١ ١ Chapter Chapter 4 Cyl Blo Cyl Blo Codes Codes Ammar Abu-Hudrouss Islam Uversty Gaza Spr 9 Slde ٢ Chael Cod Theory Cyl Blo Codes A yl ode s haraterzed as a lear blo ode B( d wth the addtoal property that for eah ode word All the yl shft versos of b s also vald ode words (.e. All are vald odewords ad of ourse all the lear ombatos betwee them. G G G 4 G G
Cyl Blo Codes Example f we defe a eerator odeword as G = the all the possble yl odewords are. All zeros 4. C (5. G 5. C (6. G ( 6. D =G+C ( 4. G ( 5. G ( 6. G (4 7. G (5 8. G (6 9. C =G + G (. C (. C (. C (. C (4 Chael Cod Theory Slde ٣ Eod by ovoluto Ths property a be formulated by us polyomal. Assume we a represet the ode word G as follow ( For our Example G = or ( If we tae the eerator polyomals ad ts frst - left shft we fd that we have learly depedet sequee. These a form the eerator matrx for the yl ode the G Chael Cod Theory Slde ٤ ٢
٣ Slde ٥ Chael Cod Theory Eod by ovoluto Go ba to the eeral otatos So that G b ( Slde ٦ Chael Cod Theory Eod by ovoluto The the eerator matrx a be wrtte as assume the formato bts are ve by The ( ( G IG b I
٤ Slde ٧ Chael Cod Theory Eod by ovoluto But as formato vetor a be wrtte as polyomal also The Or Ths forms o-systemat way for yl od. But t proves that ( dvdes ( ad ( s of order ( - ( ( Slde ٨ Chael Cod Theory Cyl Blo Codes The yl property a be formulated by us polyomal represetato If we yle shft by oe bt. Ths eerates But as Or ( ( (.. ( (. mod (
Cyl Blo Codes By exteso we a wrte m m ( mod Example f = by us prevous equato fd ( The ( ( 5 6 ( 4 mod ( s ve by Chael Cod Theory Slde ٩ Cyl Blo Codes But as eerator polyomal s vald ode word Or (. But as ( = ( (. Ths leads Whh proves that the eerator polyomal ( should be a fator of + Chael Cod Theory Slde ١٠ ٥
Geerator Polyomal The eerator sequee/polyomal for yl ode must have the follow propertes Geerator polyomal s a fator of + Deree of the eerator polyomal s So f we ow the eerator polyomal we a fd Number of the party bts whh s the order of ( The ode leth whh a be alulated from mmum value of ; where + dvdes (. For our example = 7 + dvdes ( the = 7; Chael Cod Theory Slde ١١ Systemat blo ode I systemat eod the messae s part from the ode Or C I polyomal format p p p Or p p p p( p( But as ( dvdes ( ad p ( of order - the p mod ( x Chael Cod Theory Slde ١٢ ٦
Systemat blo ode Example: The (74 Hamm ode s defed by the eerator polyomal ( Eode the follow messae = ( The osystemat eod wll lead to 4 6 ( Whle the systemat od wll ve b( mod 4 6 Chael Cod Theory Slde ١٣ Tme Doma Eod Aother method of do lo dvso s by us shft rester as show -- + + + + - u( Chael Cod Theory Slde ١٤ ٧
Lo Dvso us Shft Rester Example: For our example + + - u( To see how ths rut wor let us osder the eod of u = We start the shft rester by tal value of The remader s otet of the shft rester whe the fal bt s shfted Chael Cod Theory Slde ١٥ Lo Dvso us Shft Rester The follow table ves the shft rester after eah shft Iput - Rester Cotet The odeword s Chael Cod Theory Slde ١٦ ٨
Lo Dvso us Shft Rester A possble rtsm of the rut of prevous rut s that after the formato has bee etered a further - shfts are requred before the sydrome s formed. Ths a be overome by the follow ofurato + + U( Chael Cod Theory Slde ١٧ Lo Dvso us Shft Rester The follow table ves the shft rester after eah shft Iput - Rester Cotet The odeword s Chael Cod Theory Slde ١٨ ٩
SYNDROME OF A CYCLIC CODE It s farly easy to show that f we dvde a error sequee by the eerator ad tae the remader the result s the sydrome. Proof To do ths osder the reeved sequee r( as osst of the sum of the ode sequee b( ad a error patter e(: r b e( The sydrome equal the remader result from dvd r( by ( or r q s But as b( = u(( ths leads to e q ( S Chael Cod Theory Slde ١٩ SYNDROME OF A CYCLIC CODE The same as blo odes all possble sydrome a be assoated to error patters. s( = e( mod ( But as e the e a s e e e a x e s Whh meas that e (( mod ( equal s( mod ( Whh meas yle shft the error patter s equvalet to yl shft of the sydrome the shft rester. Chael Cod Theory Slde ٢٠ ١٠
SYNDROME OF A CYCLIC CODE The same as blo odes all possble sydrome a be assoated to error patters. s( = e( mod ( For our example ths ves the follow table Error posto 6 5 4 e S Chael Cod Theory Slde ٢١ SYNDROME OF A CYCLIC CODE But f we start from ad shft the rester wth feedba the result s Error posto 4 5 6 S The relato wth the prevous table s apparet Chael Cod Theory Slde ٢٢ ١١
