degree of

any non ero codeword polynomial

ence@ s(+) 6 8 c(+) Q(+) g(+)

"heorem 300 ,g(+) di#ides +n '-.


144/257


"heorem 300 ,g(+) di#ides +n4 '-.

"here is a cyclic code of !lock length 9n: with

generator polynomial g(+) if and only if g(+) di#ides (+n4

'), 5roof omitted -

Following this theorem@ n4 ' g(+) h(+) parity check polynomial

Further@( ) ( )

"n

xR h x e x

=

as@ h(+) c(+) h(+) g(+) a(+) (+n4 ') a(+) some

polynomials


145/257


I) %67 is in)or%a$ion po'(no%ia' o) gr 6> -17

or 'ss,

&67 H %67 g67 non s(s$%a$i& )or%

n L >%67 O r67, s(s$%a$i& )or%, hr,

r67 H *%ainr po'(no%ia'

H - *g67n->%67D

" ' ' ) ' i ' i 'i

BC C?/E& (Bose%ChaudhuriocKuenghem)


146/257


" 'arg &'ass o) %u'$ip' rror &orr&$ing, 'inar,

'o&>, &(&'i& &os."$$ra&$i &aus, 6i7 goo &os is$ )or

%ora$ 'o&> 'ng$h 6$hough, as(%p$o$i&a''(, $/n is

poor,76ii7 r'a$i'( si%p' n&oing J &oing,6iii7

&on$ains a r( i%por$an$ su&'ass o) *-S &os,

6i7 '' s$ru&$ur J hn&, i$s s$u( )or%s a asis)or goo unrs$i%a$ing o) a'grai& &os.

)ini$ion o) C CosD: =$ !3 an !% gin


147/257


)ini$ion o) C CosD:=$ 3 an % gin

an '$ ! an( '%n$ o) 63%7 o) orr !n 6i..n H 17. Thn )or an( posi$i in$gr !$ an an(

in$gr !0, $h &orrsponing C &o is $h

&(&'i& &o o) 'o&> 'ng$h !n i$h $h gnra$or

po'(no%ia'

g67 H =C# W )067, )0O167, Y )0O2$ L 167X

hr ) 67 is $h %ini%a' po'(no%ia' o) .


148/257


Lsually@ one chooses S6 '@ which leads to the smallest

degree of g(+)

Further@ a practical code reKuires a large !lock length

9n: (&hanon:s "heorem7) 8 hence@ 9: is chosen as the

primiti#e element@ whose order is the largest

For such a selection of 9:@ the BC code is called a

9primiti#e BC code:8 in this case@ n Km4 ' (the order

of the primiti#e element)

=et@ r(+) Recei#ed polynomial rii

6 7 ' i ' in

i


149/257


&67 H &o or po'(no%ia' H &ii

67 H rror po'(no%ia' H ii

r67 H &67 O 67.

Bo, &67 is a %u'$ip' o) g67

So, r67 H &67 O 67

H 67, i) is a Aro o) &67, i.. a Aro o) g67r67 H 67 H , H 1, 2, YY, r )or a''

s $ha$ ar Aro o) g67.

"

ii

i

e x

=

"

ni

i ji

e

=

This is a s$ o) !r 3ua$ions ino'ing $h rror


150/257


This is a s$ o) r 3ua$ions ino'ing $h rror

pa$$rn J no$ $h &oors.To so' $hs 3ua$ions, $h s(nro%sar

)in as,

SH r67, H 1, 2, Y, r 2$ or 0, 0O1, 0O2,

Y.., r 0 O 2$ L1.

gr o) g67 H n L > %.$ )or a inar( C &osign is$an& H 2$ O 1 J %in62$ O 17 or .



151/257


p 6 7

627: Th pri%i$i po'(no%ia': p6A7 H A

4

O A O1Eponn$ia'


inar(

Bo$a$ion&i%a'

Bo$a$ion#ini%a'

o'(no%ia'

0

0

1

2

?

0

1

A

A2

A?

00000001

0010

01001000

01

2

48

- O1

4O O1

4

O O 14O ?O 2O O 1

Eponn$ia' o'(no%ia' inar( &i%a' #ini%a'


152/257


Eponn$ia'


inar(

Bo$a$ion&i%a'

Bo$a$ion#ini%a'

o'(no%ia'

4

5

8

9

10

11

A O1

A2OA

A?O A2

A?O A O1

A2O 1

A?O A

A2O A O1

A?O A2OA

0011

0110

1100

1011

0101

1010

0111

1110

?

12

11

5

10

14

4O O 1

2O O 1

4O ?O 2O O 1

4O ?O 1

4O O 1

4O ?O 2O O 1

2 O O 1

4 O ?O 1

Eponn$ia' o'(no%ia' inar( &i%a' #ini%a'


153/257


Eponn$ia'

Bo$a$iono'(no%ia'Bo$a$ion Bo$a$ion Bo$a$ion o'(no%ia'

12

1?

