54676499-DC-Error-Correcting-Codes.ppt

8/17/2019 54676499-DC-Error-Correcting-Codes.ppt

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Department of E lectrical and C omputer E ngineering

Digital Communication and ErrorDigital Communication and Error

Correcting CodesCorrecting Codes

Timothy J. SchulzTimothy J. SchulzProfessor and Chair Professor and Chair

Engineering ExplorationEngineering Exploration

Fall, 2004Fall, 2004


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Digital Coding for Error Correction 0 0 1 0 1 0 0 1 1 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 0 0

Digital DataDigital Data

• ASCII Text

A 01000001B 01000010C 01000011D 01000100E 01000101F 01000110

. .

. .

. .


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Digital Sampling Digital Sampling

000001

010

011

111

110

101

100

00001000100000101101101101000011111011111111


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Digital CommunicationDigital Communication

• Example: Frequency Shift eyin! "FS# $ Tran%mit a t&ne 'ith a frequency (etermine( )y each )it:

( ) ( ) ( ) ( )0 1cos 2 1 cos 2 s t b f t b f t π π = + −


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Digital ChannelsDigital Channels

0

1

0

1

p

p

1-p

1-p

Binary Symmetric Channel

Error probability: p


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Error Correcting CodesError Correcting Codes

inf&rmati&n )it% channel )it%

01

000111

* channel )it% per 1 inf&rmati&n )it: rate + 1,*

channel )it% inf&rmati&n )it%

000001010011100101110111

00010111

(ec&(e )&&-

enc&(e )&&-


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0000010

0

0000000

0

1111100

0

0000001

0

11111101

inf&rmati&n )it%channel c&(ereceie( )it%(ec&(e( )it%

5 channel errors; 1 information error


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0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

channel error probability

b i t e r r o r

p r o b a b i l i t y


• An err&r 'ill &nly )e ma(e if the channel ma-e% / &rthree err&r% &n a )l&c- &f * channel )it%

ccc no errors (1-p)(1-p)(1-p) = 1-3p+3p2-p3

cce one error (1-p)(1-p)(p) = p-2p2+p3

cec one error (1-p)(p)(1-p) = p-2p2+p3

cee two errors (1-p)(p)(p) = p2-p3

ecc one error (p)(1-p)(1-p) = p-2p2+p3

ece two errors (p)(1-p)(p) = p2-p3

eec two errors (p)(p)(1-p) = p2-p3

eee three errors (p)(p)(p) = p3

%ituati&n pr&)a)ility

err&r pr&)a)ility + 3p2 – 2p3


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EE576 Dr. Kousa Linear Block Codes 15

Linear Block Codes


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Basic Definitions

• Let u be a k -bit inforation se!uence

v be t"e corres#ondin$ n-bit code%ord.

& total of 2k

n-bit code%ords constitute a 'n,k ( code.• Linear code) *"e su of an+ t%o code%ords is a code%ord.

• ,bseration) *"e all-ero se!uence is a code%ord in eer+

linear block code.


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/enerator atri

• &ll 2k code%ords can be $enerated fro a set of k linearl independent

code%ords.

• Let g0, g1, …, gk -1 be a set of k inde#endent code%ords.

• v u3/

=

=

−−−−

−

141141041

1400100

nk k k

n

! ! !

! ! !

1-k

0

g

g

G


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+steatic Codes

• &n+ linear block code can be #ut in s+steatic for

• n t"is case t"e $enerator atri %ill take t"e for

/ 8 9 k :

• *"is atri corres#onds to a set of k code%ordscorres#ondin$ to t"e inforation se!uences t"at "ae asin$le nonero eleent. Clearl+ t"is set in linearl+inde#endent.

n-k

chec- )it%

k

inforation bits


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EE576 Dr. Kousa Linear Block Codes 1;

/enerator atri 'cont


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9arit+-C"eck atri

• Aor G [ P | Ik ]4 define t"e atri H 8In-k P*:

• '*"e sie of H is 'n-k (n(.

• t follo%s t"at GH* 0.

• ince v u•G4 t"en v•H* u•GH* 0.

• *"e #arit+ c"eck atri of code C is t"e $enerator atri

of anot"er code Cd 4 called t"e dual of C.

=

1110100

0111010

1101001

H


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Encodin$ sin$ atri

'9arit+ C"eck E!uations(

[ ]

765

65>2

76>1

765

65>2

76>1

765>21

0

00

101111110011

100010001

""=""

""=""""=""

""+""

""+""""+""

"""""""

++++ ++⇒

=++=++ =++

=

0

inforation


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Encodin$ Circuit


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inimum Di%tance

• DF: *"e #a$$in! wei!ht of a code%ord v 4 denoted b+w'v(4 is t"e nuber of nonero eleents in t"e code%ord.

• DF: *"e $ini$%$ wei!ht of a code4 win4 is t"e sallest

%ei$"t of t"e nonero code%ords in t"e code.win in Fw'v() v ∈ C@ v G0H.

• DF) ain$ distance bet%een v and w4 denoted b+d'v,w(4 is t"e nuber of locations %"ere t"e+ differ.

Iote t"at d'v,w( w'vJw(• DF: The iniu distance of t"e coded in in Fd'v,w() ",w ∈ C4 G 0H

• TH3.1: n an+ linear code4 d in win


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EE576 Dr. Kousa Linear Block Codes 2>

inimum Di%tance "c&nt5(#

• TH3.2 Aor eac" code%ord of ain$ %ei$"t l t"ere

eists l coluns of H suc" t"at t"e ector su of t"ese

coluns is ero. Conersel+4 if t"ere eist l coluns of

%"ose ector su is ero4 t"ere eists a code%ord of%ei$"t l .

• COL 3.2.2 *"e d in of C is e!ual

to t"e iniu nubers of

coluns in H t"at su to ero.

• E=)

=

1110100

01110101101001

H


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Decodin$ Linear Codes

• Let v be transitted and be receied4 %"ere

v J e

e ≡ error #attern e1e2..... en4 %"ere

*"e %ei$"t of e deterines t"e nuber of errors.

• e %ill atte#t bot" #rocesses) error detection4 and error

correction.

e ii

th

= 1 if t"e error "as occured in t"e location0 ot"er%ise

Jv

e


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Err&r Detecti&n

• Define t"e sndro$e

! H* 's04 s14 4 sk-1(

• f ! 04 t"en v and e 04

• f e is siilar to soe code%ord4t"en ! 0 as %ell4 and t"e error is undetectable.

• E= .>)

( ) [ ] 0=

=

101

111

110011

100

010

001

r r r r r r r 765>E21210 s s s

65>22

5>E11

65E00

r r r r sr r r r sr r r r s

++++++ +++


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Err&r C&rrecti&n

• ! H* 'v J e( H* vH* J eH* eH*

• *"e s+ndroe de#ends onl+ on t"e error #attern.

• Can %e use t"e s+ndroe to find e4 "ence do t"ecorrectionM

• +ndroe di$its are linear cobination of error di$its.

*"e+ #roide inforation about error location.

• nfortunatel+4 for n-k e!uations and n unkno%ns t"ere are2k solutions. "ic" one to useM


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Example *.6

• Let 1001001

• s 111

• s0 e0JeJe5Je6 1

• s1 e1JeJe>Je5 1

• s2 e2Je>Je5Je6 1

• *"ere are 16 error #atterns t"at satisf+ t"e aboee!uations4 soe of t"e are0000010 1101010 1010011 1111101

• *"e ost #robable one is t"e one %it" iniu %ei$"t.ence v" 1001001 J 0000010 1001011


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tandard &rra+ Decodin$

• *ransitted code%ord is an+ one of)

v14 v24 4 v2k

• *"e receied %ord r is an+ one of 2n n-tu#le.

• 9artition t"e 2n %ords into 2k disNoint subsets D14 D244 D2k

suc" t"at t"e %ords in subset Di are closer to code%ord vi

t"an an+ ot"er code%ord.

•Eac" subset is associated %it" one code%ord.


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Stan(ar( Array

• TH 3.3 Io t%o n-tu#les in t"e sae ro% are identical.

Eer+ n-tu#le a##ears in one and onl+ one ro%.

k k nk nk nk n

k

k

k

22E2222

2EEE2EE

22E2222

2E21

eeee

eJeJee

eJeJee

0

---- +++

+

+

=


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Stan(ar( Array Dec&(in! i% inimum

Di%tance Dec&(in!• Let t"e receied %ord fall in Di subset and l t" coset.

• *"en el J vi

• %ill be decoded as vi. e %ill s"o% t"at is closer to vi

t"an an+ ot"er code%ord.

• d',vi( w' J vi( w'el J vi J vi( w'el (

• d',v j( w' J v &( w'el J vi J v &( w'el J v s(

• &s el and el J vs are in t"e sae coset4 and el is selected to

be t"e iniu %ei$"t t"at did not a##ear before4 t"enw'el ( ≤ w'el J v s(

• *"erefore d',vi( ≤ d',v j(


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EE576 Dr. Kousa Linear Block Codes

tandard &rra+ Decodin$ 'cont( ain$ code

Q of correctable error #atterns 2

Q of sin$le-error #atterns 7

*"erefore4 all sin$le-error #atterns4 and onl+ sin$le-error

#atterns can be corrected. 'Oecall t"e ain$ Bound4

and t"e fact t"at ain$ codes are #erfect.


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EE576 Dr. Kousa Linear Block Codes >

tandard &rra+ Decodin$ 'cont

6>2

651

"=""

"=""

"=""

+++

Code%ords

000000

110001

101010

011011

011100

101101110110

000111

d in

=


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tandard &rra+ Decodin$ 'cont


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*"e +ndroe• u$e stora$e eor+ 'and searc"in$ tie( is re!uired b+ standard arra+

decodin$.

