Step-by-step decoding algorithm for Reed-Solomon codes

Step-by-step

decoding algorithm for

Reed-Solomon

codes

T.-C.Chen, C.-H.Wei and S.-W.Wei

Abstract: A i i e w step-by-step dccodiiig algorithm for dccoding Reed-Solonion cotlcs ovcr CiF(2"') is presented. Based on scvcral properties of' thc syndrome miitrices, thc iicw sicp-by-step decoding algorithm can directly dclcnnine whether every rcccived synibol is an error locator. The detection of error Iocation is bascd oiily on the detcrminant of a Y x v syndrome matrix, where 1' is the number or crrors. When an error location is f'ound, ils corrcsponding error vahc can also be determined by performing a deteiminant division operation bctwcen two syndrome matrices. The new d w d i n g algorithm can significantly reduce computation complexity and improve the decoding spccd compared with tlie conventioiial slcp-hy-step decoding algorithm.

1 Introduction

Among the many crror-correcting codes, R e d S o l o m o n (RS) codes arc Ihc most frequently employed in digital communication and storage systems. Many dccoding tech- niques have been proposed for decoding RS codcs, such as the Berlekmp-Mwssey algorithm [l-41, the Euclidcaii algo- rithm [l--51, and Ihc stcp-hy-step decoding algorithm [6, 71. The step-by-step dccoding algorithm was first prcscnted by Massey in 1965. The differeiice between Lhc sLep-by-step mclhod and the standard algebraic method is that tlie skp- by-step mctliod decodes every potential crror location and crror value directly, instead of searching the error locators and cvaluating the error valucs.

Tne conventiond step-by-step decoding algorithiii cor-

rects thc cri-om in terms of the diffeimccs between tlie orig- inal syndromc matrix and the. Icmporarily changed

syndrome matrices. This idca is based on the Fdct thal tlic weight of error patterns can be distinguished from each other by ushg thc syndrome matrices. Tlvrcfore, the con- vcntional step-by-step algorithm adds in order d1 possiblc

2"' - 1 nonzei-o elements of GF(29 to every syinboi of thc received word to deteimiw whether the weight of thc crror pattcm has been iduced. If the weight of the error paltcrn is reduced, both thc error location and tbc corrcsponding error value arc found. When the iiumber of errors v in the received word

is

determined, the decoding procedure only needs t o dctcct whether a determinanl of the v x

v changd

syndmmc matrix vanishes for every nonzero element of GF(Zfn) added in every detected symbol. If the determinant vanishes, the deteckd syinbol is an error and h e added noiiicro element is its error valuc. Compared with other decoding algorithms, the slcp-by-step algorithm offeis thc

D l l x , 2w

IEE Proceediitgs online no. 20000149 PO6 IO. 1049!iip-coin:20000145,

t'apcr first rmivcd 1st February atid in rcviscd fomi 27th Ociober I999 T.-C. Chen and C,-H. Wci ai'o with the Deparlment of Elcctronia Engi-

nccdng, Nationnl Chiao Tung University, Hsiiichu, Ihiaan 30050, Republic of China

S.-W. Wci is with the Depimmenr of Electrical Enginmring, Cbiing Hua

University, Hsinchu, Tfiiwm 30067, Kcpblic OF China

advantage of a simple decoding conccpt because the step-

by-s~cp algaritlim only depends on H v x v syndrome malrix. However, 2In -

L

iterations must bc performed for

evcry rcccived symbol in tile convcnlional step-by-step decoding algorithm. In order to spced up the decoding process, a new step-by-skp dccoding algorithm is devcl- oped in this papcr. 111 the new decoding algorithm, instesid of trying cvcry potential element, ii new ~ncthod for directly searching the error locators is pitscnted. The new error locator scarching method is bascd on scvcral properties of the syndroine matrix to detect dircctly whether every received syinbol is an m o r . The detection o f error location

is only based on whctlicr a v x v syndrome matrix is singu- lar or not, where I' is thc number of errors. Whcn an error

is found by thc ncw searching method, the corrcspouditig error value can casily bc o h t a i n d by performing a detemi- mant division opcr'ation of two syndrome matrices.

2 Properties of syndrome matrix

For a t-error-coriwting RS codc wilh syiuhols from the Galois Iicld GF(29, the codeword can be cxpressed as

Due to the presence or thc channel noise, the receivcd word

r(x) in thc mciver may be different froin thc ciicoded code-

word &) in the transmitter. Let 1hc crror polynomial causcd by the channel noise havc OIC [om

Thcn Ihe received word ciiii be cxprcsscd as

C(Z) = eo

+

e13:

+

e J x z

+

. . .

