CONSTRUCTION OF REGULAR QUASI CYCLIC-LOW DENSITY PARITY CHECK CODES FROM CYCLIC CODES

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1 VOL 3, NO 3, FEBRUARY 8 ISSN ARPN Joul of Egeeg d Appled Scece 6-8 A Reech Publhg Netwok ARPN All ght eeved wwwpjoulcom CONSTRUCTION OF REGULAR QUASI CYCLIC-LOW DENSITY PARITY CHECK CODES FROM CYCLIC CODES Bouchb Aylj, Motf Belkm, Sd Nouh 3 d Hmd Zouk Deptmet of Mth, LIMA Lb, Fculty of Scece, Choub Doukkl Uvety, El Jdd, Moocco SIME Lbo, ENSIAS, TIM Lb, Mohmmed V Uvety, Rbt, Moocco 3 Fculty of Scece Be M'k, H II Uvety, Cblc, Moocco E-Ml: bouchb_ylj@yhoof ABSTRACT Low Dety Pty Check Code LDPC e cl of le eo-coectg code whch hve how blty to ppoch o eve to ech the cpcty of the tmo chel Th cl of code ppoche ymptotclly the fudmetl lmt of fomto theoy moe th the Tubo Covolutol code It del fo log dtce tmo tellte, moble commucto d t lo ued toge ytem I th ppe, ew method fo cotuctg qucyclc low dety pty-check QC-LDPC code deved fom cyclc code peeted The popoed method educe the cdece vecto, by elmtg the cojugte le pty-check mtx of the deved cyclc code to cotuct ccult hftg ub-mtce I the ed, th method poduce lge cl of egul LDPC code of qu-cyclc tuctue hvg vey low dety, hgh codg te d Te gph whch hve o hot cycle wth gth of t let 6 Pefomce wth compute multo e lo how th wok fo ome cotucted code Keywod: LDPC code, cyclc code, qu-cyclc code, mmum dtce INTRODUCTION LDPC code wee veted by Robet Gllghe [] [] the ely 96; h doctol the he h popoed peudo-dom method whch geeted good LDPC code Howeve, the lck of well defed mthemtcl tuctue h mde the codg/decodg ytem vey complex fo th fmly of code Shotly, fte the veto, thee code hve bee fogotte lgely, utl 98 whe Te [3] gve them ew tepetto: gphcl epeetto H theoy h lo bee goed moe th 4 ye utl the edcovey of thee code by McKy 995 [4] Thee code hve mde bg comebck Recetly, they hve become the topc of my eech ctvte the codg theoy It h bee poved tht the LDPC code hve mmum dtce tht cee lely wth code legth [5] Mcky d Nel [6]hve demotted tht thee code c ppoch the Sho lmt moe th the tubo code, lte,, Rchdo et l [7] hve poved tht egul LDPC code outpefom the tubo code fo log code LDPC code hve bee ued my pctcl pplcto: they e ppled Code Dvo Multple Acce CDMA, Othogol Fequecy Dvo Multplexg OFDM ytem d pce-tme codg ytem Recetly, they hve bee dopted evel dgtl vdeo tmo tdd DVB-S, DVB-NGH, do tdd IEEE 86m moble, d locl do etwok IEEE 8 Alo, they e ued fo eltme pplcto uch mgetc toge, hgh-peed fo Etheet d locl e etwok WLAN A Low Dety Pty Check Code le block code tht the pty check mtx H h low dety of [] The pee of the H mtx h evel dvtge codg, decodg, mmum dtce I th wok we e teeted the poblem elted to the mpovemet of the eo-coectg pefomce of LDPC code Th mpovemet c be mde dug the cotucto of the LDPC code o dug the decodg opeto We focu o mpovg the cotucto tem of vou pmete tht cue poblem ode to cotuct good LDPC code: dety, codg te, mmum dtce d gth of Te gph Th cotucto mpovemet mde o the pecl ubcl of LDPC code, med qu-cyclc QC LDPC code tht e well tuded the ltetue fo vou mthemtcl popete ued by the cyclc tuctue of code I ddto, they