• Oecall t"e s+ndroe

! H* 'v J e( H* eH*

• *"e s+ndroe de#ends onl+ on t"e error #attern and not on t"e transitted

code%ord.

• TH 3.& &ll t"e 2k n-tu#les of a coset "ae t"e sae s+ndroe. *"e s+ndroes of

different cosets are different.

'el J i (* el

* '1st 9art(

Let e N and el be leaders of t%o cosets4 NRl . &ssue t"e+ "ae t"e sae

s+ndroe.e N

* el * 'e & Jel (

* 0.

*"is i#lies e & Jel i4 or el e N Ji

*"is eans t"at el is in t"e Nt" coset. Contradiction.


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*"e +ndroe 'cont


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EE576 Dr. Kousa Linear Block Codes

+ndroe Decodin$

Decodin$ 9rocedure)

1. Aor t"e receied ector 4 co#ute t"e s+ndroe ! H*.

2. sin$ t"e table4 identif+ t"e coset leader 'error #attern( el .. &dd el to to recoer t"e transitted code%ord v.

• $%)

1110101 ? ! 001 ? e 0010000

*"en4 v 1100101

• +ndroe decodin$ reduces stora$e eor+ fro n2n to

2n-k '2n-k (. &lso4 t reduces t"e searc"in$ tie considerabl+.


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EE576 Dr. Kousa Linear Block Codes ;

ard%are #leentation

• Let r 0 r 1 r 2 r r > r 5 r 6 and ! s0 s1 s2

• Fr&m the H atri)

s0 r 0 J r J r 5 J r 6 s1 r 1 J r J r > J r 5

s2 r 2 J r > J r 5 J r 6

• Aro t"e table of s+ndroes and t"eir corres#ondin$

correctable error #atterns4 a trut" table can be constructed.& cobinational lo$ic circuit %it" s0 4 s1 4 s2 as in#ut and

e0 4 e1 4 e2 4 e 4 e> 4 e5 4 e6 as out#uts can be desi$ned.


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EE576 Dr. Kousa Linear Block Codes >0

Decodin$ Circuit for t"e '74>( C


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Err&r Detecti&n Capa)ility

• & code%ord %it" d in can detect all error #atterns of %ei$"t d in S 1 or

less. t can detect an+ "i$"er error #atterns as %ell4 but not all.

• n fact t"e nuber of undetectable error #atterns is 2k -1 out of t"e 2n -1

nonero error #atterns.• DF) &i ≡ nuber of code%ords of %ei$"t i.• F&i@ i04144nH %ei$"t distribution of t"e code.

• Iote t"at &o1@ & & 0 for 0 R & R d in

∑=−−=

n

d i

inii% p p '

in

(1'


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• $%) ndetectable error #robabilit+ of '74>( C

&0 &7 1@ &1 &2 &5 &60@ &&>7

9%'E( 7 p'1- p(> J 7 p>'1- p( J p7

Aor p 10-2 9%'E( 710-6

• Define t"e %ei$"t enuerator)

•*"en

• Let pP'1- p(4 and notin$ t"at &01

∑=

=n

i

i

i ) ' ) '0

('

∑∑==

−

−−=−=

n

i

i

i

i

nn

i

ini

i% p

p p ' p p p ' (

11 1(1'(1'

−

−−=

−=−

− ∑= 11(1'@111 1 p

p ' p ( p

p ' p

p '

n

%

n

i

i

i


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• *"e #robabilit+ of undetected error can as %ell be found fro t"e

%ei$"t enuerator of t"e dual code

%"ere B' ( is t"e %ei$"t enuerator of t"e dual code.

• "en eit"er &' ( and B' ( are not aailable4 9% a+ be u##er bounded

b+

9% T 2-'n-k ( 81-'1- p(n:

• Aor $ood c"annels ' p 0( 9% T 2-'n-k (

(1'(21'2 nk n% p p B −−−= −


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EE576 Dr. Kousa Linear Block Codes >>

Err&r C&rrecti&n Capa)ility

• &n 'n,k ( code of d in can correct u# to t errors %"ere

• t a+ be able to correct "i$"er error #atterns but not all.

• *"e total nuber of #atterns it can correct is 2n-k

t"e code is #erfect

2P(1' in −= d t

∑=

−=

t

i

k n

i

n

0

2f

∑∑=

−

+=

− −

−=−

= t

i

inin

t i

ini

% p pin p p

in (

01

(1'1(1'


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ain$ Codes

• ain$ codes constitute a fail+ of sin$le-error correctin$ codes

defined as)

n 2$-14 k n-$4 $ ≥ • *"e iniu distance of t"e code d in

• Construction rule of H)

H is an 'n-k (n atri4 i*e. it "as 2$-1 coluns of $ tu#les.*"e all-ero $ tu#le cannot be a colun of H 'ot"er%ise d in1(.

Io t%o coluns are identical 'ot"er%ise d in2(.

*"erefore4 t"e H atri of a ain$ code of order $ "as as its

coluns all non-ero $ tu#les.

*"e su of an+ t%o coluns is a colun of H. *"erefore t"e su of

soe t"ree coluns is ero4 i.e. d in.


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Sy%tematic 7ammin! C&(e%

• n s+steatic for)

8 $ U:• *"e coluns of U are all $-tu#les of %ei$"t ≥ 2.• Different arran$eents of t"e coluns of U #roduces

different codes4 but of t"e sae distance #ro#ert+.

• ain$ codes are #erfect codes

Oi$"t side 1Jn@ Left side 2$ nJ1

∑=−

=

t

i

k n

i

n

0 2


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Decodin$ of ain$ Codes

• Consider a sin$le-error #attern e'i(4 %"ere i is a nuber

deterinin$ t"e #osition of t"e error.

• ! e'i( H* Hi* t"e trans#ose of t"e ith colun of H.

• Ea#le)

[ ] [ ]0 1 0 0 0 0 0

1 0 00 1 00 0 11 1 00 1 11 1 11 0 1

0 1 0

=


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Decodin$ of ain$ Codes 'cont


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EE576 Dr. Kousa Linear Block Codes >;

9ei!ht Di%tri)uti&n &f 7ammin! C&(e%

• *"e %ei$"t enuerator of ain$ codes is)

• *"e %ei$"t distribution could as %ell be obtained fro t"e

recursie e!uations)

&014 &10

'iJ1(&iJ1 J &i J 'I-iJ1(&i-1 C I

i i14244I• *"e dual of a ain$ code is a '2$-14$( linear code. ts

%ei$"t enuerator is

{ }2P(1'2 (('1'(1'1

1(' −−−++

+= nn n

n '

12(12'1('−

−+= $

B $


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7i%t&ry

• In the late 1;05%


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=%e%

• 7ammin! C&(e% are %till 'i(ely u%e( in c&mputin!3telec&mmunicati&n3 an( &ther applicati&n%.

• 7ammin! C&(e% al%& applie( in

$ Data c&mpre%%i&n

$ S&me %&luti&n% t& the p&pular pule The 7at>ame

$ Bl&c- Tur)& C&(e%


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A ?@3; )inary 7ammin! C&(e

• 8et &ur c&(e'&r( )e "x1 x/ x@# F/@

• x*3 x63 xD3 x@ are ch&%en acc&r(in! t& the me%%a!e"perhap% the me%%a!e it%elf i% "x* x6 xD x@ ##.

• x; :+ x6 xD x@ "m&( /#

• x/ :+ x* xD x@• x1 :+ x* x6 x@


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?@3; )inary 7ammin! c&(e'&r(%

A ?@ ; )i 7 i C (


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A ?@3; )inary 7ammin! C&(e• 8et a + x; x6 x x@ "+1 iff &ne &f the%e )it% i% in err&r#

• 8et ) + x/ x* x x@

• 8et c + x1 x* x6 x@• If there i% an err&r "a%%umin! at m&%t &ne# then a)c 'ill )e

)inary repre%entati&n &f the %u)%cript &f the &ffen(in! )it.

• If "y1 y

/ y

@# i% receie( an( a)c 0003 then 'e

a%%ume the )it a)c i% in err&r an( %'itch it. Ifa)c+0003 'e a%%ume there 'ere n& err&r% "%& ifthere are three &r m&re err&r% 'e may rec&er the'r&n! c&(e'&r(#.


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Definiti&n: >enerat&r an( Chec-

atrice%• F&r an ?n3 - linear c&(e3 the !enerat&r matrix i% a

-Gn matrix f&r 'hich the r&' %pace i% the !ien c&(e.

• A chec- matrix f&r an ?n3 - i% a !enerat&r matrix f&r

the (ual c&(e. In &ther '&r(%3 an "n4-#G- matrix M f&r'hich Mx + 0 f&r all x in the c&(e.


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A C&n%tructi&n f&r )inary

7ammin! C&(e%• F&r a !ien r3 f&rm an r G /r 41 matrix 3 the c&lumn% &f 'hichare the )inary repre%entati&n% "r )it% l&n!# &f 13 3 /r 41.

• The linear c&(e f&r 'hich thi% i% the chec- matrix i% a ?/r 413 /r 41 $ r )inary 7ammin! C&(e + Hx+"x1 x/ x n# : MxT + 0.

Example Chec- atrix• A chec- matrix f&r a ?@3; )inary 7ammin! C&(e:


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Syn(r&me Dec&(in!

• 8et y + "y1 y/ yn# )e a receie( c&(e'&r(.

• The %yn(r&me &f y i% S:+Lr yT. If S+0 then there 'a%n& err&r. If S 0 then S i% the )inary repre%entati&n&f %&me inte!er 1 J t J n+/r 41 an( the inten(e(c&(e'&r( i%

x + "y1 yr 1 yn#.