14

15H 1

A?O A2O A O1

A?O A2O1

A?O 1

1111

1101

1001

15

1?

9

4 O ?O 2O O 1

4O ?O 1

4O ?O 1

$ote. 9:@ 9:@ ( ) 2@ all ha#e the same

minimal polynomial +2D + D '

&imilarly 0@ ( 0)


154/257


polynomial).

I) )67 is $h %ini%a' po'(no%ia' or 637 o)

6an '%n$ o) 63%7D, $hn )67 is a'so $h

%ini%a' po'(no%ia' o) 3.)ini$ion 6Conuga$s7:

To '%n$s o) 63%7 $ha$ shar $h sa%

%ini%a' po'(no%ia' or 637 ar &a''

&onuga$s.r.$. 637D.

"he 5eterson *orenstein Oierler /ecoder. (5*O)


155/257


( )

"n( $&hni3u )or &oing a &(&'i& &o &an us $o &o C &os u$ a'' o) $h% ar no$

3ua''( goo )or pra&$i&a' i%p'%n$a$ion.

Th a'gori$h% is asir $o unrs$an an i$

prois $h asis )or $h %or ))i&in$ &oing

s&h%s.

or $h sa> o) si%p'i&i$(, '$, 0H 1.

So, g67 H =C# )167, )267, Y )2$67D


156/257


+hr )67 is $h %ini%a' po'(no%ia' o) . is an('%n$ o) 63%7 L %a( or %a( no$ pri%i$i.

So, n H &o 'ng$h H orr o) or 63%7.

$ H rror &orr&$ing &apai'i$( o) $h &o.

!> H no. o) %ssag s(%o'sU i&$a$ ( %, na

an $.

Bo, $h rror po'(no%ia' %a( ri$$n as, 67 H

n-1n L 1O n L2

n-2O Y O 1 O0.

r, a$ %os$ !$ &o))i&in$s ar non-Aro.


157/257


=$, $h a&$ua' no. o) rrors in a r&i or

!.

Th &oing a'gori$h% i'' or> i) 0 $. I) $h

rrors ar a$ 'o&a$ions i1, i2, Y i, $h rrorpo'(no%ia' &an a'so ri$$n as,

E67 H i1i1O i2i2O Y O ii , ! un>non.

S1H r67 H &67 O 67 H 67

H i1i1O i2i2O Y. O ii

+ r-ri$ $h 3ua$ion as


158/257


+ r ri$ $h 3ua$ion as

S1 H F1P1O F2P2O YY O FPhrF'H i'is $h rror %agni$u 61 ' 7 an P'H i'

is $h rror 'o&a$ion nu%r.

Si%i'ar'(, &an )in $h o$hr s(nro%s S,

)or 1 2$ as

SH r6 H i7 H &6i7 O 6i7 H 6i7Thn ha $h )o''oing s$ o) !2$ si%u'$anous

eKuations in the 9: unknown error location '@ @


159/257


and the 9: unknown error magnitudes T'@ T@ @ T&' T''D T D D T

& T''DT

D D T

&t T''tD T

tD D T

t

"his set of 9t: simultaneous nonlinear eKuations must

ha#e at least one solution !ecause of the way the

syndromes ha#e !een defined

Th so'u$ion is uni3u.


160/257


To so' $hs 3ua$ions, an in$r%ia$

po'(no%ia' 67, &a'' $h rror 'o&a$or po'(no%ia'

is )in as,

67 H O -1-1O -2-2O Y.. O 1 O 1 YYYY 67

67 has Aros a$ $h inrs rror 'o&a$ions P'L1

)or ' H 1, 2, Y,. i..

67 H 61 L P17 61 L P27 YY 61 L P7.

Bo, $h i%%ia$ ai% is $o )in $h &o))i&in$s o)


161/257


67, i.. 1, 2, Y

)ro% $h >no'g o) !2$

s(nro%s.

#u'$ip'(ing o$h si o) 67 ( F'P'O, g$,

F'P'O61 O 1 O 22O Y.. O -1-1O 7 H

F'P'O.67

Bo, '$ H P'L1

. So,Tll

SD(' D 'l4'D l

%D D

%'l%(%')D

l% ) 6

Qr,


162/257


Tl, lSDD 'l

S D%'D lSD% D D

%'lS D'D

lS- 6

This r'a$ionship is a'i )or a&h '6 1 ' 7

an a&h .

