Decoding of the Triple-Error-Correcting Binary Quadratic Residue Codes

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1 Automatc Control and Informaton Scences, 04, Vol., No., 7- Avalable onlne at Scence and Educaton Publshng DOI:0.69/acs--- Decodng of the rple-error-correctng Bnary Quadratc Resdue Codes Hung-Peng Lee *, Hsn-Chu Chang Department of Computer Scence and Informaton Engneerng, Fortune Insttute of echnology, Kaohsung, ROC *Correspondng author: Receved February 4, 0; Revsed January 7, 04; Accepted February 09, 04 Abstract In ths paper, a more effcent syndrome-weght decodng algorthm (SWDA), called the enhanced syndrome-weght decodng algorthm (ESWDA), s presented to decode up to three possble errors for the bnary systematc (,, 7) and (, 6, 7) quadratc resdue (QR) codes. In decodng of the QR codes, the evaluaton of the error-locator polynomal n the fnte feld s complcated and tme-consumng. o solve such a problem, the proposed ESWDA avods evaluatng the complcated error-locator polynomal, and has no need of a look-up table to store the syndromes and ther correspondng error patterns n the memory. In comparson wth the SWDA developed by Ln-Chang-Lee-ruong (00), the smulaton results show that the ESWDA can serve as an effcent and hghspeed decoder. Keywords: syndrome, error pattern, Golay code, quadratc resdue code Cte hs Artcle: Hung-Peng Lee, and Hsn-Chu Chang, Decodng of the rple-error-correctng Bnary Quadratc Resdue Codes. Automatc Control and Informaton Scences, vol., no. (04): 7-. do: 0.69/acs---.. Introducton he trple-error-correctng bnary QR codes nclude (,, 7) QR code and (, 6, 7) QR code, respectvely. he bnary (,, 7) QR code s also called the bnary Golay code, whch s a perfect code. he Golay code s frst ntroduced by Golay n 949 [4]. If an overall party check s used, the rate s exactly /, so that most of the known QR codes are the best-known codes. Among them, the extended (4,, 8) Golay code was utlzed to act as an error control on the Voyager I and II spacecraft msson, provdng clear remote pctures of Jupter and Saturn []. Several algebrac decodng algorthms (ADAs) had been developed to decode the bnary Golay code [,]. he key dea of decodng the Golay code by usng ADA s to compute the unknown syndrome for determnng the coeffcents of the error-locator polynomal. One of the representatve methods s nverse-free Berlekamp-Massey algorthm [0]. In [], Reed et al. developed the ADA of the extended (, 6, 8) QR code wth reducble generator polynomal. However, such an algorthm s quet complcated. Ln et al. [6] thus proposed a modfed ADA to reduce the decodng complexty. For ADAs, once the coeffcents of the error-locator polynomal are obtaned, the error postons can be determned by usng the Chen search algorthm [], whch s an exhaustve search over all the elements n the fnte feld. In the decodng procedure of ADAs, ths step s the most tme-consumng and need plenty of multplcaton and dvson operatons over the fnte feld. Most recently, table-lookup decodng algorthms (LDAs) [,7,9] have played an mportant role n forward error correcton. hese types of decoders are effcent wth mnmum decodng delay; however, the LDAs requre a memory space n the decoder chp and ncrease the decodng cost rapdly when the code length s large. he SWDA proposed by Ln et al. [7] used the refned lookup table (RL) to decode the trple-error-correctng bnary Golay code and (, 6, 7) QR code. For decodng the Golay code, the RL conssts of 4 syndromes and ther correspondng coset leaders, and t only needs 68 bytes memory sze. For decodng the (, 6, 7) QR code, the RL conssts of 7 syndromes and ther correspondng coset leaders, and t only requres 88 bytes memory sze. In ths paper, the proposed ESWDA has faster decodng speed than the SWDA, and t does not need a memory sze to store the lookup table. he key dea of the proposed ESWDA s based on the weght of syndrome dfference between the syndrome of the receved word and the row vector of the transpose of the party-check matrx. herefore, the error cases can be swftly determned. he applcaton of the syndrome weght and the syndrome dfference can consttute a hgh-speed decodng algorthm. Moreover, no complcated computaton n the fnte feld s requred n the proposed ESWDA. wo examples demonstrate the decodng procedure of the proposed ESWDA. he proposed ESWDA s applcable to decode other cyclc codes such as the bnary (,, 7) BCH code; however, the decodng steps of the proposed ESWDA need to make some slght adjustments. he decodng steps of the bnary (,, 7) BCH code are demonstrated n a smple example. Computer smulaton result shows that the decodng tme of the proposed ESWDA s superor to the SWDA.

