High-Speed Decoding of the Binary Golay Code

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1 Hgh-Speed Decodng of the Bnary Golay Code H. P. Lee *1, C. H. Chang 1, S. I. Chu 2 1 Department of Computer Scence and Informaton Engneerng, Fortune Insttute of Technology, Kaohsung 83160, Tawan *hpl@fotech.edu.tw 2 Department of Electronc Engneerng, Natonal Kaohsung Unversty of Appled Scences, Kaohsung 807, Tawan ABSTRACT Recently, some table-lookup decodng algorthms (TLDAs) have been used to correct the bnary Golay code. Ths paper proposes an effcent hgh-speed TLDA called message-syndrome decodng algorthm (MSDA) by usng the message syndrome to correct the bnary systematc Golay code. The proposed MSDA s based on the novel message-syndrome lookup table (MSLT). The MSLT merely conssts of 12 canddate syndromes and ts memory sze s sgnfcantly smaller than other exstng lookup tables. Computer software smulaton results show that the decodng speed of the proposed MSDA s superor to other proposed TLDAs. Keywords: Golay code, weght, syndrome, error pattern 1. Introducton The well-known bnary Golay code, also called the bnary (23, 12, 7) quadratc resdue (QR) code [2], was frst dscovered by Golay [1] n It s a very useful perfect lnear errorcorrectng code; partcularly, t has been used n the past decades for a varety of applcatons nvolvng a party bt beng added to each word to yeld a half-rate code called the bnary (24, 12, 8) extended Golay code. One of ts most nterestng applcatons s the provson of error control n the Voyager mssons [2]. The Golay code can allow the correcton of up to t ( d 1)/ 2 = ( 7 1)/ 2 3 errors, where x denotes the greatest nteger less than or equal to x, t s the error-correctng capablty, and d s the mnmum Hammng dstance of the code. In the past few decades, several decodng technques have been developed to decode the bnary Golay codes, for example, the famous algebrac decodng algorthm (ADA) developed by Ela [3], the notable shft-search algorthm proposed by Reed et al. [4] after that, and the nverse-free Berlekamp Massey algorthm used to obtan the error-locator polynomal by Chen et al. [5]. The above s some representatve lterature on ADAs used to correct Golay codes. Recently, the TLDAs gven n [6-9] have acqured a promnent role n error-correctng decodng. A TLDA wth a refned lookup table (RLT) has been proposed by Ln et al. [8]; the RLT only needs 168 bytes memory sze. The memory sze of the RLT s the smallest memory sze requred to correct the Golay code untl now. In general, the TLDA makes consderably faster decodng speeds possble for the cyclc codes, n comparson wth the ADA. However, the TLDA requres an extra memory space n the decoder chp or software and t ncreases the decodng cost rapdly when the code length s large. Although the ADA does not need any lookup table, t requres a large number of complcated computatons n the fnte feld. These complcated computatons wll degrade the decodng performance and lead to a serous decodng delay as the code length ncreases []. Chen et al. [7] proposed the fast lookup table decodng algorthm (FLTDA) wth a lookup table of 1.35K bytes by usng the shft-search algorthm [4] to decode the Golay code. The famous syndrome decodng algorthm (SDA) wth a reduced-sze lookup table (RSLT) gven n [2, p9] requres 89 syndromes and ther correspondng error patterns, whereas the RSLT requres 445 bytes of memory. Journal of Appled Research and Technology 331

2 Hgh Speed Decodng of the Bnary Golay Code, H. P. Lee et al. / Ln et al. [8] proposed the syndrome-weght decodng algorthm (SWDA) wth RLT whch needs 168 bytes of memory, and the smulaton result shows that the decodng speed s approxmately 9.7 tmes faster than the mproved Ela s decodng algorthm (EDA). In ths paper, an effcent hgh-speed MSDA wth a MSLT s proposed to decode the bnary systematc Golay code. The novel MSLT conssts only of 12 canddate syndromes whch have one error n the message part of the receved vector, and t merely requres 24 bytes of memory. Software smulaton results show that the average decodng speed s the fastest among the