So,

or,

SO SO-1 S )or$ J 1

$ $

&&&&& "j j j j jl l l l l l l

Y X X X X X

+ + + +

=

+ + + + + =

&&&&& "j j jl ll l ll l l

Y X Y X Y X

+ + = = =

+ + + =

So, ha a s$ o) 3ua$ions r'a$ing $h i-s


163/257


, 3 g i

i$h S-s:1S O-1O 2S O-2O Y. O SH -SO

H 1, Y..,

This is a s$ o) 'inar 3ua$ions r'a$ing $h

s(nro%s i$h $h &o))i&in$s o) 67. I$ is asir

$o so' $h%.+ri$$n in a %a$ri )or%:

S S S S S S


164/257


$ #

$ # 0

# 0 ' $

$ $ $ $

&&&&&&&&&

&&&&&&&&&

&&&&&&&&&

&&&&&&&&&

M

S S S S S

S S S S S

S S S S S

S S S S S

+

+ +

+ +

M M M M M M

1 4 4 4 4 4 4 2 4 4 4 4 4 4 3

$

$ #

$

S

S

S

S

+

+

+

=

M M

Th i-s %a( )oun ( %a$ri inrsion i) $h S-

%a$ri is non singu'ar. I$ is ra''( so i)$.

"heorem 30; ,?n Nander monde matri+-.Th


165/257


$

$ $ $

$

# # #

$

$

X X X

X X X

A

X X X

X X X

=

L L

L L

L L

L L

M M M

L L


166/257


Pi )or i H 1, 2, Y.., ar is$in&$.roo): $ot included !ut can !e gi#en using the principle ofinduction

Thor% Z? Qn non-singu'ari$( o) s(nro%

%a$riD:

Th %a$ri o) s(nro%s

S S S


167/257


$

#$

$

S S S

SSSM

S S S

+

+

=

M M M

is non singu'ar i) is 3ua' $o, $h a&$ua' no. o)

rrors. Th %a$ri is singu'ar i) \.roo):?mitted

So, ( %a$ri inrsion, &an )in 1, 2, Y..,


168/257


So, ( %a$ri inrsion, &an )in 1, 2, Y..,

( s$$ing H $ )irs$ an &h&>ing a&h $i%h$hr $ 6#7 0.

B$ )in Aros o) 67 $o )in $h rror 'o&a$ion

P1, P2, Y, P.

I) $h &o is non-inar(, ri$ $h s$ o) 2$ 'inar

3ua$ions in Fi-s:S1H F1P1O F2P2O Y. O FP

S2H F1P12O F2P2

2O Y.. O FP2


169/257


S2$H F1P12$

O F2P22$

O Y. O FP2$

.Th )irs$ ! 3ua$ion &an so' )or rror

%agni$us i) $h $r%inan$ o) $h %a$ri o)

&o))i&in$s is non Aro.

Bo, s $ha$, $

$ $ $ $

$

det

X X X

X X X

X X X

L

L

L

L


170/257


( )

$

$ $ $ $ $

$

&&&&& det

XX X

X X X X X X

X X X

=

L

L

L

L

Th 'as$ %a$ri is a


171/257


g

a'gori$h%:Find the syndromes

Find the #alue of starting with a trial #alue of t and

then reducing it in steps of '

$ow we can determine the coefficients of the error

locator polynomial !y a matri+ in#ersion

$e+t@ we can determine the error locations !y a searchprocedure (Chien &earch)

En$r r67

Co%pu$ s(nro%sS H r67, H 1,2, Y.,

2$

"he flow chart for the 5*O decoder


172/257


H $

$ #D H 0 -1

in rror 'o&a$ion P' H 6' H 1, 2, Y.., ( )ining Aros o) 67

67 H r67 O b67

a'$.

Fs

Bo

C'rinsar&h

ornrs ru' in Chin sar&h:


173/257


67 H 6 YY 666 O -17O -27 O -?7O ..Y O 07

I$ ns on'(%u'$ip'i&a$ions anai$ions $o

&o%pu$ 67.

S&a'ar:

=ogic Circuits for Finite Field Arithmetic


174/257


S&a a

h h.

un&$ion o) a sing' inpu$ aria'

#u'$ip'is i/p i$h a )i '%n$ o) 637

"r:


175/257


"r:

O

D

.

un&$ions o) $o inpu$s

)ro% 637.or $h inar( &as, $h(ar EP-Q* ga$ J "Bga$s rsp&$i'(.

#u'$ip'ir:

n L s$ag shi)$ rgis$r:


176/257


Ea&h s$ag &on$ains on '%n$ o) 637 6a$ $h

ou$pu$7a&i%a' shi)$ rgis$rs %a )ro% inar(

&o%ponn$s:

Sria':


177/257


ara'''

"i$ion o) $o )i' '%n$s:

O

sria'


178/257


O OO

O

ara'''

#u'$ip'i&a$ion ( a &ons$an$ )i' '%n$ or


179/257


624

7:=$, H A? J H ?A?O 2A2O 1A O 0

_H ?AO 2A5O 1A4O 0A?

