Commucatos ad etwork, 20, 3, 7-22 do:0.4236/c.20.3003 Pubshed Oe February 20 (http://www.scrp.org/joura/c) Statstca Recogto Method of Bary BCH Code Abstract Jafeg Wag, Yag Yue, Ju Yao Isttute of Eectroc Egeerg Cha Academy of Egeerg Physcs, Mayag, Cha E-ma: hugeghost@hotma.com Receved October 30, 200; revsed Jauary 5, 200; accepted Jauary 6, 20 I ths paper, a statstca recogto method of the bary BCH code s proposed. The method s apped to both prmtve ad o-prmtve bary BCH code. The bock egth s frst recogzed based o the cycc feature uder the codto of the frame egth kow. Ad the caddate poyomas are acheved whch meet the restrctos. Amog the caddate poyomas, the most optma poyoma s seected based o the mmum rue of the weghts sum of the sydromes. Fay, the best poyoma was factorzed to get the geerator poyoma recogzed. Smuato resuts show that the method has strog capabty of at-radom bt error. Besdes, the agorthm proposed s very smpe, so t s very practca for hardware mpemetato. Keywords: Bary BCH Code, Bd Recogto, Code Legth, Geerator Poyoma. Itroducto Wth the deveopmet of the Commucato Coutermeasures, the methods of the commucato coutermeasures have trasferred from the sga eve to the formato eve. The bd recogto of chae codg s the foudato of gettg the message, therefore has become the key techoogy the area of the Commucato Coutermeasures ad gets more ad more atteto. Besdes, the bd recogto of chae codg aso has mportat appcatos cooperatve areas such as the teget commucato. The ear chae codg has two categores: bock code ad covoutoa code. We ca see that most researches focus o the bd recogto of covoutoa code [-2], whe terature about the bd recogto of bock code s rare. Lterature [3] gves a bd recogto method of the bock code wth ow ecodg rate, ts esseta prcpe s to resove equatos to recogze the sydrome matrx, ad the get the geerator matrx. Lterature [4] dscusses a method of bd recogto of prmtve RS code, ts esseta prcpe s usg reduced row echeo form of the matrx (RREF), faut-toerat matrx decomposg (FTMD) ad Gaos fed Fourer trasform (GFFT) to recogze ecodg parameters ad sydrome matrxes. Both methods eed matrx computatos o the fte fed, whe code egth s og, the EMS memory eeded w be very arge, so t s ot appcabe to hardware mpemetato. At the same tme, the error code capabty s ot dea too. I ths paper we study the bary BCH code whch s a mportat subcass of cycc codes. Accordg to the crcuar feature, we proposed a smpe statstca recogto method uder the codto that the frame egth s kow. The method has strog at-error capabty. Besdes, there s o eed for matrx computato; therefore t s sutabe for hardware mpemetato. The reset of ths paper s orgazed as foows: Secto troduces the basc feature of bary BCH code; Secto 2 dscusses the recogto method of the bock egth; Secto 3 dscusses the method of recogto of the geerator poyoma; Secto 4 observes the recogto performace of ths agorthm through smuato; fay, the paper s brefy summarzed. 2. Bary BCH Code Here we smpy troduce the features of bary BCH code. The defto of bary BCH code ad the detas of ecodg ad decodg are dscussed terature [5]. Cosder (,k) BCH code defed o GF(2), s the bock egth ad k s the message egth. Let m m0, m,, mk be the message word before ecodg, c c 0, c,, c be the code word after ecodg, because t s the cycc code, the message word Copyrght 20 ScRes.