2 8 Automatc Control and Informaton Scences he remander of ths paper s organzed as follows: he background of the bnary QR codes s brefly revewed n Secton. he proposed ESWDA s descrbed n Secton. In Secton 4, we use three examples to demonstrate the proposed ESWDA. Computer smulaton results of the proposed ESWDA and the SWDA are gven n Secton. Fnally, ths paper concludes wth a bref summary n Secton 6.. Background of the Bnary QR Codes he bnary QR codes are a nce famly of lnear cyclc codes. Let (n, (n+)/, d) denote the bnary QR codes wth generator polynomal g(x) over GF(). he length of ths code s a prme number of the form n 8l ±, where l s some nteger. Also, let k (n+)/ denote the message length or nformaton length, and d denote the mnmum Hammng dstance of the code. he set Q n of quadratc resdues modulo n s the set of nonzero squares modulo n; that s, Q n {j j x mod n, x n }. If n, then ts quadratc resdue set s Q {,,, 4, 6, 8, 9,,, 6, 8}. If n, then ts quadratc resdue set s Q {,, 4,, 7, 8, 9, 0, 4, 6, 8, 0,, 8}. Let the symbols C and C denote the bnary Golay code and bnary (, 6, 7) QR code, respectvely. Let α be a root of prmtve polynomal p r (x) x +. If α, then the roots of p r (x) are α, α 4, α 6, α 6, α 4 60, α 064, α 6 60, α 7 8, α 8 80, α 9 0, α Let the element β α u be a prmtve rd root of unty n GF( ll ), where u ( ) / 89. herefore, β α 89, where s the decmal representaton of β. he total roots are lsted n able. able. he total roots of x (n decmal number) β 0 β β 74 β 64 β 4 48 β 48 β 6 67 β 7 94 β β 9 8 β 0 β 67 β 0 β 48 β 4 8 β β 6 78 β 7 78 β β 9 9 β 0 β 876 β 08 he generator polynomal of C s defned by g ( ) ( ) Q β () x +. Now let α be a generator of the multplcatve group of all nonzero elements n GF( ). hen, the element β α u, where u ( )/. he x can be factored nto seven prmtve mnmal rreducble polynomals; that s, x (x + )(x 4 + )(x + )(x 4 + )(x + )(x 4 + )(x + ). hen, the generator polynomal of C s reducble, and s defned by g Q ( x ( x x ( x β 4 + )( x ) g g + ) g + ) 7 +, () where g (x), g (x), and g 7 (x) are the mnmal polynomals of x. hat s, g r (x) (x ß r )(x ß r ) (x ß 4r ) and ß r GF( ) for 0 4 are the roots of g r (x), where r,, and 7. If α, then the total roots of x are lsted n able. able. he total roots of x (n decmal number) β 0 β β 4 β 8 β 4 6 β β 6 0 β 7 0 β 8 β 9 6 β 0 7 β 7 β 4 β 8 β 4 9 β β 6 7 β 7 9 β 8 β 9 6 β 0 β 4 β β β 4 0 β β 6 β 7 β 8 β 9 9 β 0 8 Because the mnmum Hammng dstance of C and C s d 7, the nequalty v + 7 s vald, where v s the actual number of errors to be corrected. Hence, the error-correctng capablty s t ( d ) /, where x denotes the greatest nteger less than or equal to x. he codeword s a multple of generator polynomal g(x); n- that s, C C x m( x) g( x, where C GF() 0 ) k 0 for 0 n, and m m x denotes the message polynomal, where m GF() for 0 k. n Let k p 0 p x be the party-check polynomal, where p GF() for 0 n k. Let n k p p x be the party-check polynomal, where 0 p GF() for 0 n k. Let p(x) m(x)x n k mod g(x), then one obtans m(x)x n k q(x)g(x) + p(x). he term (p(x) + m(x)x n k ), whch s a multple of g(x), s a codeword polynomal gven by n- n k cx ( ) cx ( ) ( ), 0 px + mxx () where c GF() for 0 n. In ths paper, the systematc encodng method s utlzed. Now, let a codeword be transmtted through an addtve whte Gaussan nose (AWGN) channel to obtan a receved word wth the form r(x) c(x) + e(x), where e(x) s the polynomal of the receved error pattern expressed as n e 0e x, where e GF() for 0 n. he n-k- syndromes polynomal s expressed as s 0 s x. o smplfy the polynomal expressons mentoned above, let the message, codeword, error pattern, receved word, and syndrome polynomals be expressed as the bnary vector forms m ( m0 m mk ), c ( c0 c cn ), e ( e0 e en ), r c + e ( r 0 r r n ), and s ( s0 s s k ) n, respectvely. he systematc codeword of the vector form s gven by c mg (4) where G s called the systematc generator matrx. Let P be a k (n k) matrx and I k be a k k dentty matrx, and G can be expressed as G P k ( n k ) I k () k n he party-check matrx H can be expressed as H I, where P denotes the (n k) k [ n k P ( n k) k ]( n k) n transpose matrx of P. he vector form of the syndrome s defned by