H 6?O 07.A?O 6?O 27.A2O 62O 17.AO 1,reduced

with the help of the primiti#e polynomial p() 2D

D'-


180/257


O?2

1

0

O

O

ara'''


181/257


? 2 1 0

OO

4

?

Sria'This &>$. iis ( $h)i po'(no%ia' A4OAO1

This &>$ pr)or%s $h$o s$ps o)%u'$ip'i&i$is

Shi)$ *gis$r Cir&ui$s 6igi$a' )i'$rs7 us)u' in $h


182/257


&ons$ru&$ion o) n&ors / &ors:6i7

0 1 2 n-1 n

C(&'i&a''( shi)$s a po'(no%ia' o) gr 6n L 17.

I$ &o%pu$s .67 6%o nL 17 in on shi)$

(ii) A general =F&R (=inear Feed!ack&hift Register)


183/257


O O O

h1

O O

-1 h2 h? h4

h= - 1 h=

p -s

Y., p?, p2,p1,p0

*

L

j i j i

i

p h p j L=

=

=oa p0$o p=-1J g$ p, =. 6unning7

p"pp12 p12$

6iii7 "n au$orgrssi )i'$r or a r&ursi )i'$r 6aarian$ o) $h =S*7:

=oad the shift register stage with p p p p


184/257


a>, a>-1, Y., a1, a0

O OO O

O

h1

O

-1 h2 h? h= - 1 h=

Y., p?, p2,p1,p0

=oad the shift register stage with p6@ p'@ p@ @ p =%'

$ow@ feed a6@ a'@ a@ to get the following seKuence.

p"pp12$p12

L

j i j i j L

i

p h p a j L =

= +

6i7 =inar )orarShi)$ *gis$r 6a non

r&ursi )i'$r or an I* )i'$r7:


185/257


O O O

O O

g= g1 g0

= s$aga0, a1,a2,Y.,a>-1, a>6inpu$ in rrs orr7

O

0, 1,2,Y.,=O>-1, =O>

g=-2g=-1 g=-? g=-4

"n $rna''( gnra$ s3un& is us as

Linear Feed forwardShift Register


186/257


inpu$ $o $h shi)$ rgis$r=$ $h $ap igh$s o) $h ) )orar shi)$

rgis$r no$ (:

g67 H g==O g=-1=-1O Y.. O g1 O g0

=$ $h inpu$ an ou$pu$ s3un&s ,

a67 H a>

>O Y O a1

O a0

J 67 H >O= >O=O Y. O 1 O 0

Thn $h shi)$ rgis$r opra$ion gnra$s,

Linear Feed forwardShift Register


187/257


67 H g67.a67 or

$ote. (a) "he shift register initially contains ero and

(!) An input 9a6: is followed !y 9=: eros

Because of this multiplication property@ the circuit is

also called a multiply%!y g(+)circuit

1t may !e noted that in this scheme the contents of the

shift register are not altered

"

* " "L

j i j i n

i

b g a a j L k =

= = +

E+ 3'. =et@ g(+) +D +>D +2D +D + D ' o#er *F()

!(+) a(+)g(+)

!n E"ample


188/257


Inpu$a67

OO

OO O

a67

678 4 2 1

An alternative configuration where contents of shift registerare altered:

a67

42 1

O OO O O

67

8

b(x) = a(x).g(x)

(#) Circuit to di#ide an ar!itrary polynomial a(+) !y a

fi+ed polynomial g(+).

=$ us assu% a %oni& po'(no%ia' as $h


189/257


=$ us assu% a %oni& po'(no%ia' as $h

iisor po'(no%ia':

g67 H n->O gn->-1n->-1O YY O g1 Og0

( )

( )$

$

$

$

" $ "

$

&&&

&&& &&&&

&&&

n

k k

n n n n k

n k n k n n

n k n n

xnn n n n k

a x a a g x

x g x g x g a x a x a x g

a x a a a

+ +

+ + + + + + +

+ +

*&ursi'(, '$,[6r767: 3uo$in$ po'(no%ia' a$ $h r-$h

r&ursion ,with@ Q(6)(+) 6-


190/257


r&ursion ,with@ Q (+) 6-

*6r767: r%ainr po'(no%ia' a$ $h r-$hr&ursion ,R(6)(+) a(+)-

Thn,

an

.Coeff of +n%rin the remainder polynomial R(r%')(+)

, - , - , -, - , -r r r k r n rQ x Q x R x

= +

, - , - , -, - , - & , -

r r r k r

n rR x R x R x g x

=

( )rn r

R


191/257


a67 Y., an-2, an-1

O O O O O O

g0 g1 g2 gn->-1-1

Ini$ia''( 'oa i$hAro.

>-21, >-1YY.

ro% 6n->7 $h $o n-$h shi)$, $h 3uo$in$ passs ou$ o)


192/257


$h shi)$ rgis$r an $h r%ainr is ')$ in $h shi)$rgis$r.