8 J. F. WAG ET AL. ad code word each has a correspodg message poyoma ad code poyoma defed o GF(2), as foows k k2 m x m x m x m xm () 0 k2 k2 c x c x c x c xc (2) 0 k2 From terature [5], we kow that the (,k) bary BCH code has foowg features, ) Bock egth equas 2 m or s a factor of 2 m, where m 3. If equas 2 m, the t s prmtve bary BCH code; otherwse, t s o- prmtve oe. 2) The code word c satsfes the crcuar feature, that s, after crcuar shftg the code word c for j tmes, get the code word j, j,, k, 0,, j3, j2 c c c c c c c ad c aso beogs to (, k) bary BCH code set. 3) m x ad c x satsfy foowg reatoshp, where (2), mxgx c x k (3) g x s the geerator poyoma defed o GF k k g x x g x g x (4) k 4) There exsts sydrome poyoma h x defed o GF(2), k k h x x h x h x k whch satsfes foowg reatoshps, (5) h x c x 0mod x (6) hxgx x 5) g x s rreducbe. 3. Recogto of Bock Legth (7) The recogto of the bary (, k) BCH code s to recogze three parameters: the bock egth, the sydrome word egth k, ad the geerator poyoma g x. But we ca see from formua (4) that the order of g x s -k, the as og as the bock egth ad the geerator poyoma g x are recogzed, the vaue of k ca be ascertaed. The oy the bock egth ad the geerator poyoma g x eed to be recogzed. I ths secto we troduce the recogto of bock egth, ad ext secto we troduce the recogto of geerator poyoma g x. The recogto s based o the assumpto that frame egth f s kow. Ths assumpto s ratoa, because usuay practce, frame head s ot ecoded, so t s easy to get. Because f s kow, ad there exsts at east two code words oe frame, so we coud get foowg cocusos: f ) 3, 2, where meas foor fucto. 2) f s dvsbe by, that s, s a factor of f. f The factor umbers of f the rag 3, 2 may ot be oy oe, assume s a factor of f ad the there are two stuatos, ) Uder ths stuato, f we take as the bock egth, we coud get code words. Assume cp x s the code poyoma of the p th code word, the through eft crcuar shftg for j tmes we get j code poyomas cp x, cp2 x,, cpj x, where j. Because the bary BCH code satsfy the crcuar feature, f there s o error code c p, the the code words correspodg to cp x, cp 2 x,, cpj x ad the code word correspodg to c p x beog to the same (, k) bary BCH set, so ther geerator poyomas are the same. Accordg to formua (3) whch s the reatoshp betwee the code poyoma ad the geerator poyoma, we kow there s a commo factor for cp x, cp x, cp2 x,, cpj x. Let c x c x, the the foowg reatoshp exsts, p0 p gcd cp0 x, cp x,, cpj x j (8) We ca the code word that satsfes fomua (8) as a vad code word. Assume there are c vad code words amog the code words. Obvousy, whe there s o error code, c, that s, the percetage of the vad code words of a the code words f c s, c fc (9) 2) Uder ths stuato, f we take as the bock egth, the bockg s wrog. Assume we get code words, there exst code words that do ot satsfy formua (8) evtaby, so c, that s c fc (0) Above a, uder the codto that o error code exsts, f take formua (8) as the rue to judge f the code word s vad or ot, the for a the possbe bock egths, whe, the percetage of vad code words s fc ; whe, the percetage of vad code words s fc. Of course, whe there are error codes Copyrght 20 ScRes.
J. F. WAG ET AL. 9 exst, eve f, f c coud be ess tha. But we ca predct that f c shoud get the maxmum whe. I addto, accordg to feature (), s odd, so we coud get the recogto formua of the bock egth, arg max f () 3, f 2 rem,2 rem f, 0 where meas the maxmum operato; rem f, meas the remader of f dvded by. The rue to judge the vadty of the code word s formua (8). The we ca get the recogto process of bary BCH code bock egth : ) Let 3 ad taze the vaue of j; 2) If ca ot dvde f, the tur to (6); 3) Let be the bock egth ad get code words; 4) Accordg to formua (8), cacuate the umber of vad code words c ; c 5) Compute fc ad save; c max 6) 2 ; 7) If f 2, tur to (2); 8) Compare a the saved f c, the estmato of s that made f c has maxmum vaue; 9) The recogto process s over. We ca see from step () ad step (8), the recogto capabty of bock egth s reevat to ad, j the more the vaue of ad j, the better the recogto capabty at the frst gace; but arge vaues mea that the egth of the receved code stream seres s aso arge, ad the computato compexty w crease, so we must cosder ths tradeoff accordg to the practca stuatos. 