c be ecoded wth le complexty ug hft egte d o eed to toe full G o H mtx of QC-LDPC fo ecodg o decodg opeto, tht why th cl of code the mot effcet hdwe mplemetto A ledy metoed, the ft cotucto of LDPC code h bee popoed by Gllge [] 96 The cotucto of the pty check mtx H of LDPC code Gllghe cot ftly to buld mtx H hvg colum weght w c d ow weght equl to w The we defe dom pemutto Γ j of colum of th ub-mtx H to fom othe ub-mtce, the egul H mtx Gllge the fomed follow H H H w H c Vou lgebc o dom method fo the cotucto of egul o egul LDPC code of qucyclc tuctue hve bee popoed, the utho [8-] hve ledy peeted uvey of wok whch hve bee 98

2 VOL 3, NO 3, FEBRUARY 8 ISSN ARPN Joul of Egeeg d Appled Scece 6-8 A Reech Publhg Netwok ARPN All ght eeved wwwpjoulcom ced by lgebc cotucto bed o fte geomete, elemet of fte feld d RS code Y et l [3] fo the pt hve gve lgebc method fo cotuctg egul QC-LDPC code bed o owee-pmtve BCH code, th cotucto method eulted cl fee of cycle of legth 4 O the othe hd, Tomlo et l [4] hve how how to cotuct lgothm to ech fo by dempotet whch my be ued to cotuct by LDPC code Fooe [5] h vetgted cotucto of LDPC code fom ccult pemutto mtce, d he h how tht uch code cot hve Te gph epeetto wth gth lge th I [6] S Aly et l hve peeted two lgebc method fo cotuctg egul LDPC code whch wee deved, oe fom opmtve ow ee BCH code d the ecod dectly fom cyclotomc coet I [7] the utho hve peeted cotucto of Hgh-Rte Regul Qu-Cyclc LDPC Code bed o cyclc dffeece fmle Autho [9] hve popoed «ew code tuctue fo QC LDPC code wth multweght ccult mtce by toducg ovelppg mtce» I th ppe, we peet lgebc method fo cotuctg by egul LDPC code bed o cyclc code, whch we med MC-LDPC The cotucto method poduce cl of LDPC code of qu-cyclc tuctue hvg good mmum dtce, hgh codg te d t dtguhed fom the pevouly cotucto gve the ltetue [3-7] by the followg popete: Geelzto of the cotucto: mkg devto fom y cyclc code by ug o-pmtve elemet of exteo Glo feld GFq m A vey low complexty of codg d decodg by educg the pty mtx by 5%, whch effcet becue f the pty mtx check mtx futhe educed, the bette of eo-coectg pefomce obted A vey low dety whch mple tht the gph wll cot o cycle of legth 4 d wth gth of t let 6 The dmeo d the mmum dtce e bouded fo cotucted code The emde of th ppe ogzed follow O the ext ecto, we gve toducto o cyclc, qu-cyclc code d BCH code I ecto 3 we peet the popoed cotuctg method of egul QC- LDPC code The popete of geeted QC-LDPC code e peeted ecto IV Ad we peet the pefomce of cotucted code wth compute multo ecto V Flly, cocluo d poble futue decto of th eech e outled ecto VI CODES OF CYCLIC STRUCTURE Cyclc code e cl of eo coectg code tht c be effcetly ecoded d decoded ug mple hft egte d combtol logc elemet, o the b of the epeetto ug polyoml [8] Let C, k deote le block code The code C cyclc f d oly f evey cyclc hft of code wod othe code wod A mpott popety of cyclc code tht ll code polyoml e multple of uque polyoml, gx clled the geeto polyoml of the code It c be how tht the geeto polyoml gx dvde x - Theefoe, to fd geeto polyoml, the polyoml x - mut be fctoed to t educble fcto φj x, j =,,, l, x = φ xφ x φ l x