This &on)igura$ion o) $h ii-(-g67 &>$ %oi)is

$h &on$n$ o) $h shi)$ rgis$r. ur$hr, ai$ions ar n $o in$rna' s$ags o)

$h shi)$ rgis$r.

E. =$, g67 H 8O O 4 O 2O O 1. or 627.


193/257


, g6 7 6 7

0 42

O OO O

Qu$pu$

O

Inpu$

a67

An alternati#e configuration of di#ide%!y%g(+) ckt where

internal addition is not needed.

Th ia is $o 'a( $h su$ra&$ions un$i' a''


194/257


Th ia is $o 'a( $h su$ra&$ions un$i' a''

$h su$ra&$ions in $h sa% &o'u%n &an on

a$ $h sa% $i%. o''oing $h ia, on &an

%oi)( $h prious'( ri$$n i$ra$i prssions

as: *6r767 H a67 L [6r767 g67

"n hn&,

( ) ( )

n

r rn r n r n r i i

i

R a g Q

==

ur$hr,( ) ( ) ( ) ( ) ( )

r r r k rQ Q R +


195/257


&chematic of di#ide%!y%g(+) circuit using (n%k) stage

shift register.

( ) ( ) ( ) ( ) ( ) r r r k r

n r

Q x Q x R x

= +

O O O

O O

-1

on )or shi)$s )ro%6nO17 $o 6nO>7

a67... an-2, an-1

[uo$in$ uring shi)$s)ro% 6n->7 $o !n

O

gn->-1gn->-2 gn->-4 g1 g0

E: g67 H 8O O 4O 2O O 1


196/257


O O O

O

*

[a67

Th &oor &67 H %67g67.

Shi(t register encoder34x& o( a .amming ,'*- nonsystematic code3 g,x- 5 x0 6 x 6


197/257


OO

4 1 0 &67

%67

En&or g67

11 i$ in$. 4 i$ pa 15 i$ &oor

Th &orrsponing s(s$%a$i& &or:


198/257


In s(s$%a$i& )or%,&67 H n->%67 O r67 hn r67 H -*g67n->%67D.

*a'iAa$ion: a7&on&p$ua''( s$raigh$ )orar u$

&o%pu$a$iona''( in))i&in$.

@p )or 'as$ 4 i$sO O

Qpn on 'as$ )ouri$s

%67


199/257


@p )or 'as$ 4 i$s

on )or )irs$ 11 i$s6a)$r 4 ini$ia' shi)$s7

C676n->7 Arospa a$ $hn o) %67

To$a''(, 4 O 11 O 4 H 19 shi)$s ar n $o n&o.Multiplication !y +2 is implicit in the timing of the circuit

"he di#ision operation does not !egin till the first four !its are in

position in the register &o@ an additional 2 !it !uffer is used to

ensure that the first !it is sent to the channel Sust when the first

step of the di#ision occurs

")$r 'n i$ra$ion o) iision opra$ion, $h


200/257


r%ainr is ')$ in $h ii-(-g67 &ir&ui$ $o shi)$ ou$ $o $h &hann'.

uring $hs 'as$ 4 shi)$s, $h )a&> pa$h is

ro>n.So, in a'', i$ $a>s 19 shi$s $o gnra$ a 15 i$

&oor.

*a'iAa$ion: 7


201/257


Si%p'i)i&a$ions possi' in $h ao ra'iAa$ion: ?!ser#e that the last four !its of +2m(+) are always

ero as the information is only '' !its long &o@ the last

four !its need not !e added to the remainder

"he incoming information !its do not immediately

enter the di#ide%!y%g(+) circuit@ !ut are entered at the

right time to form the feed!ack signal "he following

modified realiation does the same thing as in

realiation a) !ut in '; clock cycles

OO

Qpn on 'as$ )ouri$s

@p )or 'as$ 4 i$s


202/257


OO

on )or )irs$ 11 i$s%67

This s&h% r3uirs on'( 11 O 4 H 15 shi)$s J

hn& is )as$r.


203/257


Fig3 4ncoding 7ith an ,n28-stage shi(t register

r" rn282r$r6 66 6

g n282g g$

S7itch

n28 shi(t registers9,:-

m,:-

A syndrome decoder for a nonsystematic amming

(';@ '') code.

"his is a single%error correcting code


204/257


"his is a single error correcting code

"he recei#ed polynomial r(+) c(+) D e(+)

s(+) syndrome polynomial (degree 0)

Rg(+),r(+)- Rg(+),c(+) D e(+)-

Rg(+)G c(+)H D Rg(+)Ge(+)H Rg(+),e(+)-

"he most likely error patterns for all possi!le

syndrome patterns are stored in a look up ta!le

r67


205/257


O O

1 or ( 15 i$ *Q#

15 i$ shi)$ *g.