3 Automatc Control and Informaton Scences 9 s rh, (6) where H denotes the n (n k) transpose matrx of H; that s, H can be expressed as H h0 I n k hn k. Pk ( n k) n ( n k) h n For C, H has the followng form: (7) I H (8) Decodng Algorthm and heorems In ths secton, the proposed ESWDA s used to decode the C and C. For the development of the proposed ESWDA, the followng defnton, theorem and corollary gven n Ln et al. [8] are needed. Defnton : he Hammng weght of a bnary vector a s denoted by w(a), and the Hammng dstance between a and b s denoted by d(a, b) w(a + b). heorem : Let a (a 0 a a n- ) and b (b 0 b b n- ) be two bnary vectors, then n w( a+ b) w( a) + w( b ) ab. (9) Corollary : If a b 0 for n, then ( + ) + ( ). w a b w a w b (0) he followng heorem s useful to compute the syndrome of the receved word when the receved word shfts one bt to the left. For more detaled proof of ths theorem, see [, p8]. heorem : Let s(x) be the syndrome polynomal correspondng to a receved polynomal r(x). Also, let r () (x) be the polynomal obtaned by cyclcally shftng the coeffcents of r(x) one bt to the left. hen the remander obtaned when dvdng xs(x) by g(x) s the syndrome s () (x) correspondng to r () (x). However, for each cyclcal shft of the receved word, we have to dvde xs(x) by g(x). If the syndrome cyclcally shfts many tmes, then the syndrome computaton s rather tme-consumng for dvdng xs(x) by g n (x) many tmes. he followng theorem provdes an effcent method to compute s () for 0 n, and t can save a lot of computatonal tme. heorem : For the bnary QR codes, let r j be an element of r and h j be the jth row vector of H for 0 j n. hen the syndrome s () of r () for 0 n has the form () n- s r 0 jh j, () [ + j] where the suffx [x] of h denotes x mod n. Proof: Let r (r 0,,r n- ) and r () (r,, r n-, r 0,, r - ) for 0 n. By (7), we have s () r () H n- n- r h r h. he proof s thus completed. j 0 [ j ] j j 0 j [ + j] heorem : reveals that the syndrome of r () can be fast computed by the vector addton. heorem 4 also provdes an effcent method to smplfy the decodng step by usng the syndrome weght. For a detaled proof, see [8]. heorem reveals that the syndrome of r () can be fast computed by the vector addton. heorem 4 also provdes an effcent method to smplfy the decodng step by usng the syndrome weght. For a detaled proof, see [8]. heorem 4: For the bnary QR codes, t s assumed that there are v errors n the receved word, where v t. All v errors are n the party-check bts f and only f the weght of syndrome w(s) v. By usng heorem 4, we can develop the followng useful theorem. heorem For the bnary QR codes, f v errors are n the nformaton bts of the receved word, where v t and t ( d )/, then the weght of the correspondng syndrome polynomal or syndrome vector satsfes ( ( )) ( ) ( ) ( ) w s x d v or w s d v. () Proof: Let error polynomal e(x) present the v errors n the nformaton bts; that s, w(e(x)) v. Snce s(x) r(x) e(x) (mod g(x)), we have e(x) + s(x) 0 (mod g(x)). hs mples that e(x) + s(x) s a codeword and hence the codeword must satsfy w(e(x) + s(x)) d. By Corollary, w(e(x) + s(x)) w(e(x)) + w(s(x)) d