Simplified Algorithm and Hardware Implementation for the (24, 12, 8) Extended Golay Soft Decoder Up to 4 Errors

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1 The Interntonl Arb Journl of Informton Technology, Vol., No., Mrch 04 Smplfed Algorthm nd Hrdwre Implementton for the (4,, 8 Extended Goly Soft Decoder Up to 4 Errors Dongfu Xe College of Mechncl nd Electrcl Engneerng, Jxng Unversty, Chn Abstrct: The purpose of ths pper s to present soft decodng lgorthm orentng to hrdwre mplementon for the (4,, 8 Goly code, nd mplement such n lgorthm n Feld Progrmmng Gtes Arry (FPGA. The soft decodng lgorthm devsed by Ln s et l. for the (4,, 8 s not sutble for hrdwre mplementton becuse of nvolvng mny rthmetc opertons, such s multplctons nd dvsons. To remove the complexty rthmetc opertons, the bsolute vlue of the chnnel nformton nsted of the bt-error probblty s employed to ndcte the chnnel confdence. Moreover, the rchtecture developed for relzng the proposed lgorthm s verfed n FPGA prototype. The BER performnce obtned by the proposed decodng lgorthm equls to the one obtned by Ln s lgorthm. At the sme tme, 5% of ddtons, 00% of multplctons, nd 00% of exponents re reduced for computng the chnnel confdence. In ddton, -db codng gn cn be obtned by Ln's lgorthm t the cost of the double of hrdwre complexty compred to El's lgorthm. Keywords: Soft decodng lgorthm, goly code, FPGA. Receved July 7, 0; ccepted My, 0; publshed onlne Jnury 9, 03. Introducton In 949, perfect lner error-correctng code nmed Goly code ws frst dscovered by Goly [4]. Lter, severl mproved lgorthms decodng the Goly code effcently re gven such s the lgebrc decodng lgorthm [], the stndrd rry decodng [6], nd so on. Moreover, the (4,, 8 extended Goly code s generted by dded prty chec bt to Goly code for the deep spce communcton [9]. Nowdys, ths code s lso used for hshng s shown n [3]. To mprove the decodng performnce, Boyrnov et l. [] proposed decodng lgorthm up to four errors for specl constructon of the (4,, 8 extended Goly code. Unfortuntely, the smulton results of ths method show tht few of the four-error ptterns cn be corrected. More recently, decodng lgorthm proposed by Ln et l. [5] ws developed to fcltte further decodng of the (4,, 8 Goly code by usng Reed s method [8] to estmte the ndvdul bt-error probbltes on receved word. It s shown tht nerly -db code gn s obtned compred to hrd decodng lgorthm. However, the reserch of the (4,, 8 Goly code soft decodng prmrly focuses on the desgn of the lgorthm. Few topcs bout hrdwre mplementton re dscussed. It s obvous tht hrdwre mplementton plys n mportnt role n pplcton. The dffcultes of hrdwre mplementton for the decodng lgorthm proposed n [7] re tht: The complexty of computng the Emblemtc Probblty Vlue (EPV defned n (6 of [7], whch requres mny rthmetc opertons, such s multplctons nd dvsons n the decodng lgorthm. In ths pper, smplfed decodng lgorthm whch sgnfcntly reduces the complexty of computng the EPV s proposed. By utlzng the cyclcl chrcterstcs of the Goly (4,, 8 code, decodng rchtecture bsed on the smplfed decodng lgorthm s nvestgted. Menwhle, the Feld Progrmmer Gte Arry (FPGA prototype of the decoder rchtecture s mplemented nd nlysed n ths pper.. Prelmnres The soft decodng lgorthm proposed n [8], s summrzed s follows: Step : Obtnng the error pttern E P from the receved codeword r bsed on the lgebrc Hrd Decodng (HD lgorthm for the (3,, 7 Goly code proposed by El []. Step : The decoder detects four errors usng the lgorthm proposed by Reed et l. [7]. Step 3: After detectng four errors, the sx possble error vectors, tht s, ẽ, ẽ, ẽ 3, ẽ 4, ẽ 5 nd ẽ 6, re determned by serchng looup tble of weght-7. C, C, C 3, C 4 nd C 5 ndcte fve weght-7 code words. ẽ for 6 re the possble four error pttern, nd Loc(w {w 0} {0,,,n-} s set