TLDAs as mentoned above. In the four-error soft-decson decoder n [], the mproved EDA s the role of the hard-decson decoder. We substtuted the proposed MSDA for the mproved EDA n the four-error soft-decson decoder. Apparently, the decodng speed of the soft-decson decoder was faster than before. The remander of ths paper s organzed as follows: The background of the bnary systematc Golay code s brefly revewed n Secton 2. Some related TLDAs and related theorems are brefly ntroduced n Secton 3. The proposed MSDA s presented n Secton 4. Computer smulaton results are gven n Secton 5. Fnally, ths paper concludes wth a bref summary n Secton Background of the bnary systematc Golay code The bnary (23, 12, 7) Golay code can be defned as a lnear cyclc code or QR code [2]. For the bnary Golay code over fnte feld GF(2 ), ts quadratc resdue set s gven by Q 23 = { j 2 mod 23 for 1 j 22} = {1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18}. (1) The generator polynomal g(x) of the bnary Golay code s defned by g( x) x Q23 x ( x β 9 x 7 ) x 6 x 5 x 1, (2) where s a prmtve 23rd root of unty n GF(2 ). n Let the codeword polynomal 1 C ( x) 0C x, where C GF(2) for 0 22, and the message k polynomal 1 m ( x) 0 m x, where k = 12 s the message length and m GF(2) for 0. C(x) s a multple of g(x), namely, C(x) = m(x)g(x). Also, let the product m(x)x dvde by g(x), then one has m(x)x = q(x)g(x) + d(x). Multplyng both sdes of last expresson by x 12 and usng x 23 = 1, one yelds d(x)x 12 + m(x) = (q(x)x 12 )g(x). The term (d(x)x 12 + m(x)), whch s a multple of g(x), s a systematc codeword gven by c(x) = d(x)x 12 + m(x) = p(x) + m(x), (3) 22 where p ( x ) 12 p x denotes the party-check polynomal [12] of a codeword and p GF(2) for The bnary systematc Golay code s practcally appled n the standard of the communcaton system for automatc lnk establshment (ALE) by the Unted States Department of Defense [13]. Now, let a 12-bt message be encoded nto a 23- bt systematc codeword and be transmtted through a nosy channel to obtan a receved word of the form r(x) = c(x) + e(x). 22 e ( x ) 0 e x s the error pattern polynomal, where e GF(2) for The syndromes of the code are defned by s r( ) c( ) e( ), (4) e( ) where Q j j 0 e j ( ) To smplfy the polynomal expressons above, the message, codeword, error pattern, receved word, and syndrome polynomals can be expressed as the bnary vector forms m = (m... m 1 m 0 ), c = (c c 1 c 0 ), e = (e e 1 e 0 ), r = c + e = (r r 1 r 0 ), and s = (s... s 1 s 0 ), respectvely. For the bnary systematc Golay code, t follows from [2, p85] that the systematc generator matrx G can be expressed as follows: 332 Vol., June 2013

3 Hgh Speed Decodng of the Bnary Golay Code, H. P. Lee et al. / I12 G PI , (5) where P and I 12 denote the 12 matrx and the 1212 dentty matrx, respectvely. The systematc vector form of the codeword can be obtaned by c mg ( p p p m m 0 ). (6) m The party-check polynomal h(x) = x 12 + x + x 7 + x 4 + x 3 + x 2 + x + 1 s a factor of x The 23 party-check matrx H can be expressed as follows: T I P 23 H, (7) where P T s a 12 transpose matrx of P and I s a dentty matrx. The H T denotes the 23 transpose matrx of H; I that s, H T. The vector form of the P 23 syndrome can be defned by s = rh T. (8) The followng lemma gven n [14] shows that the mappng between the syndromes and error patterns under the error-correctng capablty s one-to-one. Lemma 1. The mappng between the syndromes of a code and the correspondng error patterns of weght t s one-to-one. 3. Related work and theorems In ths secton, some related TLDAs and theorems are brefly ntroduced n order to develop the proposed MSDA. For the TLDA, the followng defnton, theorems and corollary gven n [9] are needed. Defnton 1. Let the Hammng norm or weght of a bnary vector a = (a n-1... a 1 a 0 ) be desgnated by w(a). The Hammng dstance between two bnary vectors a and b s defned by d(a, b) = w(a + b). Theorem 1. Let a = (a n-1... a 1 a 0 ) and b = (b n-1... b 1 b 0 ) be two bnary vectors, then w(a + b) = w(a) + w(b) 2 1 a b. (9) Corollary 1. Let a and b, defned n Defnton 