and then w(s(x)) d w(e(x)). hus, the weght of the syndrome polynomal satsfes w(s(x)) d v or w(s) (d v). he proof s thus completed. Gven a receved word r, the syndrome of r () can be fast computed by heorem. Accordng to heorem 4, f w(s), then the error postons are n the partycheck bts of r. If w(s (n-k) ), then the error postons are n the nformaton bts of r. Let h denote the th row vector of H, where 0 n. Also let sd w denote the syndrome dfference between the syndromes of r and h n each decodng step w. By usng the weght of sd w, the error cases can be quckly determned. Let u 0 (, 0,, 0) be a k-tuples unt vector and u j has only one nonzero component at the jth poston, where 0 j k. By usng these propertes, the proposed ESWDA can be constructed. Let upper case P, C, and H denote the error poston n the party-check bts, center bt, and nformaton bts of r, respectvely. For C or C, there are error cases (P, PP, PPP, H, HH, HHH, C, PC, PH, PPC, PPH, PHH, CH, CHH, and PCH), whch cover all ( ) 047 and ( ) 499 error patterns. If w(s) 0, then r has no error. If w(s) or w(s (n-k) ), then there are 6 error cases (P, PP, PPP, H, HH, and HHH). Let the syndrome dfference sd (s h ) for n k n. If 0 w(sd ), then there are error cases (C, PC, PH, PPC, and PPH). Let the syndrome dfference sd 4 (s (n-k) h ) for n k n. If n k n and w(sd 4 ), then there s only error case (PHH). If n and w(sd 4 ),

4 0 Automatc Control and Informaton Scences then there are error cases (CH and CHH). Let the syndrome dfference sd ((s h n-k ) h ) for k n. If w(sd ), then there s only error case (PCH). he decodng steps of the proposed ESWDA work as follows: ). (No error, P, PP, and PPP cases.) By heorem, compute s and w(s). If 0 w(s), then the nformaton vector s m (r n-k,, r n- ). Go to step 6. ). (H, HH, and HHH cases.) By heorem, compute s (n-k) and w(s (n-k) ). If w(s (n-k) ), then the corrected nformaton vector s m (r n-k,, r n- ) + (s (n-k) >>), where >> denotes the logcal rght shft operator n programmng or the extenson by zeros on the rght n mathematcs. Go to step 6. ). (C, PC, PH, PPC, and PPH cases.) Compute the syndrome dfference sd (s h ) for n k n and w(sd ). If 0 w(sd ), then the corrected nformaton vector s m (r n-k,, r n- ) + u -(n-k). Go to step 6. 4). (PHH, CH, and CHH cases) Compute the syndrome dfference sd 4 (s (n-k) h ) for n k n and w(sd 4 ). If n k n- and w(sd 4 ), then the corrected nformaton vector s m (r n-k,, r n- ) + (sd 4 >>). If n and w(sd 4 ), then the corrected nformaton vector s m (r n-k,, r n- ) + u 0 + (sd 4 >> ). Go to step 6. ). (PCH case.) Compute the syndrome dfference sd ((s h n-k ) h ) for k n and w(sd ). If w(sd ), then the corrected nformaton vector s m (r n-k,, r n- ) + u 0 + u -(n-k). Go to step 6. 6). Stop. able lsts all the error cases and the number of error patterns n each decodng steps of the proposed ESWDA. he flowchart of the proposed ESWDA s shown n Fgure. able. he number of error patterns n each decodng step of C and C Number of error patterns Steps Cases C C 4 P ( ) PP ( ) PPP ( ) 6 H ( ) HH ( ) HHH ( ) 6 C PC ( ) PPC ( ) PH ( )( ) PPH ( )( ) 60 PHH ( )( ) 60 CH ( ) CHH ( ) PCH ( )( ) ( ) ( ) ( ) ( ) Examples Fgure. Flowchart of the proposed ESWDA In ths secton, three examples are presented to llustrate the proposed ESWDA. Example and show the decodng steps for the bnary systematc Golay code. Example shows the decodng steps for the bnary systematc (,, 7) BCH code, denoted by C ; however, the decodng steps of the proposed ESWDA have to make some slght adjustments. Example : Let a message m ( ) be encoded nto a C codeword c ( ). If the receved word r ( ), then the error pattern e ( ), whch means a HHH error case. he decodng steps are shown below. ). Compute s (000000) and w(s). Snce w(s) >, go to step. ). Compute s () ( ) and w(s () ). he corrected nformaton word m (000000) + ( ) ( ). Go to stop. Example : hs example demonstrates the worst decodng case. Let a message m ( ) be encoded nto a C codeword c