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Example =%in! 8*

• Supp&%e "1 0 1 0 0 1 0# i% receie(.

100 i% ; in )inary3 %& the inten(e( c&(e'&r( 'a% "1 01 1 0 1 0#.


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Exten(e( ?K3; )inary 7amm.

C&(e• A% 'ith the ?@3; )inary 7ammin! C&(e: $ x*3 x63 xD3 x@ are ch&%en acc&r(in! t& the me%%a!e. $ x; :+ x6 xD x@

$ x/ :+ x* xD x@ $ x1 :+ x* x6 x@

• A(( a ne' )it x0 %uch that $ x

0 + x

1

x

/ x

* x

; x

6 x

D x

@. i.e.3 the ne' )it

ma-e% the %um &f all the )it% er&. x0 i% calle( aparity chec-.


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Exten(e( )inary 7ammin! C&(e

• The minimum (i%tance )et'een any t'& c&(e'&r(%i% n&' ;3 %& an exten(e( 7ammin! C&(e i% a 14err&rc&rrectin! an( /4err&r (etectin! c&(e.

• The !eneral c&n%tructi&n &f a ?/r3 /r41 4 r exten(e(c&(e fr&m a ?/r $13 /r $1 $ r )inary 7ammin! C&(ei% the %ame: a(( a parity chec- )it.


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Chec- atrix C&n%tructi&n &f

Exten(e( 7ammin! C&(e• The chec- matrix &f an exten(e( 7ammin! C&(e can

)e c&n%tructe( fr&m the chec- matrix &f a 7ammin!c&(e )y a((in! a er& c&lumn &n the left an( a r&'

&f 15% t& the )&tt&m.


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q4ary 7ammin! C&(e%

• The )inary c&n%tructi&n !eneralie% t& 7ammin!C&(e% &er an alpha)et A+H03 3 q3 q L /.

• F&r a !ien r3 f&rm an r G "q r 41#,"q41# matrix M &er A3

any t'& c&lumn% &f 'hich are linearly in(epen(ent.• M (etermine% a ?"qr 41#,"q41#3 "qr 41#,"q41# $ r "+ ?n3-#

q4ary 7ammin! C&(e f&r 'hich i% the chec- matrix.


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Example: ternary ?;3 / 7ammin!

• T'& chec- matrice% f&r the %&me ?;3 / ternary7ammin! C&(e%:


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Syn(r&me (ec&(in!: the q4ary

ca%e• The %yn(r&me &f receie( '&r( y3 S:+MyT3 'ill )e amultiple &f &ne &f the c&lumn% &f M3 %ay S+Mmi3 M %calar3 mi the ith c&lumn &f M. A%%ume an err&r ect&r

&f 'ei!ht 1 'a% intr&(uce( y + x "0 M 0#3 M inthe ith %p&t.


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Example: q4ary Syn(r&me

• ?;3/ ternary 'ith chec- matrix 3 '&r("0 1 1 1# receie(.

• S& (ec&(e "0 1 1 1# a%

"0 1 1 1# $ "0 0 / 0# + "0 1 / 1#.


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Nerfect 14err&r c&rrectin!

• 7ammin! C&(e% are perfect 14err&r c&rrectin! c&(e%.That i%3 any receie( '&r( 'ith at m&%t &ne err&r 'ill)e (ec&(e( c&rrectly an( the c&(e ha% the %malle%t

p&%%i)le %ie &f any c&(e that (&e% thi%.• F&r a !ien r3 any perfect 14err&r c&rrectin! linear

c&(e &f len!th n+/r 41 an( (imen%i&n n4r i% a7ammin! C&(e.


67/769


Nr&&f: 14err&r c&rrectin!• A c&(e 'ill )e 14err&r c&rrectin! if

$ %phere% &f ra(iu% 1 centere( at c&(e'&r(% c&er thec&(e%pace3 an(

$ if the minimum (i%tance )et'een any t'& c&(e'&r(% L *3%ince then %phere% &f ra(iu% 1 centere( at c&(e'&r(% 'ill )e(i%O&int.

• Supp&%e c&(e'&r(% x3 y (iffer )y 1 )it. Then x4y i% a c&(e'&r(&f 'ei!ht 13 an( M"x4y# 0. C&ntra(icti&n. If x3 y (iffer )y /)it%3 then M"x4y# i% the (ifference &f t'& multiple% &f c&lumn% &fM. 2& t'& c&lumn% &f M are linearly (epen(ent3 %& M"x4y# 03an&ther c&ntra(icti&n. Thu% the minimum (i%tance i% at lea%t *.


68/769


Nerfect

• A %phere &f ra(iu% P centere( at x i%SP"x#+Hy in An : (7"x3y# J P. 9here A i% the alpha)et3Fq3 an( (7 i% the 7ammin! (i%tance.

• A %phere &f ra(iu% e c&ntain% '&r(%.• If C i% an e4err&r c&rrectin! c&(e then

3 %& .


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Nerfect

• Thi% la%t inequality i% calle( the %phere pac-in!)&un( f&r an e4err&r c&rrectin! c&(e C &f len!th n&er Fm:

'here n i% the len!th&f the c&(e an( in thi% ca%e e+1.

• A c&(e f&r 'hich equality h&l(% i% calle( perfect.

N f N f t


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Nr&&f: Nerfect

• The ri!ht %i(e &f thi%3 f&r e+1 i% qn,"1n"q41##.

• The left %i(e i% qn4r 'here n+ "qr 41#,"q41#.

qn4r "1n"q41## + qn4r "1"qr 41## + qn.

Applicati&n%• Data c&mpre%%i&n.

• Tur)& C&(e%

• The 7at >ame


71/769


Data C&mpre%%i&n

• 7ammin! C&(e% can )e u%e( f&r a f&rm &f l&%%yc&mpre%%i&n.

• If n+/r 41 f&r %&me r3 then any n4tuple &f )it% x i% 'ithin

(i%tance at m&%t 1 fr&m a 7ammin! c&(e'&r( c. 8et> )e a !enerat&r matrix f&r the 7ammin! C&(e3 an(mG+c.

• F&r c&mpre%%i&n3 %t&re x a% m. F&r (ec&mpre%%i&n3(ec&(e m a% c. Thi% %ae% r )it% &f %pace )ut

c&rrupt% "at m&%t# 1 )it.

The 7at >ame


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The 7at >ame• A !r&up &f n player% enter a r&&m 'hereup&n they each receie

a hat. Each player can %ee eery&ne el%e5% hat )ut n&t hi% &'n.

• The player% mu%t each %imultane&u%ly !ue%% a hat c&l&r3 &rpa%%.

• The !r&up l&%e% if any player !ue%%e% the 'r&n! hat c&l&r &r ifeery player pa%%e%.

• Nlayer% are n&t nece%%arily an&nym&u%3 they can )e num)ere(.• A%%i!nment &f hat% i% a%%ume( t& )e ran(&m.

• The player% can meet )ef&rehan( t& (ei%e a %trate!y.

• The !&al i% t& (ei%e the %trate!y that !ie% the hi!he%tpr&)a)ility &f 'innin!.


73/769

EE 661,;613 Fall3 /00@

C&mmunicati&n Sy%tem%

V"u an

De#artent of Electrical and Co#uter En$ineerin$

Class 25

Dec. 6t"4 2007

OutlineOutline


74/769

EE 6;1,;61 Fall /00@

OutlineOutline

9roNect 2

&OU Oeie%

Linear Code

S ain$ Code Oeisit

S OeedSuller code C+clic Code

S COC Code

S BC Code

S O Code

ARQ FEC ECARQ FEC EC


75/769

EE 6;1,;61 Fall /00@

ARQ, FEC, EC ARQ, FEC, EC

&OU

Aor%ard Error Correction 'error correct codin$(

+brid Error Correction

tx rxErr&r (etecti&n c&(e

AC,2AC

tx rxErr&r c&rrecti&n c&(e

tx rx

Err&r (etecti&n,C&rrecti&n c&(e

AC,2AC

amming Codeamming Code


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EE 6;1,;61 Fall /00@

amming Codeamming Code

'n4k() k inforation bit len$t"4 n oerall code len$t"

n2W-14 k2W--1)

'74>(4 rate '>P7(@ '15411(4 rate '11P15(@ '1426(4 rate '26P1(

'74>() Distance d4 correction abilit+ 14 detection abilit+ 2.

Oeeber t"at it is $ood to "ae lar$er distance and rate.

Lar$er n eans lar$er dela+4 but usuall+ better code

amming Code Exampleamming Code Example


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amming Code Exampleamming Code Example

'74>(

/enerator atri /) first >-b+-> identical atri

essa$e inforation ector #

*ransission ector

Oeceied ector r

and error ector e

9arit+ c"eck atri

Error CorrectionError Correction


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EE 6;1,;61 Fall /00@

Error CorrectionError Correction

f t"ere is no error4 s+ndroe ector eros

f t"ere is one error at location 2

Ie% s+ndroe ector is

%"ic" corres#onds to t"e second colun of H. *"us4 an error

"as been detected in #osition 24 and can be corrected

ExerciseExercise


79/769

EE 6;1,;61 Fall /00@

ExerciseExercise

ae #roble as t"e #reious slide4 but #'1001(< and t"e error

occurs at location > instead. 9ause for 5 inutes

i$"t be 10 #oints in t"e finals.

mportant !amming Codesmportant !amming Codes


80/769

EE 6;1,;61 Fall /00@

mportant !amming Codesmportant !amming Codes

ain$ '74>4( -code. t "as 16 code%ords of len$t" 7. t can

be used to send 27 12 essa$es and can be used to correct 1error.