SYNDROME OF A CYCLIC CODE Let us loo at some examples of ths prple operato. Let the reeved sequee equal. There are two way to solve for the sydrome S( = R(x mod ( = + or The odeword orrespod to the reeved formato s alulated the sydrome a the be alulated by as + = I both ases ths meas the error patter s The oral word s Chael Cod Theory Slde ٢٣ SYNDROME OF A CYCLIC CODE If we repeat the same for. There are two way to solve for the sydrome S( = R(x mod ( = + or The odeword orrespod to the reeved formato s alulated.. Or s = The error patter. If s shfted the rester we et. whh meas he error s shfted oe bt to the left. Chael Cod Theory Slde ٢٤ ١٢
SYNDROME OF A CYCLIC CODE Double error orreto a be doe by the same method ( 8 4 If a eoder prevously dsussed s used to eerate the sydrome for sle error sequee. It ves Error posto Sydrome Error posto Sydrome 4 5 6 8 9 4 Chael Cod Theory 7 Slde ٢٥ SYNDROME OF A CYCLIC CODE The sydromes to loo for are those result from a error bt 4 ether o ts ow or ombato wth oe other bt. Ths ves rse to the lst show Error posto Sydrome Error posto Sydrome 4 4 4 4 44 45 46 48 49 4 4 4 4 4 47 Chael Cod Theory Slde ٢٦ ١٣
SYNDROME OF A CYCLIC CODE Now suppose that the errors are postos ad 5. By add the sydromes of those sle errors we et a sydrome value as be omputed by the eoder. It s ot o the sydrome lst So shft oe ad oe aa. Whh the lst 4 shft = the error posto We therefore orret bt ad vert the leftmost bt of the sydrome to leave. A further seve shfts ma e all wll produe the patter date aother orretable error bt 5. Chael Cod Theory Slde ٢٧ SYNDROME OF A CYCLIC CODE Suppose the errors are bts ad. The sydrome alulated by us the eoder wll be. Two shfts wll br ths to. We orret bt ad vert the frst bt of the sydrome leav. Two further shfts produe the sydrome dat a error bt. Chael Cod Theory Slde ٢٨ ١٤
Shorted Cyl Code Suppose we have a ( yl ode shorteed to (. We reeve a sequee r( ad wsh to ompute the sydrome of j r( where j s the sum of (umber of bts removed ad (the usual amout by whh the sydrome s pre-shfted. If s ( s the sydrome of r( ad s ( s the sydrome of j the the requred sydrome s s (s ( mod (. We therefore multply the reeved sequee by s ( mod ( by feed t to the approprate pots of the shft resters. Chael Cod Theory Slde ٢٩ Shorted Cyl Code Cosder for example the (5 ode eerated by 4 + + shorteed to ( 8. Frst we ompute 7 mod( whh s foud to be + +. Now we arrae the feed of the reeved sequee to the shft resters as show suh that there s a feed to the ad resters. If a sequee s fed to ths arraemet the sydrome s. If the frst trasmtted bt s error that fat wll therefore be dated mmedately. Ay other sydrome wll date a eed to shft utl s obtaed or the error s foud to be uorretable Chael Cod Theory Slde ٣٠ ١٥
Shorted Cyl Code Example Cosder (5 ode eerated by 4 + + shorteed to (8 Frst we ompute 7 mod ( whh s + + Now we feedba the reeved sequee as show + + + Chael Cod Theory Slde ٣١ Reeved sequee Shorted Cyl Code If a sequee s fed to the arraemet the resultat otets of the shft rester are Iput Rester Cotet Iput Rester Cotet - Chael Cod Theory Slde ٣٢ ١٦
EPURGATED CYCLIC CODES Expurato s the overso of formato bt to party bt ad eep the leth the same. If a yl ode has a odd value of the mmum dstae multply the eerator polyomal by + has the effet of expurat the ode ad reas d m by For example ( ( 4 Chael Cod Theory Slde ٣٣ EPURGATED CYCLIC CODES Ay odeword of the ew ode osst of ode word of the oral ode multpled by +. (shfted left ad added to tself. For example the sequee from the oral odeword beome + = Chael Cod Theory Slde ٣٤ ١٧
BCH Codes May of the most mportat blo odes for radom-error orreto fall to the famly of BCH odes amed after ther dsoverers Bose Chaudhur ad Hoquehem. BCH odes lude Hamm odes as a speal ase. The ostruto of a t-error orret bary BCH ode starts wth a approprate hoe of leth: = m - (m s teer It s ve that - mt ad d m t + Chael Cod Theory Slde ٣٥ Cyl Codes for Burst Error Correto May of Chael Cod Theory Slde ٣٦ ١٨
Fre odes Fre odes are yl odes that a orret sle burst errors wth sydrome that a be splt to two ompoets for faster deod. The form of the eerator polyomal for Fre ode whh s apable of orret burst of leth up to l s ( l h Where h ( s rreduble polyomal of leth m l whh s ot a fator of l +. I.E the order p of h ( s ot fator of l-. the leth of the ode wll be the lowest ommo multple of p ad l-. Chael Cod Theory Slde ٣٧ Fre odes A example s h ( = 4 + + s ot a fator of 7 + the ( 7 4 Ths eerator polyomal eerates (594 Fre ode whh a orret burst of leth l or less. Chael Cod Theory Slde ٣٨ ١٩