4. Recogto of Geerator Poyoma I ths secto, we dscuss how to recogze the geerator poyoma g x uder the codto of the bock egth s rghty recogzed. Assume we receve code words, cp x s the code poyoma for the p th code word, ad assume that there s o error code. Accordg to secto II, eft crcuar shft cp x to get code poyomas cp x, cp 2 x,, cp x, ad the geerator poyomas of these code poyomas ad cp x are the same. I other words, the geerator poyoma g x s a commo factor of p x, cp x, c p 2 x,, x. Aso et c x c x, ad cp p c p0 p gcd,,, f x c x c x c p0 p pj x (2) f p x s the mutpe of The g x. From the receved code words, we ca get M ( M ) dfferet poyomas accordg to formua (2), cosder there are error codes, these M poyomas beog to oe of the four stuatos beow: ) Equa ; 2) ot equas, but ot equas g x ; 3) Equa g x ; 4) Equa the mutpe formua of g x. mutpe formua of g x, aso ot the Stuatos () ad (2) ustrate there are error codes exst. Stuatos (3) ad (4) ustrate there are o error codes exst code words, or the error codes costtute aother code words the same sub code set. We ca get the geerator poyoma from the M caddate poyomas foowg three steps beow. Step. Accordg to the costrat codtos of g x satsfed, remove the poyomas that do ot satsfy the codtos Accordg to the feature of the bary crcuar code, g x eeds to satsfy the foowg restrctos [5], ) q 2) g x x, where q s the postve teger ad q 3) g x ca dvde x Accordg to these three restrctos to remove the poyomas uder stuatos () ad (2). Step 2. Accordg to the mmum rue of the weghts sum of the sydromes choose the most optma poyoma Assume after the step there are L caddate poyo- mas rema, ad g x (, 2,, L ) s the th poyoma, ad ts correspodg sydrome poyoma s, h x x g x (3) The jth sydrome of the code word s, rj h xcj xmod x (4) The code weght s, w j j weght r (5) For a the receved code word compute the sydrome code weght, w wj, 2,, L (6) j The caddate poyoma made most optma poyoma, deotes as g b x arg m g ( x),,2,, L mmum s the g x, w b w (7) Step 3. From the most optma poyoma g get the estmated geerator poyoma g x. 0 b x to Copyrght 20 ScRes.
20 J. F. WAG ET AL. Whe the reorgazato s correct, the most optma poyoma gb x s ot ecessary equas g x, but t s deftey the mutpe of g x, so further steps are eeded usg gb x to estmate the geerator poyoma. Frsty accordg to the rues [6], judge whether g x s reducbe or ot; f t s rreducbe, the b 0 b g x g x (8) If t s reducbe, get a the rreducbe factor of v gb x that satsfy step, the foow step 2 to get the most optma poyoma as the estmated geerator poyoma g0 x. The method of factorzato ca drecty get from the rues [6], we do ot commet o ths because the mtato of space. From the above recogto process, theoretcay, as og as there s a code word wthout error code, t s of very hgh possbty to correcty recogze the geerator poyoma, the ths recogto method has strog at-error capabty. Fgure. (5,) code bock egth recogto capabty. 5. Smuato The smuato has three steps: frst, we smuate for the recogto capabty of the bock egth; secod, we smuate for the recogto capabty of the geerator poyoma; thrd, accordg to the frst two steps we get the tota recogto capabty. I the smuato of the recogto capabty, we ru the smuato 000 tmes for (5,) prmtve bary BCH code ad (2,2) o-prmtve bary BCH code respectvey to get the statstca correct recogto rate. The geerator poyoma of (5,) prmtve bary 4 BCH code s g x x x; the geerator poyoma of (2,2) o-prmtve bary BCH code s g x x words oe frame, ad we choose the crcuar shftg tme as for recogzg the bock egth. 9 3 x ;assume that there are fve code 5.. Smuato Resuts Fgure 2. (5,) code geerator poyoma recogto capabty. Fgure 3. (2,2) code bock egth recogto capabty. The recogto smuatos are doe accordg to the method descrbed Sectos 2 ad 3. The resuts are show Fgures -6. The tota recogto capabty s computed accordg to the bock egth recogto capabty ad geerator egth recogto capabty, the computato formua s as foows, p p p2 (9) where p s the tota correct recogto rate, p s the correct recogto rate of bock egth, p2 s the correct recogto rate of geerator poyoma. Fgure 4. (2,2) code geerator poyoma recogto capabty. Copyrght 20 ScRes.