1, be two bnary vectors. If a b = 0 for 1 n, then w(a + b) = w(a) + w(b). () Theorem 2. For the bnary systematc (n, k, d) QR codes, let us assume that there are v errors n the receved word, where 1 v t. All v errors occur n the n k party-check bts f and only f the weght of syndrome w(s) = v. As mentoned above, the error-correctng capablty of the bnary Golay code s t =3. The full lookup table (FLT) n the conventonal table lookup decodng 3 23 algorthm (CTLDA) requres n 1 = 2,047 syndromes and ther correspondng error patterns by Lemma 1. Each syndrome and error pattern needs 2 bytes and 3 bytes, respectvely, to store n the FLT. Thus, the total memory sze needed for the FLT s (2047 (2 + 3)) =,235 bytes. However, such a large memory s less effcent n practce. Recently, some useful TLDAs have been presented n the lterature to reduce the memory sze of the FLT. The LTDA [7] used the shftsearch method [4] to reduce the memory sze of the FLT and the bnary-search method to reduce the search tme n the FLT. The shft-search method deletes all v = t error patterns and keeps 1 v t 1 error patterns. Journal of Appled Research and Technology 333

4 Hgh Speed Decodng of the Bnary Golay Code, H. P. Lee et al. / t Therefore, the LTDA needs syndromes and ther correspondng error patterns; that s, the FLT needs 276 (2 + 3) = 1,380 bytes memory sze. Nevertheless, the memory sze of the FLT s stll large and the smulaton result shows that the average decodng tme of the proposed MSDA s about.6 tmes faster that the LTDA. To develop the followng TLDAs, Theorem 3 s useful to reduce the memory sze of the FLT. The well-known SDA wth RSLT [2, p9] has the followng useful theorem of the cyclc codes to dramatcally reduce the memory sze of the FLT. For more detaled proof of ths theorem, see [2, p8]. Theorem 3. Let s(x) be the syndrome polynomal correspondng to a receved polynomal r(x). Also, let r (1) (x) be the polynomal obtaned by cyclcally shftng the coeffcents of r(x) one bt to the left. Then the remander obtaned when dvdng xs(x) by g(x) s the syndrome s (1) (x) correspondng to r (1) (x). Decodng the bnary systematc Golay code by Theorem 3, the RSLT of the SDA only needs 1/23 memory sze of the FLT, namely,235/23 = 445 bytes memory sze [8]. Nonetheless, the memory sze of the RSLT s stll large and can be further reduced. Smlarly, by Theorem 3, r (j) (x) s the polynomal obtaned by cyclcally shftng the coeffcents of r(x) j bts to the left. Next, the syndrome dfference s d s defned by s d = s s for 1 N, where s s the syndrome of r, s s the syndrome n lookup table, and N s the table sze. ( j ) Smlarly, the syndrome dfference s s defned by s ( j ) d s ( j ) s for 1 j n, where s (j) d s the syndrome of r (j). By usng the property of the syndrome dfference, the RLT of the SWDA only requres 168 bytes. However, the decodng speed s not very outstandng accordng to the smulaton result. To overcome ths problem, the MSDA s presented by usng the syndrome that only one error occurred n the message part of the receved word. 4. MSDA for the bnary Golay code In ths secton, an effcent hgh-speed MSDA wth a MSLT for decodng the bnary systematc Golay code s developed to further reduce the decodng tme and the memory sze of the RLT. By Theorem 2 and (8), t s obvous that at least one error s n the message bts f the weght of syndrome w(s) > 3. Defnton 2. The message-syndrome lookup table (MSLT) s a table of all the error patterns occurred n the message part wth weght 1 v t / 2 together wth ther correspondng syndromes, where x denotes the smallest nteger less than or equal to x. Therefore, the novel MSLT merely conssts of 12 canddate syndromes whch are the syndromes of havng one error n the message part; that s, s s the syndrome of e whch s one error located at bt for The MSLT s lsted n Table 1. Syndrome Syndrome s 1 = (00) s 2 = (0) s 3 = (00) s 4 = (0) s 5 = (1) s 6 = (000) s 7 = (00) s 8 = (00) s 9 = (001) s = (01) s = (01) s 12 = (01) Table 1. MSLT for the Bnary Systematc Golay Code. To develop the proposed MSDA by usng the MSLT, the followng lemma and propertes are