2 The Interntonl Arb Journl of Informton Technology, Vol., No., Mrch 04 of the loctons of non-zeros components n vector w{w 0,w,,w n- }. Step 4: The ndvdul bt-error probbltes, nmely p 0, p,, p, re frst determned by n extenson of des gven by Reed [8] from receved word of length 4, fter the determnton, of ll possble error ptterns. The probblty p for 0 3 s gven by: p,, < x < exp( A x Where x, A, nd re the prtculr relztons of dscrete-tme condtonl Gussn rndom sequence, the receved bt mpltude estmted from the sequence x, nd the chnnel nose power, respectvely. Step 5: The EPVs of the sx possble error vectors re defned s: Pˆ,,,,6 ', p Loc ( e ɶ ( Step 6: The most lely weght-4 error pttern correspondng to the mxml vlue of s chosen. 3. A Smplfed Soft Decodng Algorthm for the (4,, 8 Extended Goly Code Up to Four Errors Obvously, note tht through obtnng the EPV of ech error pttern, fndng the most lely four error pttern s the ey pont to correctng four error ptterns. The vlue EPV s defned s the product of four ndvdul bt-error probbltes correspondng to the loctons of the respectve errors. Nevertheless, the process of obtnng the EPV results n hghly computtonl complexty nd consequently severely lowers the decodng speed for mny rthmetc opertons nvolved, such s multplctons nd dvsons. Therefore, ths pper proposed smple scheme to estmte the EPV vlue whch s redefned s the logrthm of the product of four ndvdul precson mesurement of the chnnel confdence rther thn tht used to be defned prevously, nd where the precson mesurement s defned by the rto of the bt-error probblty nd the bt-correct probblty. Now, redefne the chnnel confdence p s p log(p E /P C.P E, clled the bt-error probblty, s gven by the followng formul: E { K ( x } P P A sgn x A A sgn( x x A x + exp( exp( ' ( (3 P C, clled the bt-correct probblty: P P{ A sgn( x A x } C A sgn( x x exp( A x exp( Where the defnton of, A, x, s the sme s the ones mentoned bove, nd the sgn (x denotes the sgn functon on x, respectvely: P ( P log P / E A x exp( log ( A x exp( A x Now, combned wth equton 5, the EPV descrbed n equton cn be modfed s: Loc ( e ɶ Loc ( e ɶ C p A x Where s clled the modfed EPV. To cheve the most lely weght-four error pttern, the mxmum of the EPV, denoted s mx( ˆp, needs to be clculted s follows: A mx ( Pˆ mn ( x, 6 Loc ( e ɶ (7 If nd A re constnts n receved word, the gretest EPV cn be redly smplfed s: mx( P ˆ m n ( x, 6 Loc ( e ɶ (4 (5 (6 (8 In fct, nd A re ssumed to be constnts mong receved code word n rel communcton systems. The number of opertons requred by usng equtons, 7 nd 8 for computng the EPV nd the mxmum of the modfed EPV re shown n Tble, respectvely. One observes from Tble tht 5% of ddtons, 57% of multplctons, nd 00% of exponents re reduced by usng equton 7 when compred wth the decodng lgorthm of Ln et l. [5]. Specfclly, f nd A re constnts n receved word from equton 8, the number of ddtons needed to compute the mxmum of the EPV s eghteen. Now, the soft decodng lgorthm proposed n [5] cn be modfed s follow: Step : The sme opertons re performed tll the determnton of ll possble error ptterns n the soft decodng lgorthm descrbed n [5]. Then, n vew of nd A re ssumed to be constnts mong recevng code word n rel