Alo, ote tht ove the feld of by umbe A coequece of the bove, the polyoml gx gve by gx x 3 jj,,, l j Defto : The opeto of multplyg by q dvde the tege mod N to et clled the cyclotomc coet mod N The cyclotomc coet C cotg cot of C={,q,q,,q j } Whee j the mllet potve uch tht q j = modn Ech cyclotomc cl et of cojugte oot the powe of the pmtve oot Defto : We defe mml polyoml m octed wth cyclotomc coet C, by the followg equto: j m x 4 j C Cyclotomc coet c deteme the umbe of educble fcto of x N - They llow to fd ll mml polyoml m of x N - ll fcto of x N - B Qu-Cyclc code Qu-cyclc QC code e geelzto of cyclc code whee cyclc hft of codewod by p poto eult othe codewod Theefoe, cyclc code e QC code wth p = The QC code c be decbed by ccult mtce A ccult mtx defed to be que mtx A of the fom A Cyclc code 983

3 VOL 3, NO 3, FEBRUARY 8 ISSN ARPN Joul of Egeeg d Appled Scece 6-8 A Reech Publhg Netwok ARPN All ght eeved wwwpjoulcom A 5 Whee ech ucceve ow cyclc hft of the pevou oe C BCH code BCH code, cl, e oe of the mot kow poweful eo-coectg cyclc code The mot commo BCH code e chctezed follow: pecfclly, fo y potve tege w 3, d t < w -, thee ext by BCH code wth the followg pmete: Block legth: = w - Numbe of mege bt: k -wt Mmum dtce: d t + Thee BCH code e clled pmtve becue they e bult ug pmtve elemet of GF w 3 CONSTRUCTING METHOD OF REGULAR QC- LDPC CODES A Decpto of The popoed cotuctg method Let GFq deote fte Glo feld of q elemet, d let GFq m t exteo feld, N=q m -, whee m the multplctve ode of q modulo N d deote pmtve oot of GFq m Fo y pmtve elemet α of GFq m, we octe the fucto δ, defed by:,,,, : N m GF q GF 6 δα h the vlue poto d the vlue ywhee, wth gog ove et {,,, q N } Exmple : δ 3 =,,,,, We c defe the vecto δ 4 the ght cyclc hft of the vecto δ 3 We defe the ccult mtx ove GF N*N of vecto δ, follow: N A 7 The ccult mtx α cot ech of t ow d t colum α the detty mtx of ode N Ad α wtte follow: A 8 Let β be o-pmtve elemet of GFq m, α pmtve elemet of GFq m theefoe N=q m - t multplctve ode Let be the legth of cyclc code defed ove GFq m o dvde N I ou Cotucto we chooe whch vefe the followg equlty q m/ < q m - β h the multplctve ode ove GFq m We defe by the equlty =N/ the β=α th ce β th oot of uty Whe cyclc code pecfed by oot β, β, β ove GFq m [8], The pty check mtx of th code c be defed follow H 9 The popoed cotucto method MC-LDPC cot the ft plce, of educg the pty mtx of the deved cyclc code C whoe mtx wtte the fom of H to ew pty mtx H whch mplfed, by pecedg elmto of ome ow follow: Whe we hve the pty check mtx H of cyclc code C, we toduce oe oot fom the et {, q, q } mtx H d we clculte cyclotomc coet of C to deteme cojugte oot of β ove GFq m ode to mplfy the mtx H by emovg the cojugte ow Wth β=α d *, the mtx mplfed H follow: H