necessary. Lemma 2. For the bnary Golay code, let e be an error pattern, and let e p and e m be ts party-check part and message part, respectvely. Let s p and s m be the syndromes correspondng to e p and e m, respectvely. Assume that w(e) 3, then we have () w(s p ) = w(e p ); () w(s m ) d w(e m ), where d = 7; () ((s m << 12) + e m ) s always a codeword, where << denotes the logcal left shft operator n programmng or the extenson by zeros on the rght n mathematcs. Proof: By Equaton (8), () s obvous. Agan, by Equaton (8), gven that ((s m << 12) + e m )H T = ((s m I << 12) + e m ) I = (sm << 12) I P + e m P = s m P + s m = 0, ((s m << 12)+ e m ) s thus a codeword, whch proves (). Hence, w((s m << 12) + e m ) = w(s m ) + w(e m ) d, that s, w(s m ) d w(e m ). The proof s thus completed. 334 Vol., June 2013

5 Hgh Speed Decodng of the Bnary Golay Code, H. P. Lee et al. / By Lemma 2, f v t, then one can obtan the followng two propertes. Property 1. For the bnary Golay code, the weght of the syndrome dfference w(s d ) has the followng cases: Case 1: If w(s) 3, then all three errors are n the party-check part and no error s n the message part. Case 2: If w(s) > 3 and w(s d ) < 3, then only 1 error s n the message part. Case 3: If w(s) > 3 and w(s d ) > 3, than at least 2 errors are n the message part. Next, for Case 3 of Property 1, the r and s are cyclcally shfted left by bts, then one obtans r and s, respectvely. Thus, one yelds the followng property. Property 2 For the bnary Golay code, f w(s) > 3 and w(s d ) > 3, then the weght of s and s d have the followng cases: Case 1: If w(s ) = 2 or 3, then r has at least 2 errors n the message part and no error s n the party-check part. d Case 2: If w(s ) > 3 and w( s ) 2, then r has at least 2 errors n the message part. d Case 3: If w(s ) > 3 or w( s ) > 3, then 2 errors are n the message part and one of them s at the poston of bt 0. By usng the MSLT, the decodng process of the MSDA s as follows: (1) Gvng a receved vector r. (2) Compute the syndrome s = rh T and ts weght w(s). (3) If w(s) 3, then return c = r (s << 12), and go to step 13. (4) For 1 12, compute the syndrome s d = s s and ts weght w(s d ). (5) If w(s d ) 2, then return c = r (s d << 12) (1 << ( 1)), and go to step 13. (6) Cyclcally shft r and s left by bts obtanng r and s, and then compute w(s ). (7) If w(s ) = 2 or 3, then return c = ((r (s << 12)) << 12) ((r (s << 12)) >> ), and go to step 13. (8) For 1 12, compute the syndrome d = s s and ts weght w( s ). d s d (9) If w( s ) = 1 or 2, then return c = ((r ( s d << 12) (1 << ( 1))) << 12) ((r d ( s << 12) (1 << ( 1))) >> ), and go to step 13. () Compute r = r 1 and s = s s 1. For 2 12, compute the syndrome s d = s s and ts weght w(s d ). (12) If w(s d ) = 1, then return c = r (s d << 12) (1 << ( 1)). (13) Stop. An example s gven below. By (6), a message vector m = ( ) s encoded nto a bnary systematc Golay code c = ( ). Assume that a nose vector s e = ( ), then the receved word becomes r = c + e = ( ). The decodng steps proceed as follows: (1) The receved vector s r = ( ). (2) Compute the syndrome s = (00000) and ts weght w(s) = 5. (3) Gven that w(s) > 3, go to step 4. (4) For = 1, compute the syndrome s d1 = s s 1 = () and ts weght w(s d1 ) = 7. Journal of Appled Research and Technology 335

6 Hgh Speed Decodng of the Bnary Golay Code, H. P. Lee et al. / (5) Gven that w(s d1 ) > 2, go to step 4. repeatng steps 4 and 5 (4) For = 12, compute the syndrome s d12 = s s 12 = (00) and ts weght w(s d12 ) = 7. (5) Gven that w(s d12 ) > 2, go to step 6. (6) Cyclcally shft r and s left by bts obtanng r = ( ) and s = (0), and then compute w(s ) = 7. (7) Gven that w(s ) 2 or 3, go to step 8. (8) For = 1, compute the syndrome s d1 = s s 1 = (001) and ts weght d1 w( s ) = 7. d1 (9) Gven that w( s ) > 2, go to step 8. repeatng steps 8 and 9 (8) For = 12, compute the syndrome s d12= s s 12 = (000) and ts weght d12 w( s ) = 5. d12 (9) Gven that w( s ) > 2, go to step. () Compute r = r 1 = ( ) and s = s s 1 = (). For = 2, compute the syndrome s d2 = s s 2 = (00) and ts weght w(s d2 ) = 5. (12) Gven that w(s d2 ) > 1, go to step. repeatng steps and 12 (13) For = 8, compute the syndrome s d8 = s s 8 = ( ) and ts weght w(s d8 ) = 1. (12) (14) If w(s d8 ) = 1, then return the correct codeword c = r1 (s d8 << 12) (1 << ( 1)) = ( ) ( ) (000000) = ( ). Go to step 13. (13) (15) Stop. Therefore, the correct message can be retreved from the receved vector. 