+

elk-l F1

(4

r ( x )

= C(X) -t- e(.)

= 1'0

+

r1r

+

r2x2 -k

. . .

+

T,-1Zn-' ( 3 )

The weight or ihc crror polynomial c(x) would bc the numbcr. or crrors in the received polynomial r ( , ~ ) , The syn- drotne values or a received word with JI errors can bc

obtained from

= e(afj

=

r,

x,"

i = 1,2, , . . , 2 t

3=1 8

where

4

indicates lhc m o r location of the jlll crroneous symbol aiid Y, is the correspondiiiing crror value. Thcrelbrc, the decoding task is, givcrl the syndrome valucs, Lo Find the error locations and error vulues,

Thc convcntionnl step-by-step dsoding algnrithm cor- rects the errors directly in tcnns of the differeiicc bctwcen ihe original syndrome matrices and the temporarily changed syndrome matriccs. The various weights of crror palterns mi1 bc distinguished tYom Ihc syndrome matrices.

A k x k syndrome matrix, IV,, lias the following rclation wit11 Lhc syndrome values [2, 5-81

Is,

S k + l

. . .

S Z k - l l

For f-error-correcting R S codes, the syndrome niatrix IV, is singular if the numbcr o f mors in i.(.lc) is k - 1 or lcss; the

syndrome ninlrix Nk is izonsingular if thc iiumher of errors is IC [2, 5-81, That is, thc deterininant of thc syndrome niatrix det(iVi) equiils zcro if llic number of errors is

IC

~ I

or lcss and det(Nk)

*

0 if thc nurnbcr of errors is lc. Thc convenlional skp-by-step decoding algorithm adds in order all 2"' - 1 nonzero elemcnts or GP(2"') to every symbol of

the received word 10 determine whether thc wcight of the error pattern has been reduced.

lr

thc weight of the crror pattern is r e d u d , both the error location and the corre- sponding crror value are found.

For a rcccivcd word ~(x), we add a nonzcro element

0

of GF(2"') to the firs1 syinbol

ro.

l h e n the changed syndrome values, dcnotcd as S'( 1 s i 5 26, will bc

st

= sj

+

p

(5)

and the changed syndromnc matrices, denoted as

N;,

I s k 5 t, can bc cxpressed as

Then the valuc oTdcl(Ar6) can be obtaincd as

Now, a new k x

IC

syndrome matrix iM, is dcfincd as

dbik =

4

s,

+s3 $2 3-

s*

''I SI,

+

S k + 2 s2 f

s4

s,

+

s 5 ' '

.

SA-1-1

+

Sk+3 S k

+

Sk.l.2 4 + 1

+

Sk+J '

' . S 2 k - - I

+

S , k + l ( 8 ) rlet(,Vi) = dct(iVk)

+

fl

. d c l : , ( M k - l ) For the,jth cyclic shifted polynomial o f @), dcnoted as

(9)

1 :

Then dcl(N',,) c m be expressed as

with initial value det(Mo) = 1.

# ( x ) = ?&j - l - ' l ' 2 t - j + ] % + , . , + ' f ' , - ] P i - ]

+

4- , ,

.

+

' ? ' n - i - 1 2 " - 1 (10)

thc corresponding syndronw valucs arc dcnolcd as S/, L 5 i

s 2t. Siniilarly, tlic corrcsponding syndrome intttriccs can be exprcsscd as and

Mi

=

s;

+s.;

si

t

sj

".

si

"Si+,

si

4-

si

3;

+ s-;

' ' . 4-

si

-k

t

Sbt3

' " SiA-, t (12) Then, a nonxro element

B

i s added to the first symbol r!l-j of rcl(x) to obtain tlic IICW polynomial i ' ~ ~ ( x ) and the corre-

sponding syndroine valucs S:J, I 4 i < 22. Consequently, we

ciiii obtain the following equation

(13)

dct(NC) = i l c t ( i q )

+

p .

deL(n.1;) where

Henceforth, in ordcr to describe some theoreins and the decoding algorithm for convenience, wc lc1 thc notation

tio)(x) = #)(x> = r(x> denote the initial rcccivcd word r(x>

and A$o, Mko denote the corrcsponding syndrome matrices. Somc properties o f the new syndro~i~c matrices are pre-

scnlcd as follows.