4 VOL 3, NO 3, FEBRUARY 8 ISSN ARPN Joul of Egeeg d Appled Scece 6-8 A Reech Publhg Netwok ARPN All ght eeved wwwpjoulcom The cotucto of pty check mtx H LDPC of LDPC code c be obted fom the mplfed pty mtx H of the code C pecfed by the oot, the we pply the fucto δ equto 6 to ech elemet of the pty mtx H, d we pply the equto 7 ode to obt ccult ub- mtce to H We wll hve the followg mtx: H LDPC The obted pty mtx H LDPC h the followg popete: The ze of H LDPC * N N MC-LDPC method educe le umbe of mtx H LDPC by 5 %: become *,wth < * Ech colum h Hmmg weght equl to * Ech ow h Hmmg weght equl to H LDPC egul * N * N the dety equl N * N N * N N Pemete Gth the gph of t let 6 If the mmum dtce of the cyclc code C d c the the mmum dtce d LDPC of the devtve LDPC code bouded by d c : d LDPC d c B Popoed MC-LDPC lgothm The decpto of the lgothm of the popoed cotuctg method follow: Algothm : MC-LDPC lgothm Iput: q, m : legth of cyclc code C gx: geeto polyoml of cyclc code C Output: N, *, H LDPC Step : Clculte C cyclotomc coet modulo Step : Clculte m mml polyoml m octed to cyclotomc coet C Step 3: If gx== let commo multple LCM m,,m j The j // : umbe of the oot octed to m // EdIf Step 4: Clculte p // p: umbe cojugte oot of β // Step 5: Clculte * -p // * : umbe of H ow // Step 6: Clculte H Step 7: Clculte H LDPC We c gve the followg exmple to llutte the equece of the MC-LDPC lgothm Exmple : We tke the exteo feld GF56, wth q= d the pmtve polyoml px=+x +x 3 +x 4 +x 8 The ttg cyclc code C5,9,, t o pmtve code of geeto polyoml gx=+x +x 3 +x 5 + x 6 +x 8 +x 9 +x +x +x +x 4 +x 6 +x 9 +x +x 4 + x 3 +x 3 The we hve: m=8, N=55, =5, =5 d β=α 5 oot GF56 of multplcty ode 5 Step : The cyclotomc coet C mod 5 e follow: C ={} C ={,,4,8,6,3,3,6} C3 ={ 3,6,,4,48,45,39,7} C 5 ={ 5,,,4,9,7,4,8} C 9 ={ 9,8,36,,4,33,5,3} C ={,,44,37,3,46,4,3} C 7 ={ 7,34} C 9 ={ 9,38,5,5,49,47,43,35} Step : Mml polyoml m octed to C e epectvely: m x=x+ m x= +x+x 4 +x 5 +x 6 +x 7 +x 8 m 3 x= +x+x +x 4 +x 6 +x 7 +x 8 m 5 x= +x+x 3 +x 4 +x 8 m 9 x= +x 3 +x 4 +x 5 +x 8 m x= +x 3 +x 4 +x 5 +x 8 m 7 x= +x+x m 9 x= +x+x +x 3 +x 4 +x 7 +x 8 Step 3: gx =LCMm,,m the = Step 4-5: We hve β, β 4, β 8 cojugte oot of β, β 6 cojugte oot of β 3 d β 7, β cojugte oot of β 5 The p=6 whece *=4 d β, β 3, β 5, β 9 e oot of H Step 6-7: H LDPC So, we cotuct the egul QC-LDPC code of the mtx H LDPC of ze *35, ech ccult ubmtx α the ze 55*55 H LDPC h Hmmg weght colum equl to 4, Hmmg weght ow equl to 5 d dety, 4 The Tble- peet ome QC-LDPC code cotucted by the MC-LDPC method Algothm 985

5 VOL 3, NO 3, FEBRUARY 8 ISSN ARPN Joul of Egeeg d Appled Scece 6-8 A Reech Publhg Netwok ARPN All ght eeved wwwpjoulcom Tble- Pmete of ome egul QC-LDPC code deved fom ome cyclc code ug lgothm N= m - Code cyclc Root of H Sze of H LDPC Dety of H LDPC 3 C3,6,7 β, β 3, β 5 93, C,,5 β, β 3 6, C9,3,3 β 63, C7,9, β,β 3,β 5,β 7, β 9 635, C7,9,5 β 55, C5,9, β, β 3, β 5, β 9, C73,9, β,β 3,β 5,β 9, β, β 3 366, C33,3, β, β 3 46, C3,3, β 47, C5,8,5 β, β 3 89, PROPRIETIES OF GENERATED QC-LDPC CODES A Gth of Te gph Gth oe pmete uully tgeted fo optmzto of eo pefomce, ptcul eo floo A LDPC code wth lge gth deble, tetve decodg covege fte fo gph wth lge gth I MC-LDPC method, QC-LDPC code e cotucted by block of pe ccult mtce of weght ot moe th ech ow o colum, the, thee o two ow o two colum the me mtx o dffeet block of ccult mtce hvg moe th oe vlue commo Coequetly, the obted HLDPC mtx lo pe mtx of lowe dety, whch gve qu-cyclc LDPC code whoe the Te