• /ola+ '241247( -code. t "as > 0;6 code%ords. t can be used to

transit 60 essa$es and can correct errors.

Uuadratic residue '>742>411( -code. t "as 16 777 216code%ords and can be used to transit 1>0 77 > 55 2

essa$es and correct 5 errors.

Reed "uller codeReed!"uller code


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EE 6;1,;61 Fall /00@

Reed!"uller codeReed!"uller code

C#clic codeC#clic code


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EE 6;1,;61 Fall /00@

C#clic codeC#clic code

C+clic codes are of interest and i#ortance because

S *"e+ #osses ric" al$ebraic structure t"at can be utilied in aariet+ of %a+s.

S *"e+ "ae etreel+ concise s#ecifications.

S *"e+ can be efficientl+ i#leented usin$ si#le shift re!ister

S an+ #racticall+ i#ortant codes are c+clic

n #ractice4 c+clic codes are often used for error detection

'C+clic redundanc+ c"eck4 COC(

S sed for #acket net%orks

S "en an error is detected b+ t"e receier4 it re!uests

retransission

S &OU

"#$C"#$C DE%&'(&DE%&'(& of Cyclic Codeof Cyclic Code


83/769

EE 6;1,;61 Fall /00@

"#$C "#$C DE%&'(& DE%&'(& of Cyclic Codeof Cyclic Code

%%)E*+E&C, of C,CC C(DE$)E*+E&C, of C,CC C(DE$


84/769

EE 6;1,;61 Fall /00@

% % )E*+E&C, of C,CC C(DE$)E*+E&C, of C,CC C(DE$

E.#/PE of a C,CC C(DEE.#/PE of a C,CC C(DE


85/769

EE 6;1,;61 Fall /00@

E.#/PE of a C,CC C(DE E.#/PE of a C,CC C(DE

P(,&(/#$P(,&(/#$ oeroer %2%2qq33


86/769

EE 6;1,;61 Fall /00@

P(,&(/#$P(,&(/#$ oeroer %2 %2 qq 3 3

E.#/PEE.#/PE


87/769

EE 6;1,;61 Fall /00@

E.#/PE E.#/PE

C#clic Code EncoderC#clic Code Encoder


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EE 6;1,;61 Fall /00@

C#clic Code Encoder C#clic Code Encoder

C#clic Code DecoderC#clic Code Decoder


89/769

EE 6;1,;61 Fall /00@

C#clic Code Decoder C#clic Code Decoder

Diider

iilar structure as ulti#lier for encoder

Cyclic Redundancy Checks (CRC)


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Cyclic Redundancy Checks (CRC)

Examle &f C


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EE 6;1,;61 Fall /00@

Examle &f C


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Capa$ilit# o% CRCCapa$ilit# o% CRC


93/769

EE 6;1,;61 Fall /00@

Capa$ilit# o% CRC p #

&n error E'=( is undetectable if it is diisible b+ /'(. *"e

follo%in$ can be detected. S &ll sin$le-bit errors if /'( "as ore t"an one nonero ter

S &ll double-bit errors if /'( "as a factor %it" t"ree ters

S &n+ odd nuber of errors4 if 9'( contain a factor J1

S &n+ burst %it" len$t" less or e!ual to n-k S & fraction of error burst of len$t" n-kJ1@ t"e fraction is 1-2W'-'-n-k-1((.

S & fraction of error burst of len$t" $reater t"an n-kJ1@ t"e fractionis 1-2W'-'n-k((.

9o%erful error detection@ ore co#utation co#leit+co#ared to nternet c"ecksu

9a$e 652

&C Code&C Code


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&C Code

Bose4 Oa+-C"aud"uri4 oc!uen$"e

S ulti#le error correctin$ abilit+ S Ease of encodin$ and decodin$

S 9a$e 65

ost #o%erful c+clic code

S Aor an+ #ositie inte$er and tR2W'-1(4 t"ere eists a t-error

correctin$ 'n4k( code %it" n2W-1 and n-kRt.

ndustr+ standards

S '5114 >;( BC code in *-*. Oec. .261 Xideo codec for

audioisual serice at kbitPsY a ideo codin$ a standard used for

ideo conferencin$ and ideo #"one.

S '>04 2( BC code in &* '&s+nc"ronous *ransfer ode(

&C 'er%ormance&C 'er%ormance


95/769

EE 6;1,;61 Fall /00@

C e o a ce

Reed(Solomon CodesReed(Solomon Codes


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EE 6;1,;61 Fall /00@

Reed Solomon Codes

&n i#ortant subclass of non-binar+ BC

9a$e 65>

ide ran$e of a##lications

S tora$e deices 'ta#e4 CD4 DZD(

S ireless or obile counication

S atellite counication

S Di$ital teleisionPDi$ital Zideo Broadcast'DZB(

S i$"-s#eed odes '&DL4 DL(

ExamplesExamples


97/769

EE 6;1,;61 Fall /00@

pp

10.2 #a$e 6;

10. #a$e 6>

10.> 9a$e 651

i$"t be > #oints in t"e final

)*+) "ariner *)*+) "ariner *


98/769

EE 6;1,;61 Fall /00@

ariner u%e( a ?*/331 )eed-/uller

c&(e t& tran%mit it% !rey ima!e% &f ar%.

camera rate:

100,000 bits/second

transmission speed:16,000 bits/second

)*+*- .o#agers / // )*+*- .o#agers / //


99/769

EE 6;1,;61 Fall /00@

# g# g

Q&ya!er% I R II u%e( a ?/;31/3K olay c&(et& %en( it% c&l&r ima!e% &f upiter an( Saturn.

Q&ya!er / traele( further t& =ranu%an( 2eptune. Becau%e &f the hi!hererr&r rate it %'itche( t& the m&rer&)u%t )eed-$olomon c&(e.

"odern Codes"odern Codes


100/769

EE 6;1,;61 Fall /00@

ore recentl+

%rbo codes %ere inented4

%"ic" are used in

/ cell #"ones4'future( satellites4

and in t"e Cassini-u+$ens s#ace

#robe 81;;7S:.

,t"er odern codes) Aountain4 Oa#tor4 L*4 online codes

Iet4 net class



101/769

EE 6;1,;61 Fall /00@

ggimer"ectness &f a !ien c&(e a% the (ifference )et'een the c&(e% require( E),2& t&attain a !ien '&r( err&r pr&)a)ility "N'#3 an( the minimum p&%%i)le E),2& require( t&attain the %ame N'3 a% implie( )y the %phere4pac-in! )&un( f&r c&(e% 'ith the %ame)l&c- %ie k an( c&(e rate r .

Radio S#stem 'ropagationRadio S#stem 'ropagation


102/769

EE 6;1,;61 Fall /00@

# p g# p g

Satellite CommunicationsSatellite Communications


103/769

EE 6;1,;61 Fall /00@

Lar$e counication area. &n+ t%o #laces %it"in t"e coera$e of radiotransission b+ satellite cancounicate %it" eac" ot"er.

eldo effected b+ land disaster' "i$" reliabilit+(

Circuit can be started u#onestablis"in$ eart" station '#ro#tcircuit startin$(

Can be receied at an+ #lacessiultaneousl+4 and realie broadcast4ulti-access counicationeconoicall+' feature of ulti-access(

Zer+ fleible circuit installent 4 can

dis#erse oer-centralied traffic atan+ tie.

,ne c"annel can be used in differentdirections or areas 'ulti-accessconnectin$(.

1'S 1'S


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[ust a tier4 2> satellite

Calculation #osition

Z05>


105/769


C+clic codes

C7ANTE< *: Cyclic an( c&n&luti&n

c&(e%Cyclic c&(e% are &f intere%t an( imp&rtance )ecau%e

• They p&%%e% rich al!e)raic %tructure that can )eutilie( in a ariety &f 'ay%.

• They hae extremely c&nci%e %pecificati&n%.

• They can )e efficiently implemente( u%in! %impleshift registers4

• any practically imp&rtant c&(e% are cyclic.

C&n&luti&n c&(e% all&' t& enc&(e %tream% &( (ata")it%#.

#$S%C#$S%C &E'%%T%&E'%%T% $&$& E*$M+LESE*$M+LESZ05>


106/769


C+clic codes

•Definiti&n A c&(e C i% cyclic if •"i# C i% a linear c&(eU

•"ii# any cyclic %hift &f a c&(e'&r( i% al%& a c&(e'&r(3 i.e. 'heneer a03 an 41 ∈ C 3 thenal%& an 41 a0 an $/ ∈ C .Example"i# C&(e C + H0003 1013 0113 110 i% cyclic.

"ii# 7ammin! c&(e !am"*3 /#: 'ith the !enerat&r matrix

i% equialent t& a cyclic c&(e.

"iii# The )inary linear c&(e H00003 10013 01103 1111 i% n&t a cyclic3 )ut it i%equialent t& a cyclic c&(e.

"i# I% 7ammin! c&(e !am"/3 *# 'ith the !enerat&r matrix

"a# cyclicV")# equialent t& a cyclic c&(eV

=

1111000

01101001010010

1100001

2110

1101

''RE,-EC o" CCL%C C&ESRE,-EC o" CCL%C C&ES

Z05>


107/769

EE576 Dr. Kousa Linear Block Codes 107C+clic codes

•C&mparin! 'ith linear c&(e%3 the cyclic c&(e% are quite %carce. F&r3 example there are 11 K11linear "@3*# linear )inary c&(e%3 )ut &nly t'& &f them are cyclic.