J. F. WAG ET AL. 2 Fgure 5. (5,) code tota recogto capabty. (5,) code, the whe the code words s the same, through a crcuar shft, the percetage of satsfyg crcuar feature woud be hgher tha (5,) code, eve f ot rght bockg. 2) The frame egth of (2,2) code s 05, t has 6 odd factor: 3, 5, 7, 5,, 35; whe the frame egth of (5,) code s 75, t has 4 odd factor: 3, 5, 5, 25. Obvousy, whe the code word umber s the same, the more the factor, the worse the recogto capabty. Of course, the recogto capabty s aso reevat to the structure of the code, (5,) code s prmtve bary BCH code ad (2,2) code s o-prmtve bary BCH code. But we ca see from the smuato resut, athough whe the umber of code words s the same, the recogto capabty of (2,2) code s worse tha (5,) code, but we ca crease code words to reach the same recogto capabty. 6. Cocusos Fgure 6. (2,2) code tota recogto capabty. ote: I pctures, BER represets Bt Error Rate, CRR represets Correct Recogto Rate, ad Code repesets Code-words umber. 5.2. Aayss of Smuato Resuts From the resuts of the smuato, we kow that o matter the recogto capabty of the bock egth, the recogto capabty of the geerator poyoma, or the tota recogto capabty, w crease wth the umber of code words that are used for the recogto process. The recogto capabty w aso crease wth the crcuar shftg tme. Because of the mtato of space, the smuato resuts are ot gve here. Otherwse, we ca see that uder the codto of the umber of code words s the same, the dfferece betwee the recogto capabty of these two kds of code words s reatvey arge. For (5,) code, whe the error code rate s 0%, 00 code words are used, ts tota recogto capabty coud reach above 90%. Whe for (2,2) code, whe the error code rate s the same, to reach the same recogto capabty, we eed 000 code words. The reaso for ths dfferece s varous, ) The code egth of (2,2) code s oger tha The bd recogto method of the bary BCH code proposed ths paper s a statstca recogto method. So the recogto capabty s drecty reated to statstca bt umber, the more the statstca bts, the better the recogto capabty ad the bgger the computato compexty, especay uder the codto of og code, whch s a drawback of the agorthm proposed ths paper. But whe compared to ts recogto capabty, the bg computato compexty s acceptabe. Besdes, the agorthms ths paper do ot vove compcated computatos, ad t coud be ready apped to the hardware processor because of ts bary characterstcs. So the hardware mpemetato s easy ad the computato ca be acceerated. 7. Ackowedgmet The work ths paper s supported by the Fud of Scece ad Techoogy Deveopmet of CAEP. The umber s 2009B0403043. 8. Refereces [] F. H. Wag, Z. T. Huag ad Y. Y. Zhou, A Method for Bd Recogto of Covouto Code Based o Eucdea Agorthm, IEEE Iteratoa Coferece o Wreess Commucato etworkg ad Mobe Computg, Shagha, 2007, pp. 44-47. do:0.09/ WICOM.2007.358 [2] P. Z. Lu, L. She, Y. Zou ad X. Y. Luo, Bd Recogto of Puctured Covoutoa Codes, Scece Cha, Vo. 48, o. 4, 2005, pp. 484-498. do: 0.360/ 03yf0480 Copyrght 20 ScRes.
22 J. F. WAG ET AL. [3] J. J. Za ad Y. B. L, Bd Recogto of Low Code-Rate Bary Lear Bock Code, Rado Egeerg of Cha, Vo. 39, o., 2009, pp. 9-24. [4] J. Lu,. Xe ad X. Y. Zhou, Bd Recogto Method of RS Codg, Joura of Eectroc Scece ad Techoogy, Vo. 38, o. 3, March 2009, pp. 363-367. [5] S. L ad D. J. Costeo, Error Cotro Codg, 2d Edto, Pearso Pretce Ha, Upper Sadde Rver, 2004, pp. 36-46. [6] X. Wag, X. M. Wag ad B. D. We, A Effcet ad Determstc Agorthm to Determe Irreducbe ad Prmtve Poyomas over Fte Feds, Acta Scetrarum aturaum Uverstats Suyatse, Vo. 48, o., 2009, pp. 6-9. Copyrght 20 ScRes.