5. Smulaton results The proposed MSDA has been programmed n C++ language. On an Intel Q6600 PC, all codewords wth up to three errors, namely C 8, receved vectors, were created to check every possble error pattern. The detaled decodng tmes for decodng the bnary systematc Golay code wth dfferent decodng algorthms are gven n Table 2. The proposed MSDA has the fastest average decodng tme and works perfectly wth an average decodng tme of μs per error pattern. Algorthms Proposed MSDA SDA wth RSLT FLTDA wth table SWDA wth RLT Improved EDA Number of errors Average Table 2. Computer Decodng Tme for Decodng the Bnary Golay Code (n μs). From Table 2, the proposed MSDA has the fastest average decodng tme and s about 44.5 tmes faster than the mproved EDA. Furthermore, we substtuted the proposed MSDA for the mproved EDA n the four-error soft-decson decoder []. Apparently, the decodng speed of the softdecson decoder was about 44.5 tmes faster than before. The decodng tme of the two hard-decson decoders wth respect to the SNR values s shown n Fgure Conclusons Ths paper presented a smple and more effcent decodng algorthm for decodng the bnary systematc Golay code. The key deas of the MSDA are based on the propertes of cyclc codes, the one-error syndrome n the message part, the weght of syndrome, the weght of syndrome 336 Vol., June 2013

7 Hgh Speed Decodng of the Bnary Golay Code, H. P. Lee et al. / dfference, Property 1, and Property 2. The MSLT of the MSDA merely requres a very small memory sze of 24 bytes. These facts lead to a substantal mprovement n decodng the bnary systematc Golay code. Smulaton results show that, followng the new decodng strateges, the proposed MSDA acheves the best effcency. Hence, t s naturally sutable and proper for DSP or embedded system software mplementaton. [7] Y. H. Chen et al., Effcent decodng of systematc (23, 12, 7) and (41, 21, 9) quadratc resdue codes, J Inf Sc Eng, vol. 26, no. 5, pp , 20. [8] T. C. Ln et al., On the decodng of the (24, 12, 8) Golay code, Inf Sc, vol. 180, no. 23, pp , 20. [9] T. C. Ln et al., Hgh speed decodng of the bnary (47, 24, ) quadratc resdue code, Inf Sc, vol. 180, no. 20, pp , 20. [] S. Ln and E. J. Jr Weldon, Long BCH codes are bad, Inf Contr, vol., no. 4, pp , [] S. I. Chu et al., Fast decodng of the (23, 12, 7) Golay code wth four-error-correctng capablty, Eur Trans Telecomm, vol, 22, no. 7, pp , 20. [12] C. A. Vázquez-Fernández and G. Vega-Hernández, On the Weght Dstrbuton of the Dual of some Cyclc Codes wth Two Non Conjugated Zeros, J Appl Res Technol, vol.9, no.1, pp , 20. [13] MIL-STD-188/141B. Department of Defense Interface Standard: Interoperablty and Performance Standards for Medum and Hgh Frequency Rado Systems. STD+( )/MIL_STD_188_141B_1703. Fgure 1. Decodng Tme of Two Hard-Decson Decoders at SNR of 0~6 db. References [14] I. S Reed et al., The algebrac decodng of the (41, 21, 9) quadratc resdue code, IEEE Trans Inf Theory, vol. 38, no. 3, pp , [1] M. Golay, Notes on dgtal codng, Proc. IRE, vol. 37, p. 657, [2] S. Wcker, "Error Control Systems for Dgtal Communcaton and Storage, Prentce Hall: Upper Saddle Rver, NJ, [3] M. Ela, Algebrac decodng of the (23, 12, 7) Golay code, IEEE Trans Inf Theory, vol. 33, no. 1, pp , [4] I. S. Reed et al., Decodng the (24, 12, 8) Golay code, IEE P-COMPUT DIG T, vol. 137, no. 3, pp , [5] Y. H. Chen et al., Algebrac decodng of quadratc resdue codes usng Berlekamp-Massey algorthm, J Inf Sc Eng, vol. 23, no. 1, 2007, pp [6] H. Chang et al., A weght method of decodng the (23, 12, 7) Golay code Usng Reduced Lookup Table, 2008 Internatonal Conference On Communcaton, Crcuts and Systems (ICCCAS 2008), Xamen, Chna, 2008, pp Journal of Appled Research and Technology 337