•Triial cyclic c&(e%. F&r any fiel( % an( any inte!er n W+ * there are al'ay% the f&ll&'in! cyclicc&(e% &f len!th n &er % :

• 2&4inf&rmati&n c&(e 4 c&(e c&n%i%tin! &f Ou%t &ne all4er& c&(e'&r(.

•


108/769


•The c&(e 'ith the !enerat&r matrix

•ha% c&(e'&r(%

• c 1 + 1011100 c / + 0101110 c * +0010111

• c 1 c / + 1110010 c 1 c * + 1001011 c / c * + 0111001

• c 1 c / c * + 1100101

•an( it i% cyclic )ecau%e the ri!ht %hift% hae the f&ll&'in! impact%

• c 1 → c /3 c / → c *3 c * → c 1 c *

• c 1 c / → c / c *3 c 1 c * → c 1 c / c *3 c / c * → c 1

• c 1 c / c * → c 1 c /

=11101000111010

0011101

+LM%$LS+LM%$LS o/ero/er G'(G'())Z05>


109/769

EE576 Dr. Kousa Linear Block Codes 10;C+clic codes

+LM%$LS+LM%$LS o/ero/er G'(G'( ))

• % q? (en&te% the %et &f all p&lyn&mial% &er % "6 #.• deg "f" ## + the lar!e%t m %uch that m ha% a n&n4er& c&efficient in f234

ultiplicati&n &f p&lyn&mial% If f" #3 !" # ∈ % q? 3 thendeg "f" # !" ## + deg "f" ## deg "!" ##.

Dii%i&n &f p&lyn&mial% F&r eery pair &f p&lyn&mial% a" #3 )" #≠ 0 in %

q? there

exi%t% a unique pair &f p&lyn&mial% q" #3 r" # in % q? %uch that

a" # + q" #)" # r" #3 (e! "r" ## X deg ")" ##.

Example Dii(e * 1 )y / 1 in % /? .

Definiti&n 8et f" # )e a fixe( p&lyn&mial in % q? . T'& p&lyn&mial% !" #3 h" # are %ai(t& )e c&n!ruent m&(ul& f" #3 n&tati&n

!" # ≡ h" # "m&( f" ##3if !" # 4 h" # i% (ii%i)le )y f" #.

R%GR%G o"o" +LM%$LS+LM%$LSZ05>


110/769


•The %et &f p&lyn&mial% in % q? &f (e!ree le%% than deg "f" ##3 'ith a((iti&n an( multiplicati&n m&(ul& f" # f&rm% a rin!denoted '01x23"(x).

•Example Calculate " 1#/ in % /? , " / 1#. It h&l(%

•" 1#/ + x/ /x 1 ≡ / 1 ≡ "m&( x/ x 1#.

•7&' many element% ha% % q? , f" #V•


111/769


'%EL&'%EL& R R nn44 R R nn 55 F F 0011 x x 22 3 3 (( x x nn 66 7)7)

•C&mputati&n m&(ul& n $ 1

•Since n ≡ 1 "m&( n 41# 'e can c&mpute f" # m&( n 41 a% f&ll&':•In f" # replace n )y 13 n 1 )y 3 n / )y /3 n * )y *3

•I(entificati&n &f '&r(% 'ith p&lyn&mial%

•a0 a1 an 41 ↔ a0 a1 a/ / an 41 n 41

•ultiplicati&n )y in ) n c&rre%p&n(% t& a %in!le cyclic %hift

• "a0 a1 an 41 n 41# + an 41 a0 a1 / an 4/ n 41

$l!e8raic$l!e8raic characterizationcharacterization o"o" cycliccyclic codescodesZ05>


112/769


!! yy

• The&rem A c&(e C i% cyclic if C %ati%fie% t'& c&n(iti&n%

• "i# a" #3 )" # ∈ C ⇒ a" # )" # ∈ C

• "ii# a" # ∈ C 3 r" # ∈ ) n ⇒ r" #a" # ∈ C

• Nr&&f

• "1# 8et C )e a cyclic c&(e. C i% linear ⇒ "i# h&l(%.• "ii# 8et a" # ∈ C 3 r" # + r 0 r 1 r n 41 n 41

• r" #a" # + r 0a" # r 1 a" # r n 41 n 41a" #• i% in C )y "i# )ecau%e %umman(% are cyclic %hift% &f a" #.

• "/# 8et "i# an( "ii# h&l(

• • Ta-in! r" # t& )e a %calar the c&n(iti&n% imply linearity &f C .

• • Ta-in! r" # + the c&n(iti&n% imply cyclicity &f C .

CSTR-CT%CSTR-CT% o"o" CCL%CCCL%C C&ESC&ESZ05>


113/769


•2&tati&n If f" # ∈ ) n3 then∀ 〈f" #〉 + Hr" #f" # Y r" # ∈ ) n

•"multiplicati&n i% m&(ul& n

41#.

•The&rem F&r any f" # ∈ ) n3 the %et 〈f"x#〉 i% a cyclic c&(e "!enerate( )y f#.

•Nr&&f 9e chec- c&n(iti&n% "i# an( "ii# &f the prei&u% the&rem.

•"i# If a" #f" # ∈ 〈f" #〉 an( )" #f" # ∈ 〈f" #〉3 then

• a" #f" # )" #f" # + "a" # )" ## f" # ∈ 〈f" #〉

•"ii# If a" #f" # ∈ 〈f" #〉3 r" # ∈ ) n3 then• r" # "a" #f" ## + "r" #a" ## f" # ∈ 〈f" #〉.

Example C + 〈1 / 〉3 n + *3 6 + /.

9e hae t& c&mpute r" #"1 /# f&r all r" # ∈ ) *.) * + H03 13 3 1 3

/3 1 /3 /3 1 /.


114/769

EE576 Dr. Kousa Linear Block Codes 11>C+clic codes

CharacterizationCharacterization theoremtheorem "or "or cycliccyclic codescodes

•9e %h&' that all cyclic c&(e% C hae the f&rm C + 〈f" #〉 f&r %&me f" # ∈ ) n.

•The&rem 8et C )e a n&n4er& cyclic c&(e in ) n. Then• there exi%t% unique m&nic p&lyn&mial !" # &f the %malle%t (e!ree %uch that• C + 〈!" #〉• !" # i% a fact&r &f n 41.

Nr&&f

"i# Supp&%e !" # an( h" # are t'& m&nic p&lyn&mial% in C &f the %malle%t (e!ree.Then the p&lyn&mial !" # 4 h" # ∈ C an( it ha% a %maller (e!ree an( a multiplicati&n)y a %calar ma-e% &ut &f it a m&nic p&lyn&mial. If !" # ≠ h" # 'e !et a c&ntra(icti&n.

"ii# Supp&%e a"x# ∈ C .Then

a" # + q" #!" # r" # "deg r" # X deg !" ##

an(r" # + a" # 4 q" #!" # ∈ C .

By minimalityr" # + 0

an( theref&re a" # ∈ 〈!" #〉.

CharacterizationCharacterization theoremtheorem "or"or cycliccyclic codescodesZ05>


115/769


CharacterizationCharacterization theoremtheorem "or "or cycliccyclic codescodes

•"iii# Clearly3

• n

$1 + q" #!" # r" # 'ith deg r" # X deg !" #

•an( theref&re r" # ≡ 4q" #!" # "m&( xn 41# an(•r" # ∈ C ⇒ r" # + 0 ⇒ !" # i% a fact&r &f n 41.

>E2EE2E


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•The la%t claim &f the prei&u% the&rem !ie% a recipe t& !et all cyclic c&(e% &f !ien len!th n.

•In(ee(3 all 'e nee( t& (& i% t& fin( all fact&r% &f

• n

41.•Nr&)lem: Fin( all )inary cyclic c&(e% &f len!th *.

•S&luti&n: Since

• * $ 1 + " 1#" / 1#

• )&th fact&r% are irre(uci)le in % "/#

•'e hae the f&ll&'in! !enerat&r p&lyn&mial% an( c&(e%.

• >enerat&r p&lyn&mial% C&(e in ) * C&(e in 7 "*3/#

• 1 ) * 7 "*3/#

• 1 H03 1 3 /3 1 / H0003 1103 0113 101

• / 1 H03 1 / H0003 111

• *

$ 1 " + 0# H0 H000

&esi!n&esi!n o" o" !enerator!enerator matricesmatrices "or"or cycliccyclic codescodesTh S C i li ( f ( ( f l th ith th t l i l

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• Theorem Supp&%e C i% a cyclic c&(e &f c&(e'&r(% &f len!th n 'ith the !enerat&r p&lyn&mial

• !" # + !0 !1 !r r .

• Then dim "C # + n 4 r an( a !enerat&r matrix 1 f&r C i%

Nr&&f

"i# All r&'% &f 1 are linearly in(epen(ent."ii# The n 4 r r&'% &f repre%ent c&(e'&r(%

!" #3 !" #3 /!" #33 n 4r 41!" # "]#

"iii# It remain% t& %h&' that eery c&(e'&r( in C can )e expre%%e( a% a linear c&m)inati&n &fect&r% fr&m "]#.

In(e(3 if a"x# ∈ C 3 thena" # + q" #!" #.

Since deg a" # X n 'e hae deg q" # X n 4 r.7ence

q" #!" # + "q0 q1 qn 4r 41 n 4r 41#!" #

+ q0!" # q1 !" # qn 4r 41xn 4r 41!" #.

=

r

r

r

r

! !

! ! ! !

! ! ! !

! ! ! !

,

...0...00...00

......

0...0...00

0...00...0

0...000...

0

210

210

210

1

E*$M+LEE*$M+LE•The ta%- i% t& (etermine all ternary c&(e% &f len!th ; an( !enerat&r% f&r them

Z05>


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•The ta%- i% t& (etermine all ternary c&(e% &f len!th ; an( !enerat&r% f&r them.