Theorem I : For a receivcd word of (n, k , t )

HS

code, if the numbcr of errors is v and tb(M,L, ) = 0, then the symbol v , , ~ must be a conect symbol.

Pruuj See the Appendix (Seclion 6. I>.

Theorem 2: For an (H, k, 1) RS code, if the number of errors in a received word ~ ( x ) is v < 1, t l m thc symbol rn3

is an erroneous symbol if and only if det(M$ = 0, 1 s j 6 n.

Ottieiwise. r,,-i must bc a correct symbol. P r o 4 Scc thc hppmdix (Section 6.2).

For a general 6-error-correcting RS code, the niunber of errors v can bc pre-dclcrmincd by the syndrome matrices

IV,, i 5 t , as defined iti eqn. 4. Bascd 011 thcorcm 1 , v , ? ~ must be a correct symbol

iC

the (IJ ~ 1) x (17 ~ 1) syndrome matrix

M i l is singular, i.e. dct(Mil) = 0. However, this theorem docs not cnsum that rlr-. is an erroneous symbol if dct(M/ 1 )

*

0. Bascd on t i e o w n 2, r,[-, must be an error symbol if and only if lhe v x v syndroiim matrix M,( i s sin- gular, i t . det(M,i) = 0. However, when Y = f , S21+l, and lience M i are not calcul;ible. Therefore, when the crror number of the weived word r(x) is equill to t , thc (t - 1) x

(f - I) syndrome matrix M / . l will be first detected. If

dcl(M{.I ) = 0, the syinhol rfZ3 must be a c o r i x t symbol. IP det(Mj_, )

*

0, thc symbol Y ,inay ~ ~ possibly be an erroneous symbol. Assuming that r n j IS an crroncous symbol, we may find a m n m o clcmcnl i n GF(2")) to let the error number of r'O)(,y) become I

-

1. That is, det(N%, = dct(N'&, ) = 0.

Bascd on the equation

cIei,(ivEj) = clet(!@)

+

p .

cIet(iW:-l) =

o

only the rionzero clcrncnt /j = det(N/)/det(MC, ) would let dct(N'/) = 0. If the nonzero element

p

= det(Nj)iciet(A411 ) can also let det(N;i, ) = 0, then the symbol r,lj is an erro- neous symbol rind the valuc /3 = dc~(N/)/dct(Mj_~) i s tlic corresponding error value. Although S"i+,

,

and liencc dcl(WA.l), arc not calculable for a /-error-correctitig RS code. Ilowcvcr, det(N'j) equals zero and is the cofactor of S%+, in the syndroiiie inalrix , N'/L,. Thc valuc or d ~ t ( N y . , ~ ) is indepident of

S'i,+,

and can thus be obtaincd. Therefore, if the number of errors i n t(,v) is t, we

first calciulatc thc valiic dct(ML, ); if det(M,LI) = 0, then the symbol r,lj ir; a corrcct symbol; if det(M)Il )

+

0, then let

0

= det(N/)/det(MjLl ) and calculalc tlic valuc dct(A"j.,.l)

.

I f dc~(N!\~.,) = 0, r r r j is an error symbol and

8

= det(Nh/ dct( MLl ) is the coriwpondiiig error vdue. Consequently,

we have the following corollary.

Cornilary I : For a received word of (n, k, f ) RS code and nuinher of erroi-s U = I, the symbol r,., must be an erronc- 011s symbol if and only if a unique nonzero value

p

= del(Nj)/det(Mj_, ) cxists suck that det(Ncl,, ) = 0.

Based on theorem 2 and coi*ollary 1, 1hc crror localions can be found symbol-by-symbol. Thai, the corresponding crror valucs can casily be obtained li-om the following tlieo-

rcm

.

Thctirm 3: For a received word of (it, k , /) RS code, if the

number of errors is 1' mid the symbol rfi-; is an erroneous syiiikol, then det(Mi.l )

+

0 and the coiresponding error

value is det(N,l)ldet(M{ ).