gph h o cycle of legth 4 We clculted the gth of Te gph fo ome code cotucted by MC-LDPC deved fom BCH code The Tble- how tht MC-LDPC method educe le umbe of mtx H LDPC by 5 % d cee the gth vlue of t let 6 fom BCH code teted The gth obted by the lgebc clculto MAGMA B Dety of H LDPC It c be demotted tht the dety D of the mtx H LDPC of ze * N N D N 4 Fom o-pmtve cyclc code the dety of the obted LDPC code vey low becue the tege S We ote tht t h lowe dety f S> Tble- Gth fo ome cotucted QC-LDPC fom BCH code N= m - Root MC-LDPC Code H H LDPC Gth 5 BCH5,,3 β 5,5 No cycle 5 BCH5,7,5 β, β 3 3,5 8 5 BCH5,5,7 β, β 3, β 5 45,5 6 3 BCH3,,5 β, β 3 6, BCH63,5,5 β, β 3 6, C Rk d te of H LDPC We ote tht the le of the mtx H LDPC e ot ecely lely depedet o the by Glo feld GF I th ce, to deteme the dmeo of the cotucted code, we mut clculte the k of the mtx H LDPC The Rk of egul mtx H LDPC of ze * N N cotucted by MC-LDPC : Rk * N 5 We poved tht the Rk deped o the ozeo tege *, o we c wte tht: Rk * N - ct 6 Whee ct deote cott tht deped o * We foud the equvlet mtx of H LDPC ytemtc fom, ode to clculte the Rk of equvlet code The Tble-3 gve the k of evel tet code 986

6 VOL 3, NO 3, FEBRUARY 8 ISSN ARPN Joul of Egeeg d Appled Scece 6-8 A Reech Publhg Netwok ARPN All ght eeved wwwpjoulcom Fom the obted eult, we c lmt the vlue of the ct betwee d * - The Rk c be ewtte follow: Rk * N - * - 7 Tble-3 Rk d codg te fo ome cotucted QC-LDPC code N= m Sze of Codg - Code Rk H LDPC te 63 C9,3,3 63, /33 5 SIMULATION RESULTS The pefomce of cotucted code multed fo ome dffeet legth d te codg, ug the pmete of Tble-4 Tble-4 Pmete of multo Pmete Chel 5 BCH5,,3 5,5 5 /5 5 BCH5,7,5 3,5 9 96/5 5 BCH5,5,7 45,5 43 8/5 55 C7,9, 55, / C,9,3 63, /363 3 BCH3,,5 6,96 6 9/96 3 BCH3,6,5 7,96 75/96 63 BCH63,4,5 44, /3969 Vlue AWGN Modulto BPSK Mmum umbe of edul bt eo Mmum umbe of tmtted bloc Numbe of teto [5,5] Fo the decodg method we ue the decodg lgothm SPA um-poduct lgothm t logthmc veo [] The decodg lgothm topped whe the pedefed umbe of teto eched o whe the ydome of the eceved wod equl to zeo We computed the bt eo pobblty BER fo ech teted code d we plotted the BER cuve fucto of gl-to-oe to E b / N to lyze the eo codg pefomce Fgue- Pefomce of QC-LDPC code cotucted by MC-LDPC method The Fgue- how the eo pefomce cuve of me cotucted QC-LDPC code whch e deved follow: QC-LDPC93,96 deved fom the cyclc code C3,6,7 whch lo BCH code, the eult QC- LDPC hvg gth of 6 d dety,35 QC-LDPC 6, 33 of gth 6 d dety,5 deved fom the cyclc code C,,5 whch o pmtve BCH code QC-LDPC 55, 4335 deved fom the cyclc code C7, 9, 5 whch o pmtve BCH code, the cotucted QC-LDPC hvg gth of 8 d dety,3 QC-LDPC, 35 h gth of 8 d dety, 3 deved fom the o pmtve BCH code C5, 9, Ad QC-LDPC 635, 69 deved fom the cyclc code C7, 9, hvg gth of 8 d dety, 7 The umbe of decodg teto ued the SPA lgothm ve betwee 5 d 5 teto tem of the code ze We c obeve tht fte bout 987