•Fact&riati&n &f ; 4 1 &er % "*# ha% the f&rm

• ; 4 1 + " 4 1#" * / 1# + " 4 1#" 1#" / 1#

•Theref&re there are /* + K (ii%&r% &f x; 4 1 an( each !enerate% a cyclic c&(e.

• >enerat&r p&lyn&mial >enerat&r matrix

• 1 ;

•

• 1

• / 1

• " 4 1#" 1# + /

4 1

• " 4 1#" / 1# + * 4 / 4 1 ? 41 1 41 1

• "x 1#"x/ 1# ? 1 1 1 1

• x; 4 1 + 0 ? 0 0 0 0

−

−

−

−−

1010

0101

1010

0101

1100

0110

0011

1100

0110

0011

CheckCheck olynomialsolynomials andand arity checkarity check matrices "or cyclic codesmatrices "or cyclic codes

Z05>


119/769


•8et C )e a cyclic ?n8k 4c&(e 'ith the !enerat&r p&lyn&mial !" # "&f (e!ree n 4 k #. By the la%t the&rem!" # i% a fact&r &f n 4 1. 7ence

• n 4 1 + !" #h" #

•f&r %&me h" # &f (e!ree k "'here h" # i% calle( the chec- p&lyn&mial &f C #.

•The&rem 8et C )e a cyclic c&(e in ) n 'ith a !enerat&r p&lyn&mial !" # an( a chec- p&lyn&mial h" #.Then an c" # ∈ ) n i% a c&(e'&r( &f C if c" #h" # ≡ 0 4 thi% an( next c&n!ruence% are m&(ul& n 4 1.

Nr&&f 2&te3 that !" #h" # + n 4 1 ≡ 0"i# c" # ∈ C ⇒ c" # + a" #!" # f&r %&me a" # ∈ ) n

⇒ c" #h" # + a" # !" #h" # ≡ 0. ≡ 0

"ii# c" #h" # ≡ 0c" # + q" #!" # r" #3 deg r" # X n $ k + deg !" #

c" #h" # ≡ 0 ⇒ r" #h" # ≡ 0 "m&( n 4 1#

Since deg "r" #h" ## X n $ k k + n3 'e hae r" #h" # + 0 in % ? an( theref&re

r" # + 0 ⇒ c" # + q" #!" # ∈ C .

+LM%$L+LM%$L RE+RESET$T% o" &-$L C&ESRE+RESET$T% o" &-$L C&ES

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120/769


•Since dim "〈h"x#〉# + n 4 k + dim "C ⊥# 'e mi!ht ea%ily )e f&&le( t& thin- that the chec-p&lyn&mial h" # &f the c&(e C !enerate% the (ual c&(e C ⊥.

•


121/769


•Nr&&f A p&lyn&mial c" # + c0 c1 cn 41 n $1 repre%ent% a c&(e fr&m C if c" #h" # + 0.F&r c" #h" # t& )e 0 the c&efficient% at -33 n 41 mu%t )e er&3 i.e.

•Theref&re3 any c&(e'&r( c0 c1 cn 41 ∈ C i% &rth&!&nal t& the '&r( h- h- 41h0000 an( t&

it% cyclic %hift%.•


122/769


! ! y y p p yme%%a!e p&lyn&mial an( the !eneratin! p&lyn&mial f&r the cyclic c&(e.

•8et C )e an "n3k #4c&(e &er an fiel( % 'ith the !enerat&r p&lyn&mial•!" # + !0 !1 !r $1 r 41 &f (e!ree r + n 4 k .

•If a me%%a!e ect&r m i% repre%ente( )y a p&lyn&mial m" # &f (e!ree k an( m i% enc&(e()y

• m ⇒ c + m13

•then the f&ll&'in! relati&n )et'een m" # an( c" # h&l(%

• c" # + m" #!" #.

•Such an enc&(in! can )e realie( )y the %hift re!i%ter %h&'n in Fi!ure )el&'3 'here inputi% the k 4)it me%%a!e t& )e enc&(e( f&ll&'e( )y n 4 k 0 an( the &utput 'ill )e the enc&(e(

me%%a!e.

•Shift4re!i%ter enc&(in!% &f cyclic c&(e%. Small circle% repre%ent multiplicati&n )y thec&rre%p&n(in! c&n%tant3 ⊕ n&(e% repre%ent m&(ular a((iti&n3 %quare% are (elay element%

EEC&%G o" CCL%C C&ESC&%G o" CCL%C C&ES %%%%Z05>


123/769


• An&ther meth&( f&r enc&(in! &f cyclic c&(e% i% )a%e( &n the f&ll&'in! "%& calle(%y%tematic# repre%entati&n &f the !enerat&r an( parity4chec- matrice% f&r cyclic c&(e%.

•The&rem 8et C )e an "n3k #4c&(e 'ith !enerat&r p&lyn&mial !" # an( r + n 4 k . F&r i + 03133k 4 13 let /3i )e the len!th n ect&r 'h&%e p&lyn&mial i% /3i" # + rI 4 rI m&( !" #. Thenthe k ] n matrix / 'ith r&' ect&r% /3I i% a !enerat&r matrix f&r C .

•&re&er3 if ! /3 i% the len!th n ect&r c&rre%p&n(in! t& p&lyn&mial ! /3" # + O m&( !" #3then the r 9 n matrix ! / 'ith r&' ect&r% ! /3 i% a parity chec- matrix f&r C . If the me%%a!eect&r m i% enc&(e( )y

• m ⇒ c + m/3

•then the relati&n )et'een c&rre%p&n(in! p&lyn&mial% i%• c" # + r m" # 4 ? r m" # m&( !" #.

•[n thi% )a%i% &ne can c&n%truct the f&ll&'in! %hift4re!i%ter enc&(er f&r the ca%e &f a%y%tematic repre%entati&n &f the !enerat&r f&r a cyclic c&(e:•Shift4re!i%ter enc&(er f&r %y%tematic repre%entati&n &f cyclic c&(e%. S'itch A i% cl&%e( f&rfir%t k tic-% an( cl&%e( f&r la%t r tic-%U %'itch B i% (&'n f&r fir%t k tic-% an( up f&r la%t r tic-%.

9ammin!9ammin! codescodes asas cycliccyclic codescodes

Z05>


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EE576 Dr. Kousa Linear Block Codes 12>C+clic codes

•Definiti&n "A!ain_# 8et r )e a p&%itie inte!er an( let ! )e an r 9 "/r 41#matrix 'h&%e c&lumn% are (i%tinct n&n4er& ect&r% &f 7 "r 3 #. Then the

c&(e hain! ! a% it% parity4chec- matrix i% calle( )inary 9ammin!code (en&te( )y !am "r 3 #.

•It can )e %h&'n that )inary 7ammin! c&(e% are equialent t& cyclicc&(e%.

The&rem The )inary 7ammin! c&(e !am "r 3 # i% equialent t& a cyclic c&(e.

Definiti&n If p" # i% an irre(uci)le p&lyn&mial &f (e!ree r %uch that i% a primitie element&f the fiel( % ? , p" #3 then p" # i% calle( a primitie p&lyn&mial.

The&rem If p" # i% a primitie p&lyn&mial &er % " # &f (e!ree r 3 then the cyclic c&(e〈p" #〉 i% the c&(e !am "r 3 #.

Z05>


125/769


9ammin!9ammin! codescodes asas cycliccyclic codescodes

•Examle N&lyn&mial * 1 i% irre(uci)le &er % " # an( i%primitie element &f the fiel( % /? , " * 1#.

•% /? , " * 1# +

•H03 3 /3 * + 13 ; + / 3 6 + / 13 + / 1•The parity4chec- matrix f&r a cyclic er%i&n &f !am "3 #

=

1110100

0111010

1101001

#

+R'+R' o" o" T9EREMT9EREMTh )i 7 i ( ! " # i i l t t li (

Z05>


126/769


•The )inary 7ammin! c&(e !am "r 3 # i% equialent t& a cyclic c&(e.•It i% -n&'n fr&m al!e)ra that if p" # i% an irre(uci)le p&lyn&mial &f (e!ree r 3 then the rin! % /? , p" # i% a fiel( &f&r(er /r .•In a((iti&n3 eery finite fiel( ha% a primitie element. Theref&re3 there exi%t% an element α &f % /? , p" # %uch that

• % /? , p" # + H03 13 α3 α/33 α/r $/.

•8et u% i(entify an element a0 a1 ar 41 r 41 &f % /? , p" # 'ith the c&lumn ect&r • "a03 a133 ar 41#T

•an( c&n%i(er the )inary r ] "/r 41# matrix• ! + ? 1 α α/ α/`r $/ .

•8et n&' C )e the )inary linear c&(e hain! ! a% a parity chec- matrix.•Since the c&lumn% &f ! are all (i%tinct n&n4er& ect&r% &f 7 "r 3 #3 C + !am "r 3 #.•Nuttin! n + /r 41 'e !et• C + Hf 0 f 1 f n 41 ∈ 7 "n3 # Y f 0 f 1 α f n 41 αn $1 + 0 "/#• + Hf" # ∈ ) n Y f"α# + 0 in % /? , p" # "*#

•If f" #∈

C an( r" #∈

) n3 then r" #f" #∈

C )ecau%e• r"α#f"α# + r"α# • 0 + 0

•an( theref&re3 )y &ne &f the prei&u% the&rem%3 thi% er%i&n &f !am "r 3 # i% cyclic.