PruuJ Scc tlic Appendix (Section 6.3). 3 New step-by-step decoding algorithm

Bised on the syndrome mitrices a i d thc ncw mclliod for scarcliing lhe error locations aiid error values, B new step- by-step decoding algorithm is prcscnlcrl. Firsl, lhc syn-

drome v:ilues SF, i = 1, 2, 2t, and syndroinc malriccs

N Z , k = 'I, 2,

...,

6 , of the received word r(x) are calculiited to deccrminc lhc number of crrors (i.e., the value of v). For

a general t-error-correcting RS codc, the numbcr or C C ~ O ~ S ciiii be deteimiiied by jus1 consccutivcly testing dct(N,('), det(NPr), ,,,, until a nonzero cieterminatit, say det(N:), is round [GI. Whcn the error number v is known, the new algorithm tests every symbol of r(x) step-by-step to detcct

whclhcr or not it is an crror. I f an ei-rot' locator is found, its crror valuc can casily he obtained by calcuIating det(Ny!)/ det(ML, ). Bascd on thcsc propcrtics or syndronic matrices, two types of the new decoding algorithm, lhc parallcl vcrsion and scqucnlial version, are presented as follows. 10

3,

I

Parallel

decoding

algorithm

S # y I: Calciilatc llic initial syndrome values S/ (i = 1, 2,

.,.,

21) and Ihc dclcrminanls or Lhc initial syndrome matri- ccs d c ~ (

N,:)

(k = 1, 2,

,..,

t ) from the rcccived word A(')(x) = r(x) = ro

+

r1x

+

r y ?

+

...

+

';,

,F'. Then determine the number of errors v.

Szep 2: If v = 0, go to step 7.

Stey 3: Calculate ttie syndrome values S j = SF a's, I 5 i

5 2v, and the dekiminants del@") = cdt2 I det(Nko) Tor all

, j [see the Appendix (Sections 6.4 and 6.511. Sky> 4: If Y = t, go to step 6.

Step 5 Calculatc thc value dct(M,I), 1 5 j 5 n. For all j , if

dct(M,q = 0, lel r12-j = rlj-j + det(Nj)idct(M,&

>.

Go to step

7.

Step 6: Calculate det(M/:,), 1 s j 5 11. For all,/, if det(M/$)

# 0, let 1;!, ,i = det(NJ)/det(M/.I) and calculate the deteimi- mint drl(Nhr ). For all,j, ifdel(M{;,) ;r 0 and tlcl(Nhl) = 0,

let = r,l:i -t. rlct(N/)/dct(ML,).

Sfep 7 Tlic dccodiiig algorithm has bccti conipletcd.

3.2

Sequential decoding algorithm

Step I : Calculale the initial syndrome values SF (i =

L,

2,

...,

Zr} and determine the number of errors v from der(Nko)

(it = 1, 2,

...,

t).

Srep 2: If'v = 0, go lo step 13.

Swp 3: Let] = I .

Step 4: Calculatc tlic syiidromc valucs S! = S / - '

ai3

i = 1,

2, ,,,, 2 1 7 - I, aiid fhc dctcrminaiits dct(Nkj) = ak2

dct(Nkj-I), k = 1, 2,

...,

v. [sec 1hc Appcndix (Sections 6.4 and 6.511.

s6cp 5: If v < I , go to step 11.

Sfcp (j: Calculate det(M/ I ) from the syndrome values Si. Seep 7: ll'det(Mj_l) = 0, go to step 13.

S!oy 8: Let /j = det(N/)/det(M{l,) and f,,? = rlf?.

+

p.

Step 9: Let S;j = S/ -I-

p,

i = I , 2, ..., 21, and ciibulate

det(N'j+, ).

Sky it): if det(N'/+,) f 0, go tn step 13. Otherwise, CJcu- late det(hdkq, k = I , 2: _ _ _ , I ~ 2, and let v = v ~ 1 , mnj =

...,

v.

Go to step 13.

Step

I I:

Calculalc llic valuc dcl(M,!),

Srcy I2 If det(Mvq

+

0, go to step 13. Otherwise, c:tlculafe Ihc valucs dcl(MkJ}, k = I, 2,

,..,

v - 1 , and IcL

0

= det(iY,,)/

dct(/Wi,), rlFj = F ~ , . ~

+

/I, S/ = S,'

+

/?, Y = 1' ~ 1, dct(N,i) =

det(N,/) -t

6 .

det(M&.l), k = 1, 2, _ _ _ , i'.

Srep 13: l f j = n or v = 0, tlicn this decoding algorithm is completed. Otherwise, l e t j = j

-+

1 and go to step 4.

Thc new dccoding algorithm can also bc applied in dccoding sliorlcncd RS codcs. Consider a (U - 1, k - [, t) shorcciicd RS codc, lhc cncoded codeword c(x) and the rcccivcd word r(x) can be cxprcsscd, rcspcclivcly, as Y \ ~ + S j = SI!, dct(N$) = dcI(/V,j) +

p .

dCt(M$i1), k = 1 , 2,

C { . ) = 3. c15

+

+

.