3 Smplfed Algorthm nd Hrdwre Implementton for the (4,, 8 Extended Goly Soft... 3 communcton systems, let the chnnel mesurement be: p x (9 The EPVs of the sx possble error vectors re computed s: p Loc( e ɶ x Loc( e ɶ (0 detled mplementton for the hrd decoder s descrbed n [0]. The bloc of serchng the most possble error pttern s used for fndng the most possble four errors vector. The bloc clled detectng four errors montors n event tht the receve code word s occurred by four errors pttern. The rchtecture for detect four errors s devsed n [7]. Is Frst In Frst Out (FIFO buffer n Fgure. Step : The most lely weght-4 error pttern correspondng to the mxml vlue of P ˆ s chosen: mx( P ˆ m n( x, 6 Loc ( e ɶ ( Tble. The number of opertons requred by usng equton nd equtons 7 nd 8 proposed n ths pper for computng EPV. Operton Addton Multplcton Dvson Exponent Formul ( Formul ( Formul ( The Bt-Error Rte (BER performnces n AWGN re llustrted n Fgure. It s seen obvously tht the BER performnce obtned by the Smplfed Decodng Algorthm (SDA equls to the one obtned by Ln s lgorthm proposed n [5]. Bt Error Problty Eb/No(dB Fgure. BER performnce of the (4,, 8 extended Goly code n AWGN. 4. Hrdwre Implementton for the (4,, 8 Extended Goly Code Up to Four Errors 4.. The Top Level of Soft Decoder In ths secton, decoder hrdwre mplementton wth ppelne rchtecture for the (4,, 8 extended Goly code up to four errors s presented. Ths decoder cn ccept new receved messge every twelve clocs. In other words, the tme of ech ppelne stge s twelve clocs. Concernng the decodng lgorthm descrbed n secton 3, the functonl bloc dgrm of soft decoder s shown n Fgure. The bloc nmed clcultng p computes the chnnel mesurement of ech bt. The bloc of hrd decoder obtns E P bsed on the hrd decodng lgorthm proposed by El []. A Fgure. The functonl bloc dgrm of soft decoder. 4.. Clcultng p The bloc of clcultng p computes the chnnel mesurement p descrbed n Equton 9, before serchng the most possble error pttern Serchng the Most Possble Error Pttern If r occurred by four errors s detected by the bloc of detectng four errors, the bloc of serchng the most possble error pttern strts to wor. Once recevng E P from the hrd decoder, the bloc of serchng the most possble error pttern reds p corresponded to E P from FIFO. As shown n Fgure 3, twenty-two serch engnes produce nd serchng the 53 vld code words n twelve clocs. 53 vld code words cn be obtned by vld code words v{v } ccordng to the chrcterstcs of the cyclc codes. Ech v genertes twenty-two vld code words wthout overlp. The detled serchng of ech serch engne shown n Fgure 4 s descrbed s: through The vector s obtned from the XOR of E P nd v. Ech element of controls ts correspondng MUX, whose output s 0 for the control lne equls to 0, nd output s p for the control lne equls to. The results of twenty-three MUX re dded together to get the vlue, whch then s compred to the of the ntl vlue of 0: P ˆ f Pˆ > P ˆ otherwse e ɶ f Pˆ > P ˆ e ɶ otherwse (

4 4 The Interntonl Arb Journl of Informton Technology, Vol., No., Mrch 04 Fgure 3. The rchtecture of serchng the most possble error pttern. Fgure 4. The rchtecture of serch engne. Then nd ts correspondng re stored n the regsters z-, respectvely. It s noted tht durng the perod of genertn g one, where one new vlue v s generted from the prevous v by left shftng one bt concurrently to obtned the next. After twelve clocs, the fnl mxmum correspondng to the EP s generted. The outputs of twenty-two serch engnes re dvded nto two groups, whch re prllel-to-serlzed nto {0 0}. {0 0} nd {0 0}. Then the vlues nd {0 } re obtned by comprng the nd the {0,,,0}: Then vlue Loc( E P ˆ ˆ ˆ P f P > P otherwse e f Pˆ P ˆ ɶ > e ɶ otherwse s compred wth of ntl x, f (3 s greter thn, then Pˆ P ˆ, else eeps unchnged. The mxmum element of whch s equls to mx ( Pˆ s obtned fter the twelve clocs. 