15 i$ shi)$ *g. O O O

6 7

is $h 3uo$in$ po'(no%ia' hn is ii ( g67. , -m x

, -c x$

;c,x-

m;,x-

Reed &olomon (R&) Codes

=inear@ !lock@ cyclic non!inarycodes. i) systematic(common) ii) nonsystematic


206/257


(common) ii) nonsystematic

Multiple random as well as !urst error correctingcapa!ility

A popular su!set of the non%!inary BC codes in

which the sym!ol field *F(Km' ) and the error locatorfield*F(Km) are the same@ ie m' m'

A primiti#e R& code is characteried !y the generatorpolynomial.

g(+) d%'

i'(+%i)@ where 9: is the primiti#e

element of *F(K)

$ote. "he minimal polynomial o#er *F(K) of anelement@ say @ in the same field *F(K) is f

(+) (+%)

A R d & l C d i i di t


207/257


A Reed%&olomon Code is a ma+imum distancesepara!le (M/&) code

d t D' n%kD' dU ie n%k t

, &ingleton Bound. dU n% kD'-

$o of coded sym!ols in a codeword n K 4 '

$o of information sym!ols in a codeword k n 4 t

"he dual of a Reed%&olomon code is also an R& code(0'@ ';) (0'@'


208/257


#os$ o) $h &o%pu$a$ion r3uir $o &oC &os using a'gori$h% ar u $o $hso'u$ion o) $h %a$ri 3ua$ion.:

S1 S2YY.. S -SO1

S2 S?YY.. SO1 -1 H -SO2

S SO1Y.. S2O1 1 -S2

$o of computation needed for matri+ in#ersion 0

B%M algorithm !ypasses this matri+ in#ersion !y#iewing the pro!lem as that of an =F&R synthesis


209/257


pro!lem.=et@ (+) !e known "hen@ the first row of the matri+

eKuation defines &D' in term of &'@ &@ & and the

coefficients of (+)

&imilarly@ the second row defines &D in term of &'@ &@

&D'and the coefficients of (+) and so on

"his seKuential o!ser#ation may !e summaried !y the

following eKuation.

SH -

iH1iS-i, HO1, YY, 2.

"he following =F&R does this So! if initially loaded with

&'@ &@ &.

O O O


210/257


O

-1

O

-2

S-1 S-2YYY. S-O1 S-

O

--1 -0H1

Y..S?S2S1

SH -iH1iS-i

7hich can !e used as tap

7eights to generate the $tsyndromes&

"o design the reKuired =F&R@ we must determine (a)the shift register length 9=: and (!) the feed!ack

connection polynomial (+)

+D%'+

%'D D '+ D

' where !y design (+) = 6So% righ$%os$ s$ags %a(


211/257


'@ where !y design (+) = 6So% righ$%os$ s$ags %a(no$ $app7.

"he design procedure is inducti#e

or a&h r, s$ar$ing i$h r H1, sign an =S* )or

gnra$ing $h )irs$ r s(nro%s. =$ $h %ini%u% 'ng$hshi)$ rgis$r 6#=S*7 prou&ing S1, S2, Y., Sr no$( 6=r, 6r7677.

This rgis$r n no$ uni3u. Sra' &hoi&s%a( is$ u$ a'' i'' ha $h sa% 'ng$h.

"$ $h s$ar$ o) r-$h i$ra$ion, haa'ra( &ons$ru&$ a 'is$ o) =S*-s $i'' $h 6r-17-$h i$ra$ion:

6=1, 617677


212/257


6=2, 627677

.

.

6=r-1, 6r-17677

Th # a'gori$h% )ins a a( $o &o%pu$ a nshor$s$ 'ng$h S* 6=r, 6r7677 hi&h gnra$s S1, S2, Y.., Sr

( 6i7 using $h %os$ r&n$ S* or 6ii7 ( %oi)(ing $h $apigh$s. or 6iii7 ( in&rasing $h 'ng$h J %oi)(ing $h $apigh$s.

"$ r-$h i$ra$ion, &o%pu$ $h n$ ou$pu$ o) 6r-17-$h shi)$ rgis$r:


213/257


g

crH -n-1

H16r-17

Sr-

(Many terms are ero here as the upper limit has !een

chosen as n%' for con#enience)

=$ us no )in $h r-$h is&rpan&( as:

r H Sr- crH SrOn-1

H16r-17

Sr-

H n-1

H06r-17

Sr-

Cas-6i7 I) rH 0, s$ 6=r, 6r7

677 H 6=r-1, 6r-17

677


214/257


Cas-6ii7 J 6iii7 Q$hris, %oi)( $h $aps as )o''os:

6r7

67 H 6r-17

67 O "'6%-1767 hr " is a )i''%n$, !' is a posi$i in$gr an 6%-1767 is on o)

$h S* po'(no%ia's apparing ar'ir in $h 'is$. Bo, r&o%pu$ $h r-$h is&rpan&( 6sa( r7:

rH

n-1

H0 6r7

.Sr-H

n-1

H0W6r-17

O ".'