, ,-I- e,-1ICn--I-I r(.) = 1'0

+

rl!c

+

r.2x2

+

.

.I

+

l . l k - l x ' l ' . l . . l :ind

For the parallel decoding, only replace ' I s j 5 n' by ' I

+

I

5.j 5 II' for all stcps. For Llic scquciitial dccoding, just mod-

iry step 3 as hllows:

Step

3,:

Calculatc S/ = Sf .

a',',

I 5 i 5 2v - 1, and dct(Nkl)

=

dk

_{det(NL)). L e t j =}

_I

₊

_1.

Because the new decoding algorithm is performed symbol- by-symbol, the coinputation complexily caii bc rcduccd Tor Ow shorlciicd codcs.

4 Conclusions

The conventional step-by-step decoding algorithm corrccts llic errors in leiins of the difhwiccs bctwccn lhc original syndrome matrices and tlic tcmporaril y changed syndrome matrices. Compaicd with thc othcr decoding algoritliins, thc stcp-by-stcp algorithm offers lhc advaimgc of a simple dccoding proccss which dcpciids 011 calculating dct(N‘/)

(and det(N’j,.l) if v = t). Howcvcr, tlic conventional step- by-step algorithm must perform 2”’ - 1 itciatioils to detect

the determinant det(iV’$ (and det(N’& ) if v = I ) for cvery i e i v e d symbol. In order to speed up the decoding proccss,

a new step-by-step decoding algorithm lies been presented. A new syndrome matrix Mk was developed. Based on somc

propertiEs of the syndrome matrim, t~ new method for

searching the error locations and the corresponding error values has also becn prcsenkd. Tlic new stcpby-step decoding algorithm only dclccls (he dclci-iniiiant or Lhc v x Y matrix det(M,J) once for evei-j‘ received symbol (it delccts det(Mll ) and det(Nhl ) if v = I ) . Cornpared with thc con- ventional step-by-step algorithm, the iicw algorithm reduces the computational complexity by a i‘hclor oT 2’” - 1.

Uascd on tlic ncw sty-by-stcp dccodiiig method, a parallel decoding algorithm and a sequential decoding dgorithtn havc bccn proposed. The parallcl dccoding algorithm detects all received symbols to obtain the corresponding error pattern in parallel. Thus, a high-speed pmdlel decoder can be constructed to perform the decoding P ~ O G ess in the interval of one iteration, which is parlicularly suitable ror shortcncd RS codcs. Thc scquciilial dccoding algorithm tests one symbol at B time. The decoder has lower circuit-complexity xnd ciin complete the dccoding process with n iterations.

5 I 2 3 4 5 h 7 8 6 References

LIN,

s.,

and COSTELLO, D.: &or control coding: iliudainci~tnls and applications’ (t’rcnticc Hall, Iliiglcvood Cliffs, NJ, 1983) RLAHIJT, R.E.: ‘Theory and 11raaice or crror conlrol codcs’ (Addi-

son-Wcslcy, New York, 1983)

CLARK, G.C , and CAIN, J.B.: ’Etroralrecling coding for digital

coininuoimkms’ (Mcnum, Ncw York, 1981)

MICHNSON, A.M., mid LIYIXQUE, AH,; ‘Erlar-control tcch-

niqties for digital commuiiictition’ (Wiley, New York, 1985)

PETERSON, W.W., and WELDON; E.J.: ‘El-ror-comctiug codes’ (M IT I’ivss, Cambiidge, 1972, 2nd crln.)

MASSEY, J.L.: ‘Stcphy-stcp dccoding o f thc I~o~Chaudhuri-i-loc-

qucnghcin cutics’, JEEE T,.o!a. I$ Theory, 1965, 1T-11, (4). pp. 5X& 585

W M , S.W., ant1 Wl3, C.14.: ‘High-spd dccndcr of Rd-Solnmnn codcs’, I E l X Truttc Corrmrun., 1993,41, (I I), ~p 1588 ,1593

JU, S., tind RT, G.: ‘Fast decoding a l g o r i h

10;

RS cndca’, F/ec~roii.