#C9#C9 codescodes andand Reed6SolomonReed6Solomon codescodesZ05>


127/769


•T& the m&%t imp&rtant cyclic c&(e% f&r applicati&n% )el&n! BC7 c&(e% an(


128/769


• 'e( )*+e + ! /)++ +) e)e *+e !+e ) !eve !+e! )* + 4 !(

5+!.• Con/olution codes4 ) con/olution code CC>

:1418 221 +++= - - -,

++=

-

- -,

1

1

0

0

12

E2C[DI2> &f FI2ITE N[8\2[IA8SZ05>


129/769


• An "n3-# c&n&luti&n c&(e 'ith a - x n !enerat&r matrix > can )e u%( t& enc&(e

a• -4tuple &f plain4p&lyn&mial% "p&lyn&mial input inf&rmati&n#

• =2 0 238 12.38>8 k-1233

• t& !et an n4tuple &f crypt&4p&lyn&mial%

• C+"C0"x#3 C1"x#33Cn41"x##

• A% f&ll&'%

• C+ I . >

EZAN8ES


130/769


EZAN8ES

• EZAN8E 1

• "x* x 1#.>1 + "x* x 1# . "x/ 13 x/ x 1

• + "x6 x/ x 13 x6 x; 1#

• EZAN8E /

+++=++

-

- - - -, - - -

1

1

0

0

1(.14'(.14'

2

2

2

E2C[DI2> &f I2FI2ITE I2N=T ST


131/769


• The B 7) %(x)

• and

• C7(x) 5 C7@ B C77x B A 5 (x> B x B 7) %(x).

• The "irst multilication can 8e done 8y the "irst shi"t re!ister "rom the next

• "i!ure second multilication can 8e er"ormed 8y the second shi"t re!ister

• on the next slide and it holds

• C@i 5 / i - / i-2 , C )i 7 / i - / i() - / i(2 8• 9hat is the output streams C 0 and C ) are o$tained $# con6ol6ing the input

• stream 5ith pol#nomials o% 1 ):

E2C[DI2>

Z05>


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The fir%t %hift re!i%ter The fir%t %hift re!i%ter ⊕⊕

1 x x1 x x//

inputinput

&utput&utput

'ill multiply the input %tream )y x'ill multiply the input %tream )y x//1 an( the1 an( the %ec&n( %hift re!i%ter %ec&n( %hift re!i%ter

⊕⊕

1 x x1 x x//

inputinput

&utput&utput

'ill multiply the input %tream )y'ill multiply the input %tream )y ??14??14

E2C[DI2> an( DEC[DI2>

Z05>


133/769


⊕⊕

1 x x1 x x// II

CC00003C3C01013C3C0/0/

⊕⊕ CC10103C3C11113C3C1/1/

[utput %tream%[utput %tream%

The f&ll&'in! %hift4re!i%ter 'ill theref&re )e an enc&(er f&r theThe f&ll&'in! %hift4re!i%ter 'ill theref&re )e an enc&(er f&r thec&(e CCc&(e CC11

F&r enc&(in! &f c&n&luti&n c&(e% %& calle(F&r enc&(in! &f c&n&luti&n c&(e% %& calle(

Qiter)i al!&rithmQiter)i al!&rithm

I% u%e(.I% u%e(.


134/769


Cyclic Linear Codes


135/769


E

?1 N&lyn&mial% an( '&r(%

?/ Intr&(ucti&n t& cyclic c&(e%

?* >eneratin! an( parity chec- matrice% f&r cyclic c&(e%

?; Fin(in! cyclic c&(e% ?6 Dual cyclic c&(e%

Cyclic Linear Codes


136/769


y• ?1 N&lyn&mial% an( '&r(%

$ 1. N&lyn&mial &f (e!ree n &er

$ /. E! ;.1.1

= + + + + +

∈ =

2 3

0 1 2 3

0

[ ] { .... }

,...., , deg( ( ))

n

n

n

K x a a x a x a x a x

a a K f x n

= + + + = + + = + +

+ = + +

+ = + += + + + + + + + + +

+ + = +

3 4 2 3 2 4

2 4

2 3

2 3 2 3 3 2 3

4 2 3 7

Let ( ) 1 ( ) ( ) 1 then

( ) ( ) ( ) 1

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( )

f x x x x g x x x x h x x x

a f x g x x x

b f x h x x x x

c f x g x x x x x x x x x x x x

x x x x x x

Cyclic Linear Codes


137/769


y

$ *. ?Al!&rithm ;.1.KDii%i&n al!&rithm

$ ;. E!. ;.1.

≠

= +

= <

Let ( ) and ( ) be in [ ] with ( ) 0. Then there exist

ni!e "#$%n#&ia$ ( ) and ( ) in [ ] s'h that

( ) ( ) ( ) ( ),

with ( ) 0 #r deg( ( )) deg( ( ))

f x h x K x h x

q x r x K x

f x q x h x r x

r x r x h x

= + + + = + + +

= + = + +

= + + + +< =

2 2 4

3 4 2 3

3 4 2 3

( ) , ( ) 1

( ) , ( )

( ) ( )( ) ( )

deg( ( )) deg( ( )) 4

f x x x x x h x x x x

q x x x r x x x x

f X h x x x x x x

r x h x

Cyclic Linear Codes


138/769


$ 6. C&(e repre%ente( )y a %et &f p&lyn&mial%• A c&(e C &f len!th n can )e repre%ente( a% a %et &f p&lyn&mial% &er &f

(e!ree at m&%t n41

$

$

$ . E.! ;.1.1/

−−= + + + +

2 10 1 2 1( ) .... #*er +

nnf x a a x a x a x

−= n0 1 2 1... # $ength n in + nc a a a a

C&(e'&r( c N&lyn&mial c"x#

0000101001011111

1x/

xx*

1xx/x*

Cyclic Linear Codes


139/769


$ @. f"x# an( p"x# are e6uialent modulo h"x#

$ K.E! ;.1.16

$ . E! ;.1.1

= =≡

( ) d ( ) ( ) ( ) d ( )

. ( ) ( )(d ( ))

f x h x r x p x h x

ie f x p x h x

= + + + = + = += = + =

4 - 11

( ) 1 , ( ) 1 , ( ) 1 ( )d ( ) ( ) 1 ( )d ( )

/(x) and "(x) are e!i*a$ent d h(x)

f x x x x h x x p x x f x h x r x x p x h x

= + + + + = + + = += + = +

2 - 11 2 2

4 3( ) 1 , ( ) 1 , ( )

( )d ( ) , ( )d ( ) 1

/(x) and "(x) are T e!i*a$ent d h(x)

f x x x x x h x x x p x x x f x h x x x p x h x x

Cyclic Linear Codes


140/769

EE576 Dr. Kousa Linear Block Codes 1>0

$ 10. 8emma ;.1.1@

$ 11. E!. ;.1.1K

≡+ ≡ +

≡

( ) ( )(d ( )), then

( ) ( ) ( ) ( )(d ( ))

and

( ) ( ) ( ) ( )(d ( ))

f x g x h x

f x p x g x p x h x

f x p x g x p x h x

= + + = + + = + = +≡

+ ++ + + + = = + + + +

+ + + = + =

7 2

7 2 2

7 3

( ) 1 , ( ) 1 , ( ) 1 , ( ) 1

s# ( ) ( )(d ( )), then

( ) ( ) and ( ) ( ) 5((1 ) (1 ))d ( ) ((1 ) (1 ))d ( )

( ) ( ) and ( ) ( ) 5

((1 )(1 ))d ( ) 1 ((1

f x x x g x x x h x x p x x

f x g x h x

f x p x g x p x x x x h x x x x x h x

f x p x g x p x

x x x h x x + + +2 )(1 ))d ( ) x x x h x

Cyclic Linear Codes


141/769


• ?/Intr&(ucti&n t& cyclic c&(e%

6 1. '%'$i' shit (*)

• Q: 0101103 : 001011

6 2.'%'$i' '#de

• A c&(e C i% cyclic c&(e"&r linear cyclic c&(e# if "1#the cyclic %hift &f eachc&(e'&r( i% al%& a c&(e'&r( an( "/# C i% a linear c&(e

• C1+"0003 1103 1013 011 i% a cyclic c&(e

• C/+H0003 1003 0113 111 i% 2[T a cyclic c&(e $ Q+1003 +010 i% n&t in C/

π ( )v

10110 111000 0000 1011π ( )v 01011 011100 0000 1101

π ( )v

Cyclic Linear Codes


142/769


$ *. Cyclic %hifti% a linear tran%f&rmati&n

•

• S+H3 "#3 /"#3 3 n41"#3 an( C+XSW3

then i% a !enerat&r &f the linear cyclic c&(e C

π π π

π π

π

+ = += ∈ =

∈

Le&&a 4.2.3 ( ) ( ) ( ),

and ( ) ( ), {0,1}

Ths t# sh#w a $inear '#de 8 is '%'$i'

it is en#gh t# sh#w that ( ) 8

#r ea'h w#rd in a basis #r 8

v w v w

av a v a K

v

v

Cyclic Linear Codes


143/769


$ ;. Cyclic C&(e in term% &f p&lyn&mialπ => =>( ), ( ) ( )v v v x xv x

= + +

3

7

2 4

2

9g 4.2.11 */1101000, n/7, *(x)/1:x:x

w#rd "#$%ni&ia$(d 1:x )

;;;;;;;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

0110100 ( )

0011010 x (

xv x x x x

v = + +

= + +

= + + ≡ + + += + + ≡ + + +

= + + ≡ +

2 3 4

3 3 4

4 4 7 4 7

7

7 -

)

0001101 x ( )

1000110 x ( ) 1 d(1 )

0100011 x ( ) d(1 )

1010001 x ( ) 1

x x x x

v x x x x

v x x x x x x x

v x x x x x x x x

v x x x x + +2 7d(1 ) x x x

Cyclic Linear Codes


144/769

EE576 Dr. Kousa Linear Block Codes 1>>

$ 6. 8emma ;./.1/8et C )e a cyclic c&(e let in C. Then f&r any p&lyn&mial a"x#3c"x#+a"x#"x#m&("1xn# i% a c&(e'&r( in C

$ . The&rem ;./.1*C: a cyclic c&(e &f len!th n3!"x#: the !enerat&r p&lyn&mial3 'hich i% the unique n&ner&

p&lyn&mial &f minimum (e!ree in C.