.6%-17

67X.Sr-

H n-1

H06r-17Sr-O ".n-1

H06%-17Sr--'

Bo, i) &hoos % r su&h $ha$ % 0 an


215/257


&hoos 'H r - % an "H - r/%, $hn s $ha$,rH rL 6r/ %7. % H 0.

So, $h n S* i'' gnra$ S1, S2, Y, Sr-1, Sr.

Bo, $o sp&i)( %, hi&h gis ris $o 'os$gr 67, &hoos % as $h %os$ r&n$ i$ra$iona$ hi&h =%\=%-1. ($ote. m6 when =mV =m%')

67 H O -1-1O YY. O 1 O 1.

Forney Algorithm


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H 'H161-P'7

=$ us )in a s(nro% po'(no%ia'S67 as:

S67H 2$

H1SH 2$

H1

iH0FiPi

"'so )in an rror a'ua$or po'(no%ia'67 as,

67 H S67.67 6%o 2$

7.

"heorem 3 (?n the e+pression of the error e#aluatorpolynomial (+)).


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Th rror a'ua$or po'(no%ia' &an ri$$n as,

67 H .

iH1FiPi 'i61-P'7

roo):

67 HdS67.67 6%o 2$7

H 2$

H1

iH1FiPi.D

iH161-P'7D 6%o 2$7

H

iH1WFiPi61-Pi7.2$

H16Pi7-1

D.

'i61-P'7X 6%o 2$7

Bo,

2$ 1 2 ?


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61-Pi72$

H16Pi7-1

H61-Pi7W1OPiO 6Pi72

O 6Pi7?

OY..O 6Pi72$-1X

H 1- 6Pi72$H 1- 2$Pi2$

67 H W

iH1FiPi61-2$Pi2$7X 'i61-P'7 6%o 2$7

H .

iH1FiPi'i61-P'7X ence pro#ed.

-1 -1

"heorem("he Forney Algorithm).Th rror%agni$us ar gin (,


219/257


6P' 7 6P'-1

7F'H H -i61-PP'-17 P'-16P'-17

roo):

67 H F1P161-P2761-P?7Y.61-P7 O F2P261-P1761-P?7 Y.. 61-P7 O YY O F'P' 61-P17 61-P27 Y.61-P'-17 61-P'O17 YY 61-P7 O Y... O FP61-P17

61-P27 61-P?7 YY. 61-P-17D

6 H P'-17 H P'-1 0 O 0 OY. O F'P'61-P-1'P17 61-P-1

'P27 Y61-P-1

'P'-17 61- P-1

'P'O17Y.61-P-1

'P7D

l%',TllSl('%l

%'S)-

'


220/257


Tl

S

l('%Sl%'

)

i.. F'H 6P'-17 D/ '61-PP'-17D

Further@

(+)

+%'

D (%')%'+

%

D D+ D '

%

i'iSi('%+S)

(l%') % lSl(' 4 Sl%')

Sl('%Sl%') %l%'(l%')

F'H - 6P'-17 D/ P'-16P'-17 n& ro.

Berlekamp 4 Massey Algorithm.


221/257


Ini$ia'iA: 67H0U rH0 = H0U 67H1

r rO1

Co%pu$ rror in n$ s(nro%

drH SrO =

H1Sr-H =

H0Sr-

drH0Vos &urrn$ shi)$ rgis$rsign prou& n$s(nro%V

FES 6$aps ar QK7

BQ 6$aps %us$ &orr&$7

Co%pu$ n &onn&$ion po'(no%ia' )orhi&h drH 0U T67 H 67 - dr 67


222/257


r 6 7 6 7 r 6 7

2= r-1V

67 T67

67dr-1

67 : S$or o' shi)$ rgis$ra)$r nor%a'iAing

67 T67 : @pa$ shi)$ rgis$r= r L = : @pa$ 'ng$h.

67 67

#us$ shi)$ rgis$r 'ng$hnV

FES

BQ

r H 2$ VFES BQ


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g 67 H =

V

BQ FES

a'$- %or$han !$ rrors.

ro& $o n$s$p 6i. orn(

"'gori$h%7

MaSor steps for fast decoding of BC Codes.