5 Automatc Control and Informaton Scences ( ). If the receved word r ( ), then the error pattern e ( ), whch means a PCH error case. he decodng steps are shown below. ). Compute s (0000) and w(s) 7. Snce w(s) >, go to step. ). Compute s () (000000) and w(s () ). Snce w(s () ) >, go to step. ). Compute sd (s h ) for and w(sd ). sd (s h ) (0000) (00000) (000000). w(sd ). sd (s h ) (0000) (00000) (000000). w(sd ). sd (s h ) (0000) (0000) ( ). w(sd ) 4. sd (s h ) (0000) (00000) (000000). w(sd ). Snce every w(sd ) >, go to step 4. 4). Compute sd 4 (s () h ) for and w(sd 4 ). sd 4 (s () h ) (000000) (00000) (000000). w(sd 4 ). sd 4 (s () h ) (000000) (00000) (000000). w(sd 4 ). sd 4 (s () h ) (000000) (0000) (00000). w(sd 4 ) 6. sd 4 (s () h ) (000000) (00000) (0000). w(sd 4 ) 7. Snce every w(sd 4 ) >, go to step. ). Compute sd ((s h ) h ) (000000) h for and w(sd ). sd (s h ) h (000000) (00000) (0000). w(sd ) 7. sd (s h ) h (000000) (0000) (00000). w(sd ) 6. sd (s h ) h (000000) (0000) (00000). w(sd ) 6. sd (s h ) h (000000) (00000) ( ). w(sd ). Snce w(sd ), the corrected nformaton word m (r,, r ) + u 0 + u - (00000) + ( ) + ( ) ( ). Go to stop. PCH case s the worst case; however, only error patterns, namely.9%, wll enter step. Example : Let a message m (000) be encoded nto a codeword of C wth g(x) x , and obtan the codeword c ( ). If the receved word r (00000), then the error pattern e ( ), whch means a PHH error case. Frst, the decodng steps of the proposed ESWDA mentoned n Secton can be slghtly adjusted as follows: ). (No error, P, PP, and PPP cases.) By heorem, compute s and w(s). If 0 w(s), then the nformaton vector s m (r n-k,, r n- ), and go to step. ). (H, HH, HHH, PH, PPH, and PHH cases.) By heorem, compute s (n-k) and w(s (n-k) ). If w(s (n-k) ), then the corrected nformaton vector s m (r n-k,, r n- ) + (s (n-k) & ()), where the notaton & denotes the btwse AND operator, and go to step. ). (PH and PPH cases.) Compute the syndrome dfference sd (s h ) for n k n and w(sd ). If 0 w(sd ), then the corrected nformaton vector s m (r n-k,, r n- ) + u -(n-k), and go to step. 4). (PHH case) Compute the syndrome dfference sd 4 (s (n-k) h ) for n k n and w(sd 4 ). If w(sd 4 ), then the corrected nformaton vector s m (r n-k,, r n- ) + (sd 4 & ()), and go to step. ). Stop. For ths code, there are 9 error cases. able 4 lsts all the 9 error cases and the number of error patterns n each decodng steps. able 4. he number of error patterns n each decodng step of C Number of error patterns Steps Cases C 0 P ( ) 0 ( ) PP 0 4 ( ) PPP 0 0 H ( ) HH ( ) 0 ( ) HHH 0 PH ( )( ) ( )( ) PPH 0 ( )( ) PHH 0 0 PH ( )( ) ( )( ) 0 PPH ( )( ) ( )( ) 0 4 PHH ( )( ) ( )( ) 7 0 he decodng steps are shown below. ). Compute s (00000) and w(s). Snce w(s) >, go to step. ). Compute s () (00) and w(s () ) 8. Snce w(s () ) >, go to step. ). Compute sd (s h ) for 0 4 and w(sd ). sd (s h ) (00000) (0000) (00000). w(sd ). sd (s h ) (00000) (000) (0000). w(sd ) 6. sd (s h ) (00000) (000) (000000). w(sd ) 4. sd (s h 4 ) (00000) (0000) (00000). w(sd ). sd (s h ) (00000) (0000) (000). w(sd ) 7. Snce every w(sd ) >, go to step 4. 4). Compute sd 4 (s () h ) for 0 4 and w(sd 4 ). sd 4 (s () h ) (00) (0000) ( ). w(sd 4 ). Snce w(sd 4 ), the corrected nformaton word m (r 0,, r 4 ) + (sd 4 & ()) (0) + (( ) & ()) (000). Go to stop.