7 VOL 3, NO 3, FEBRUARY 8 ISSN ARPN Joul of Egeeg d Appled Scece 6-8 A Reech Publhg Netwok ARPN All ght eeved wwwpjoulcom 5 teto the ocllto of the decodg covege to fxed thehold whch em the me utl the ed of teto Fom Fgue-, we how tht multed QC- LDPC code pefom bette th othe QC-LDPC pevouly cotucted [3-7] d QC-LDPC, 35 d QC-LDPC 635, 69 code pefom much bette below the BER of -4 At the BER of -5, thee two code epectvely pefom t 36 db d 35 db d LDPC 635, 69 pefom 8dB fom the Sho lmt 6 CONCLUSIONS AND PERSPECTIVES I th ppe, we hve peeted cotucto method of egul QC-LDPC bed o cyclc code The fct tht we hve elmted the cojugte le ptycheck mtx of the deved cyclc code; we hve educed hlf the cdece vecto of mtx H LDPC d hve mpoved the eo-coectg pefomce of QC-LDPC code I ddto, we hve ceed the Gth vlue of t let 6 We hve demotted tht thee cotucted QC- LDPC code hve hgh dety, hgh te d the cyclc tuctue help to compute the dmeo d Rk Ou cotuctg method llow ceg the mmum dtce of the cotuctg QC-LDPC code We pl to vefy moe popete of th QC- LDPC fmly d evlute the pefomce ove dffeet commucto chel, d we wll fd ew le code bed fom by cyclc code REFERENCES [] D R Gllge 96 Low dety pty-check code IRE T Ifom Theoy pp -8 [] R Gllge 963 Low dety pty-check code MIT Pe, 6-7 My [3] RM Te 98 A ecuve ppoch to low complexty code IEEE Tcto o Ifomto Theoy 7: [4] D J C McKy d R M Nel 996 Ne Sho lmt pefomce of low dety pty check code Electo Lett 38: [5] V V Zyblov d M S Pke 975 Etmto of the eo coecto complexty fo gllge low dety code Pobl Peedch If : : 3-36 [6] D J C McKy d R M Nel 996 Ne ho lmt pefomce of low dety pty check code Electo Lett 38: [7] T J Rchdo, M A Shokollh d R L Ubke Deg of cpcty ppochg egul low-dety pty check code IEEE T O Ifom Theoy 47: [8] I Djudjevc, J Xu, K Abdel-Ghff d S L 3 A cl of low dety pty check code cotucted bed o eed-olomo code wth two fomto ymbol IEEE Commucto Lette 7: [9] S Sog, L L, S L d K Abdel-Ghff 6 Cotucto of qucyclc ldpc code bed o the pmtve elemet of fte feld Ifomto Scece d Sytem, 6 4th Aul Cofeece pp [] S Sog, L Zeg, S L d K Abdel-Ghff 6 Algebc cotucto of oby qu-cyclc ldpc code Poc 6 IEEE Itl Symp Ifom Theoy pp [] G Lv, S Sog, Y Ry W L, L Zhg d S L Deg of ldpc code: A uvey d ew eult J Comm Softwe d Sytem, 6 [] J Wg, J Le, S N QC-IRA-d Code Bed o Ccult Pemutto Mtce IEEE Commu Lett 5: 4-7 [3] Y Y, L Shobo d H Dwe 5 Cotucto of ldpc code bed o ow-ee pmtve bch code Vehcul Techology Cofeece 3: [4] M Tomlo, C J Tjh, M A Amboze d M Z Ahmed 4 By cyclc dffeece et code deved fom dempotet bed o cyclotomc coet IEEE Tcto o Ifomto Theoy [5] Mc P C Fooe 4 Qu-Cyclc Low- Dety Pty-Check Code fom Ccult Pemutto Mtce IEEE Tcto o Ifomto Theoy 58 [6] S A Aly 8 Fmle of ldpc code deved fom opmtve bch code d cyclotomc coet CORR, Febuy 8 [7] Houg Pk, Seokbeom Hog, Jog-Seo No d Dog-Joo Sh 3 Cotucto of Hgh-Rte Regul Qu-Cyclc LDPC Code Bed o Cyclc Dffeece Fmle IEEE Tcto o Commucto 68 [8] Moelo-Zgoz 6 The At of Eo Coectg Codg Secod Edto Robet, Joh Wley & So, Ltd ISBN:

8 VOL 3, NO 3, FEBRUARY 8 ISSN ARPN Joul of Egeeg d Appled Scece 6-8 A Reech Publhg Netwok ARPN All ght eeved [9] B Sh, H Pk, S Hog, J-S No d S-H Km 4 Qu-cyclc LDPC code ug ovelppg mtce d the lyeed decode AEU - Itetol Joul of Electoc d Commucto 685: [] T K Moo 5 Eo Coecto Codg - Mthemtcl Method d Algothm Wley wwwpjoulcom 989