L w , 1997,13, (17), pp, 1452-1453 Appendix A

6.1

Proofof

theorem

7

The number of errors v can be pie-determined by Lhc syn- drome matrices Ni, i 5 t as delined in cqn. 4. Bawd 011 the

properly of cyclic codes, the niiinbcr of crrors in &Y) is also v, and then det(N,!) ir 0. By adding a nonzero elcrncnt /3 to rlkj, we obtain llic ncw polynomial r’kl(x), From eqn. 13, we have

c~cI;(N;~) = clet,(!\-:) -i-

p

-

cIet(iM~-,) IC Ikc symbol T ,is an cwoncous symbol, the ~ ~ correspond-

ing crror valuc inust

bc

round, and also det(N’,!) = 0. How- ever, for all nonzero elements in GF(29, it is impossible to have the relation det(N’,!) = 0 because det(N,!’) z 0 and

IEE P ~ U C , - C ~ I J ~ I I I U I ~ , , Vol. 117, No, 1. Fcbninry 2OM

dzt(M{:l) = 0. Therefore, the symbol P , , ~ must

l

x

a coriwt symbol i T dct(ML, ) = 0.

6.2

Proof of theorem

2

First, we add an arbitray nonzero element

p

from GF(2”’) 10 tlic symbol r,f-i to obtain a new polynomial r’bl(x). If rl,:i is a correcl symbol, the number of errors in r’b)(x) should be i’ + 1. If Y ? ~ . ~ is an erroneous symbol, the nuinbcr of errors in ~ ’ ~ ~ ( x ) would bc Y - I (/3 js the crror value) or Y

(0

i s not the correct crror vaiuc). nascd 011 eqn. 13:

dCtm(llr$l) = dct(,?ruj+l) -t-

p .

cl et(^^^^)

Because det(NL,~l

1

= 0 and

p

# 0, dct(N’/+, ) ic 0 if and

only if det(M4

*

0. That is, h e symbol r,l:i is il correct

symbol originally and then the iiomcro element

p

is added to let the error number of r’o)(x) bccoinc v t. 1 if and only if det(M$

*

0. On lhc other hand, det(W{+,) = 0 if and only if det(iW,$ = 0. This incam that the error number of ~ ‘ ~ ~ ( x ) is iess than v

+

1. That is, the symbol rn-] is an erro- neoiis symbol originally and then thc nonzcro clcnicnt

p

is ttdded to let the error number of ~ ’ ~ ~ ( x )

bc

v or Y - I , and

thus det(Wi+I ) = 0 if and only if det(M,/) = 0.

83 Proof of theorem

3

Supposc thal the symbol rfid, is >in erroneous symbol and let the nonzcro element /3 in GF(2pn) be its corresponding error valuc. Then ridding the error value /3 to t-lf-j will reduce the

crror numbcr or r’b](x> to 1’ - 1 and thus

det(j\rLj) = cIet,(iY:) -t-

p

~ I ~ I ; ( M : - ~ ) =

o

Rccausc dct(N,!) ir 0 and

p

# 0, det(MJI ) $ 0 must be true

and /? = dct(iVL!)/tIe~(M,L4) is the only possible solution. TlicrcTorc, i T thc symbol is an error symboI, then dct(Mil) cannot

l

x

equal to zero, w d the corresponding error value must be det(N,/)/iIet(M,i, ),

6.4

Calculation

of syndrome

values

The syndroiiie values S / , I s i s 21, can bc obtaincd from So or S/-’ by pcrfoiining tlic following operations:

s,j

= ,ij

,s,o

xnd

si

=

2 .

$ 1 , 15 i

5

21 f i o o j

s:

= .(j) ) (.

IC=”

- - Y’PL-3

+

7‘,L--j+lT -I-

. . .

4- Tn-lXj-l = Y , , - j

+

*i.,-j+ld

+

2 ’ E - l -t- r o d $.

. . .

-t- ?,,&-j-l

+

r o d i

+

I I

.

+

~ , , - j - I d ~ - ~ ) . ~ (15)

For GF(29, ,2”’-1 1 = I and

(any

=

&

= 1 .

Then

AY./ = (,bpi

+

r,-j+lai

+

. . .

+

T,-1 ( 3 - 1) .i>

+TO&

+

. . .

+

I’, 3~ IQ’ (1)- .L) .i + , , , +,l~n-lo(w~. l1.i

- ,

-

- ?*ocy3‘~

+

. .

I

+

rvL-.j-:L &(n-l).; + . r n - j p . i

= m7.i ’ (,r0

+

Tla:

+

.

. .

+?-,-I &--L).z)

- 0 3 3 . - ”

s;

( 16) P (17) Si m i I ar I y Therefore $ 1 = &-l) , SO

= aij IS,“ = ni

.

(fli’(j. I So) =

Sj-’

(1.8)