6 Automatc Control and Informaton Scences. Smulaton Results he proposed ESWDA has been programmed n C++ language. On an Intel Q6600 PC wth XP operatng system, all k n codewords wth all ( ) error patterns were created to check every possble error pattern of C, C, and C, respectvely. In other words, the error, patterns of C, C, and C are ( ) 7 ( ) 047, and ( ) 499, respectvely. he decodng tmes of the proposed ESWDA and the SWDA are shown n the able, able 6, and able 7, respectvely. For v, t means that one error of that code nput to the decoder, and for the average decodng tme, t means that all the error patterns of that code nput to the decoder. In these three tables, the average decodng tme of the proposed ESWDA s about 0.6 tmes, 9.6 tmes, and 4 tmes faster than the SWDA, respectvely. he memory requrements of the two algorthms are also shown n able, able 6, and able 7, respectvely. It s obvous that the proposed ESWDA sgnfcantly reduces decodng tme wth the ncrease of the code length. able. Comparson of the decodng tme (n µs) and memory requrement (n bytes) for the Golay code between two algorthms Algorthms Number of errors v v v Average Memory sze ESWDA SWDA able 6. Comparson of the decodng tme (n µs) and memory requrement (n bytes) for the (, 6, 7) QR code between two algorthms Algorthms Number of errors v v v Average Memory sze ESWDA SWDA able 7. Comparson of the decodng tme (n µs) and memory requrement (n bytes) for the (,, 7) BCH code between two algorthms Algorthms Number of errors v v v Average Memory sze ESWDA SWDA Conclusons Bnary QR codes are well known for ther good features. A hgh-speed and effcent ESWDA s developed to decode C, C, and C. he proposed ESWDA nether stores large lookup table n the memory nor computes complcated algebrac computatons. By usng heorem, heorem 4, heorem, and the weght of sd w, the error cases can be quckly dentfed and corrected. herefore, the proposed ESWDA s a very effcent and low-cost decoder for decodng the trple-error-correctng QR codes. he proposed ESWDA can be extended to decode other QR codes or BCH codes; however, the decodng steps of the proposed ESWDA need to make some slght adjustments. References [] Chen, R.., Cyclc decodng procedure for the Bose-Chaudhur- Hocquenghem codes, IEEE rans. Inform. heory, 0(4) Oct [] Chen, Y.H., Chen, C.H., Huang, C.H., ruong,.k., and Jng, M.H., Effcent decodng of systematc (,, 7) and (4,, 9) quadratc resdue codes, J. Inform. Sc. and Eng., 6() Sept. 00. [] Ela, M., Algebrac decodng of the (,, 7) Golay codes, IEEE rans. Inform. heory, (). 0-. Jan [4] Golay, M.J.E., Notes on dgtal codng, Proc. IRE, 7, [] Lee, C.D., Weak general error locator polynomals for trpleerror-correctng bnary Golay code, IEEE Comm. Letters, (8) Aug. 0. [6] Ln,.C., Chang, H.C., Lee, H.P., Chu, S.I, and ruong,.k., Decodng of the (, 6, 7) quadratc resdue code, J. Chnese Insttute of Engneers, (4) June 00. [7] Ln,.C., Chang, H.C., Lee, H.P., and ruong,.k., On the decodng of the (4,, 8) Golay code, Inform. Sc., 80() Dec. 00. [8] Ln,.C., Lee, H.P., Chang, H.C., Chu, S.I, and ruong,.k., Hgh speed decodng of the bnary (47, 4, ) quadratc resdue code, Inform. Sc., 80(0) Oct. 00. [9] Ln,.C., Lee, H.P., Chang, H.C., and ruong,.k., A cyclc weght algorthm of decodng the (47, 4, ) quadratc resdue code, Inform. Sc., Aug. 0. [0] Reed, I.S., Shh, M.., and ruong,.k., VLSI desgn of nverse-free Berlekamp-Massey algorthm, IEE Proc. Comput. Dgt. ech., 8() Sept. 99. [] Reed, I.S., Yn, X., and ruong,.k., Algebrac decodng of the (, 6, 8) quadratc resdue code, IEEE rans. Inform. heory, 6 (4) July 990. [] Reed, I.S., Yn, X., ruong,.k., and Holmes, J.K., Decodng the (4,, 8) Golay code, IEE Proc. Comput. Dgt. ech., 7() May 990. [] Wcker, S.B. Error control systems for dgtal communcaton and storage, Prentce Hall, New Jersey, 99.