(e!ree"!"x## : n4-3

• 1. C has dimension k• 2. g(x), xg(x), x2g(x), …., xk-1g(x) are a basis for C• 3. f c(x) in C, c(x)!a(x)g(x) for some "olynomial a(x)

#i$h degree(a(x))%k

Cyclic Linear Codes


145/769


$ @. E! ;./.1

the %malle%t linear cyclic c&(e C &f len!th c&ntainin! !"x#+1x * X4W 100100 i%

H0000003 1001003 0100103 0010013 11011031011013 0110113 111111

$ K. The&rem ;./.1@

!"x# i% the !enerat&r p&lyn&mial f&r a linear cyclic c&(e &f len!th n if &nly if !"x# (ii(e% 1xn

"%& 1xn +!"x#h"x##.

$ . C&r&llary ;./.1K

The !enerat&r p&lyn&mial !"x# f&r the %malle%t cyclic c&(e &f len!th n c&ntainin!the '&r( "p&lyn&mial "x## i% !"x#+!c(""x#3 1xn#

$ 10. E! ;./.1

n+K3 +11011000 %& "x#+1xx*x;

!"x#+!c("1xx*x; 3 1xK#+1x/

Thu% !"x#+1x/ i% the %malle%t cyclic linear c&(e c&ntainin!

"x#3 'hich ha% (imen%i&n &f .

Cyclic Linear Codes


146/769


• ?*. >eneratin! an( parity chec- matrice% f&r cyclic c&(e $ 1. Effectie t& fin( a !eneratin! matrix

• The %imple%t !enerat&r matrice% "The&rem ;./.1*#

=


147/769


/. E! ;.*./• C: the linear cyclic c&(e% &f len!th n+@ 'ith !enerat&r p&lyn&mial

!"x#+1xx*3 an( (e!"!"x##+*3 +W - + ;

= + +

= + +

= + +

= + +

3

2 4

2 2 3

3 3 4

( ) 1

( )

( )

( )

g x x x

xg x x x x

x g x x x x

x g x x x x

1101000

0110100=/

0011010

0001101

==>

Cyclic Linear Codes


148/769


*. Efficient enc&(in! f&r cyclic c&(e%

(a/c'codelinear$enerala

of t"at%it"co#aredefficienttieore

'('('( )al$orit"Encodin$

((444'essa$esourcen$re#resenti'

(' #ol+noialessa$e

k(.-nde$ree"as$'( #ol+noial$eneratort"e'so

kdiensionandnlen$t"of codec+clica beCLet

110

1110

=

=

+++=−

−−

! ac

aaa

a aa a

k

k k

Cyclic Linear Codes


149/769

EE576 Dr. Kousa Linear Block Codes 1>;

$ ;. Narity chec- matrix

• 7 : '7+0 if &nly if ' i% a c&(e'&r(

• Sym(r&me p&lyn&mial %"x#

$ c"x#: a c&(e'&r(3 e"x#:err&r p&lyn&mial3 an( '"x#+c"x#e"x#

$ %"x# + '"x# m&( !"x# + e"x# m&( !"x#3 )ecau%e c"x#+a"x#!"x#

$ 7: i4th r&' r i i% the '&r( &f len!th n4-

+W r i"x#+xi m&( !"x#

$ '7 + "ce#7 +W c"x# m&( !"x# e"x# m&( !"x# + %"x#

Cyclic Linear Codes


150/769


$ 6. E! ;.*.@

• n+@3 !"x#+1xx*3 n4- + *

= == =

= =

= = +

= = +

= = + += = +

0

1

2 22

33

4 24

2

2

( ) 1d ( ) 1

( ) d ( )

( ) d ( )

( ) d ( ) 1

( ) d ( )

( ) d ( ) 1

( ) d ( ) 1

r x g x

r x x g x x

r x x g x x

r x x g x x

r x x g x x x

r x x g x x x

r x x g x x

< −− >< −− >< −− >< −− >< −− >

< −− >< −− >

100 010

001

110

011

111

101

==>

=

100

010

001

110

011

111101

H

Cyclic Linear Codes


151/769


• ?;. Fin(in! cyclic c&(e% $ 1. T& c&n%truct a linear cyclic c&(e &f len!th n

• Fin( a fact&r !"x# &f 1xn3 (e!"!"x## + n4-

• Irre(uci)le p&lyn&mial%

$ f"x# in ?x3 (e!"f"x## W+ 1 $ There are n& a"x#3 )"x# %uch that f"x#+a"x#)"x#3

(e!"a"x##W+13 (e!")"x##W+1

• F&r n X+ *13 the fact&riati&n &f 1xn

"%ee Appen(ix B#• Impr&per cyclic c&(e%: n an( H0

Cyclic Linear Codes


152/769


$ /. The&rem ;.;.*

$ *. C&r& ;.;.;

= +r n 2i n/2 then 1:x (1 ) r ss x

n.len$t"of codesc+clic

linear #ro#er2(1'2 aret"ere*"en

s. #ol+noialeirreduciblof #roductt"e

be1letandoddiss%"ere42Let

r −+

+=

sr sn

Cyclic Linear Codes


153/769


$ ;. I(emp&tent p&lyn&mial% I"x#

• I"x# + I"x#/ m&( "1xn# f&r &(( n

• Fin( a ^)a%icb %et &f I"x#

Ci+ H %+/ O i "m&( n# Y O+03 13 3 r

'here 1 + /r m&( n

== ∈∑

i0( ) ( ), a {0,1}

k

i iiI x a c x

∑=∈ iC &

&

i c (' %"ere

Cyclic Linear Codes


154/769


$ 6. E! ;.;.1/

$ . The&rem ;.;.1*Every cyclic code contains a unique idempotent

polynomial which generates the code.(?)

= = =

= = = + +

= = + +

==> + + ∈

≠

0

0 0

1 2 4

1 2 4 1

3 .

3 . 7 2

0 0 1 1 3 3 i

>#r n/7,

8 {0}, s# ' ( ) 1

8 {1, 2, 4} / 8 8 , s# ' ( )

8 {3, ., } / 8 / 8 , s# ' ( )

(x)/a ' ( ) a ' ( ) a ' ( ), a {0,1},

(x) 0

x x

x x x x

x x x x

x x x

Cyclic Linear Codes


155/769


$ @. E!. ;.;.1; fin( all cyclic c&(e% &f len!th

= = =

==> = = + + + + + = +

+ +

0 1 3

2 4 7 3 0 1 3

0 0 1 1 3 3

8 {0}, 8 {1,2,4,,7,}, 8 {3,}

' ( ) 1, ' ( ) , ' ( )

// (x)/a ' ( ) a ' ( ) a ' ( )

x x x x x x x x x x x

x x x

The !enerat&rp&lyn&mial

!"x#+!c("I"x#3 1x#

I(emp&tent p&lyn&mialI"x#

11xx*x;xx@

1x*

1xx/

:

1xx/x;x6x@xK

x*x

1xx/x;x6x@xK

:

Cyclic Linear Codes


156/769


Cyclic Linear Codes

• ?6.Dual cyclic c&(e% $ 1. The (ual c&(e &f a cyclic c&(e i% al%& cyclic

$ /. 8emma ;.6.1

a > a(x), b > b(x) and b’ > b’(x)xnb(x!") mod "#xn

then

a"x#)"x# m&( 1xn + 0 iff -"a#． )5+0

f&r -+0313n41

$ *. The&rem ;.6./

C: a linear c&(e3 len!th n3 (imen%i&n - 'ith !enerat&r !"x#

If 1xn + !"x#h"x# then

C?: a linear c&(e 3 (imen%i&n n4- 'ith !enerat&r x-h"x41#

Cyclic Linear Codes


157/769


Cyclic Linear Codes

$ ;. E!. ;.6.*!"x#+1xx*3 n+@3 -+@4*+;

h"x#+1xx/x;

h"x#!enerat&r f&r C ? i%

! ? "x#+x;h"x41#+x;"1x41x4/x4; #+1x/x*x;

$ 6. E!. ;.6.;!"x#+1xx/3 n+3 -+4/+;

h"x#+1xx*x;

h"x#!enerat&r f&r C ? i% ! ? "x#+x;h"x41#+1xx*x;


158/769


&(ulati&n3 Dem&(ulati&n an(C&(in! C&ur%e

Neri&( * 4 /006

S&r&ur Falahati

8ecture K

8a%t time 'e tal-e( a)&ut:


159/769

EE576 Dr. Kousa Linear Block Codes 15;Lecture

8#herent and n#n;'#herent dete'ti#ns

9*a$ating the a*erage "r#babi$it% # s%&b#$ err#r #r di@erentband"ass d$ati#n s'he&es

8#&"aring di@erent d$ati#n s'he&es based #n their err#r

"er#r&an'es.

T&(ay3 'e are !&in! t& tal- a)&ut:

8/17/2019 54676499-DC-E

54676499-DC-Error-Correcting-Codes.ppt

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Transcript of 54676499-DC-Error-Correcting-Codes.ppt