En$r 67


224/257


Co%pu$ s(nro%s: SH 67U H 1,2,Y.2$.

in 67 using $h r'>a%p-#ass( "'gori$h%

in rror 'o&a$ion P'( )ining Aros o) 67U' H 1, 2, Y.,6&hin sar&h a'gori$h%7

orn( "'gori$h% $o &o%pu$ rror a'us:

2$


225/257


67 H S6767 6%o 2$

767 H

H167-1

H -

iH1Pii61-P7

- 6P'-17an F'H

P'-16P'-17

6P'-17

H '61-PP'-17 'H 1, 2, Y.,Corr&$ $h r&i or 67


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Multi%stage Coding

&chemes

S-$R %&AL


227/257

02/01/15 SCK, IIT Kharagpur 22GSSS! ""#G$ S. %&A#RA'AR" F%

"R

* +

%,*R%&AL

"R

%,*R

(! R)

,-R

%,*R

(n! r)

,-R

* +

%,*R

*AA

S,-R%

*AA

S"#

S-$R %,*R (n ! r R) S-$R *%,*R

, SAG %,* %,%AA", S%&/


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Cono'u$iona' Coing

$ # & & & & 8?

8? stage shi(t

register

m 5 m*m$* &@* mi* @&&

Input seAuence

,shi(ted in 8 at a time-


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$ n

Code7ord seAuence-5 B*B$* @* Bi* @

here Bi5 ui* @* ui*@* uni

5 i2th code7ord !ranch

ui5 2th !inary code sym!ol

o( !ranch 7ord Bi

n modulo2$

adders

Convolutional 4ncoder 7ith constraint length ? and rate 8/n


230/257


uFirst code

sym!ol

u$Second code

sym!ol


231/257


6Y.1011017

vH60

6070

617, 1607 1607,..

s6 s'

,"-

v

,-

v

'6

&tate /iagram of (@'@0) Con#olutional Code

d


232/257


''

'6 6'

66

6'6'

'' ''

66

'6

66c

a

6

'

!

d

S" S

66

"ree /iagram of (@'@0) Con#olutional Code


233/257

02/01/15 SCK, IIT Kharagpur 2??

66

66 6'

6'

'6 ''

66 6' '6 ''

00

0011

11

10 01

11 00 01 10

(a) 6666 66 66 66 66

'' ''

"rellis /iagram of (@'@0) Con#olutional Code


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(!) 6'

(c) '6

(d) ''

(6) (') () (0) (2) (;)

'' '' '' '' ''

''

66 66 66

'6 '6 '6 '6

6' 6' 6'

'' '' '' ''

S$a$ $ransi$ions


235/257



236/257


Decoder trellis diagram ,rate 5 /$* ?5#-


237/257


dd2compare2select computations in 9iter!i decoding

The #iterbi !lgorithm for a $n, k, K%con&olutional code'


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Step () Beginning at time unit t = 1, compute the partialmetric for the ingle path entering each tate! "tore thepath #the ur$i$or% and it metric for each tate!

Step *) Increae t b& 1! 'ompute the partial metric forall 2( path entering a tate b& adding the branch metricentering that tate to the metric of the connecting ur$i$or atthe pre$iou time unit!

)or each tate, compare the metric of all 2( pathentering that tate, elect the path *ith the larget metric #the

ur$i$or%, tore it along *ith it metric, and eliminate allother path!

+he baic computation performed b& the -iterbi algorithm ithe add, compare, elect #A'"% operation of tep 2! A apractical matter, the information e.uence correponding to

1i8elihood o( s$ 1i8elihood o( s


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p,)Gs$- p,)Gs-

),T-

""" "" "" " "" " "

H2level so(t

decision

" $2level hard decision

Fig3 .ard and so(t decisions

Bncoded 02ary PM Bncoded 02ary PS? Bncoded 2ary M


240/257


Rate $/# coded H2ary PM Rate 0/' coded #$2ary MRate $/# coded H2ary PS?

Fig3 Increase o( signal set si)e (or trellis2coded modulation

u First coded !it

u$ Second coded !it

mFirst

data

!it


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u# Third coded !it

m$Second

data !it

Fig3 Rate $/# Convolutional 4ncoder

"""

Jranch 7ord

u u$ u#

ti6ti

State a 5 """"


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! 5 "

d 5

c 5 "

"

"

"""

""

""

"

""

"

""

"""

"

Fig3 Trellis diagram ,rate $/#

code-


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Fig 3 Bnger!oec8 partitioning o( an H2PS? signal set


244/257


Fig3 Bnger!oec8 partitioning o( 2M signals


245/257


Fig3 4ight2state trellis diagram (or coded H2PS?


246/257


Fig3 Jand7idth2e((iciency plane


247/257


$arallel concatenation of two RS% encoders&


248/257


Feedbac0 (urbo) *ecoder


249/257


Error r)or%an& o) so%EC Cos

*ecoded 'R vs in1ut 'R

f th (23 22) '%& d


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for the (23!22) '%& code42) ex1eri5ental (&**)!

6) anal7tical (&**)!

8) ex1eri5ental (S**)! and

9) anal7tical(S**).

$ t d di 'R bP


251/257


$ost +decoding 'R vs bo

for the (23!22) '%& code4

2) &**!

6) S**! and

8) uncoded s7ste5 1erfor5ance.

linear algebra spring 2014-2015 (1)

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