5. FPGA Prototype To verfy the decodng rchtecture proposed n secton 5, two FPGA prototypes re mplemented n xc6slx50. These prototypes re desgned for the SDA nd El s lgorthm respectvely. Ln s lgorthm s not mplemented n the FPGA pltform, becuse ts some rthmetc opertons re not sutble for hrdwre mplementton. The worng frequency of ech FPGA prototype s 00MHz. The hrdwre cost nd the correspondng thoughput of ech decoder re gven n Tble. The BER performnce comprsons of hrdwre smultons between the SDA nd El s lgorthm re descrbed n Fgure 5. It s llustrted clerly tht SDA obtns performnce gn bout - db over El's lgorthm lthough ts hrdwre complexty doubles tht of the ltter. Tble. The hrdwre cost nd the correspondng thoughput of decoders bsed on the SDA nd Chse s lgorthm. LUTs Regsters Throughput The SDA Mbps El s lgorthm Mbps Rto.47.5 Bt Error Problty Eb/N0 Fgure 5. The BER performnce comprson of softwre smulton nd hrdwre smulton between the SDA nd El s lgorthm. 6. Conclusons A smplfed soft decodng lgorthm up to four errors for the (4,, 8 extended Goly code s proposed n ths pper. Smulton results show tht the codng gn obtned by the smplfed decodng lgorthm

5 Smplfed Algorthm nd Hrdwre Implementton for the (4,, 8 Extended Goly Soft... 5 over el's lgorthm s the sme s the one cheved by Ln s lgorthm. At the sme tme, 5% of ddtons, reduced for computng the chnnel confdence. In lne wth ths smplfed lgorthm, ths pper nvestgtes n effcent hrdwre mplementton on FPGA pltform, nd presents the detled rchtecture of the soft decoder. Compred to El's lgorthm, -db codng gn cn be obtned by Ln's lgorthm t the cost of the double of hrdwre complexty. Acnowledgements The uthor would le to thn Dr. Ln wng nd Dr. Hsn-Chu Chng for ther constructve suggestons on mprovng the qulty nd the presentton of ths pper. Dongfu Xe receved hs BS degree n electronc engneerng from Hngzhou Dnz Unversty, Chn, n 006, nd PhD n crcut nd system from Xmen Unversty, Chn, n 0. He s currently n ssstnt professor n the College of Mechncl nd Electrcl Engneerng, Jxng Unversty, Chn. Hs reserch nterests ncludes error control codng, s gnl processng, nd VLSI desgn. References [] Boyrnov I., Mrtn I., nd H-onry B., Hgh- Speed Decodng of Extended Goly Code, IEE Proceedng of Communcton, vol. 47, no. 6, pp , 000. [] El M., Algebrc Decodng of the (3,, 7 Goly Code, IEEE Trnsctons on Informton Theory, vol. 33, no., pp. 50-5, 987. [3] El-Qwsmeh E., Sfr M., nd Knn T., Investgton of Goly Code (4,, 8 Structure n Improvng Serch Technques, the Interntonl Arb Journl of Informton Technology, vol. 8, no. 3, pp. 65-7, 0. [4] Goly M., Notes on Dgtl Codng, n Proceedngs of IRE, vol. 37, pp. 657, 949. [5] Ln T., Truong T., Su W., Shh P., nd Dubney G., Decodng of the (4,, 8 Extended Goly Code Up to Four Errors, Insttuton of Engneerng nd Technology Communctons, vol. 3, no., pp. 3-38, 009. [6] Mcelece R., Theory of Informton nd Codng, Addson Wesley, USA, 977. [7] Reed I., Yn X., Truong T., nd Holmes J., Decodng the (4,, 8 Goly Code, n Proceedngs of IEEE Computers nd Dgtl Technques, vol. 37, no. 3, pp. 0-06, 990. [8] Reed I., Sttstcl Error Control of Relzble Bnry Symmetrc Chnnel, Techncl Report, Msschusetts Insttute of Technology, USA, 959. [9] Wcer S., Error Control Systems for Dgtl Communcton nd Storge, Prentce-Hll, USA, 995. [0] We S. nd We C., On Hgh-Speed Decodng of the (3,, 7 Goly Code, IEEE Trnsctons Informton Theory, vol. 36, no. 3, pp , 990.