COMPARATIVE ANALYSIS OF LDPC AND BCH CODES ERROR-CORRECTING CAPABILITIES

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1 5 UDC COMPARAIVE ANALYSIS OF LDPC AND BCH CODES ERROR-CORRECING CAPABILIIES Leonid O. Uyvsky, Sehii O. Osyphuk eleommuniation Netwoks Depatment Institute of eleommuniation Systems National ehnial Univesity of Ukaine KPI Kyiv, Ukaine he eo-oeting apabilities of egula LDPC (Low Density Paity Chek) odes and BCH (Bose-Chaudhui- Hoquenguem) odes ae examined. he qualitative analysis and the quantitative assessment of eo-oeting abilities ae pefomed fo LDPC odes with ode wod length n=1000 bits and BCH odes with ode wod length n=1023 bits. he ode ates of LDPC and BCH odes ae detemined fo a known signal to noise atio in the gaussian hannel; deteted ode ates ae optimal fo pedefined modulation type and equied infomation eliability on the eeive side. Intodution Signifiant inteest has aised fo LDPC (Low Density Paity Chek) odes eently. he impotane of LDPC odes is shown by diffeent standads and eommendations, whee LDPC odes ae used: DVB-S2, IEEE et al. [1]. heoetiians and paties in eo ontol oding aea enewed a level of inteest to LDPC odes ove the wold. Many sientifi publiations devoted to LDPC: [2-4] (Geat Bitain), [5] (USA), [1] (Japan) and othes. LDPC odes ae blok stutued linea divisible odes. he LPDC odes ae intodued fo the fist by R. Gallaghe in 1962 [6], but inteest was not attated to them so muh at that time. hese odes have been fogotten fo seveal tens of yeas. Hee is the next explanation [7] of the eason why LDPC odes exploation was held up afte Gallaghe s publiations and esumed in ubo odes wee disoveed in the middle of 1990 and have iteative deoding poedues with attative eo-oeting haateistis; wheeas LDPC odes have iteative deoding poedues as well [8], an inteest was aoused fo these odes too. It was assumed that LDPC odes stand as well losely to Shannon limit as tubo odes, and this was ooboated in elevant eseahes [2, 7]. BCH odes (Bose-Choudhuy-Hoquenguem), in tun, ae one of the best blok odes. he haateistis of BCH odes ae shown in [9]. he goal of this eseah is LDPC and BCH odes ompaison. Citeia fo ompaison ae the next: idential ode wod length, equal shift keying manipulation, known hannel paamete SNR (Signal to Noise Ratio), same equied bit eo pobability on the eeive end. Poblem statement he enty paametes fo task ae below: Channel paamete: SNR = 0 14 db; Shift keying manipulation: QPSK; Code wod length fo antinoise oding: n=1000 fo LDPC odes and n=1023 fo BCH odes; Requiement to the bit eo eliability on the eeive side: Output paametes ae LDPC and BCH oding ates: R LDPC and R BCH. o eah the goal of eseah, the antinoise ode ates R LDPC and R BCH ae found to ahieve equied infomation eliability on the eeive side if the desibed enty paametes above ae known; given ode ate values ae ompaed and the best eooeting method {LDPC, BCH} on the iteion {R MAX, d MAX } is hosen. his task an be shematially pesented as shown on the Fig. 1. So, the main task is a seah of antinoise ode with maximal ode ate R and ode distane d values, and this is a fundamental poblem of oding theoy [10]. h 2 =onst, db QPSK р bit =10-2 LDPC: n=1000 BCH: n=1023 p bit_eq =10-6 {R max,d max } Fig. 1. Statement of the poblem h 2 =onst, db QPSK p bit_eq =10-6 R {LDPC, BCH}? he next subtasks wee set up to ahieve the goal: Development and implementation the seah poedue of minimal LDPC ode distane when the ode length and hek matix paametes ae pedefined; Detemination of positions the LDPC and BCH odes points in oodinates R = f (d/2n); ISSN , Infomation and eleommuniation Sienes, 2014, Volume 5, Numbe , National ehnial Univesity of Ukaine Kyiv Polytehni Institute

2 6 INFORMAION AND ELECOMMUNICAION SCIENCES VOLUME 5 NUMBER 1 JANUARY JUNE 2014 Definition the maximum antinoise ode ate that is able to povide equied bit eo eliability. LDPC and BCH odes haateistis BCH odes ae haateized by the possibility to fom the antinoise ode with pedefined eooeting abilities suh as minimal ode distane d. he BCH ode exists fo any values m and t=(d-1)/2 with ode length n=2 m 1 that oets all ombinations of t o less eos numbe; this ode has mt oetive bits in the ode wod. hus, the BCH ode length an not be hosen andomly and depends fom the paamete m; BCH ode length always has an odd value. he popeties of some BCH odes with paamete m=10 and ode length =1023 ae shown in the able I. ABLE I. BCH CODES n k n-k=mt t R As follows fom example, BCH ode with ode length n=1023 an be fomed with ode ate step Heewith the BCH ode ate deeases linealy wheeas eo-oeting apability ineases: R = 1 0, 0097t, o (1) 1 R t = (2) 0, 0098 Inauay of (1) and (2) is lowe than 2.2%. Let s tun to the LDPC odes haateistis. LDPC odes ae not analytial and this is one of the diffeenes fom BCH odes. LDPC ode popeties annot be defined analytially as a esult of this. A lot of LDPC ode modifiations exist, and most of them ae not exploed in full. ogethe with this, all LDPC odes ae lassified by two goups: egula and non-egula. hese two goups ae diffeentiated by the hek matix onstution that used fo enoding and deoding ode wods. Non-egula LDPC odes ae built based on egula LDPC odes [8]. It is shown in [11] that egula LDPC odes moe often demonstate bette haateistis than non-egula LDPC odes. It s shown in [2] that egula LDPC odes have bette popeties in Gaussian hannel than nonegula LDPC odes. ogethe with this, the onditions ae pesented in [5] when non-egula LDPC odes have bette haateistis atually. heeby, eithe egula LDPC odes o non-egula LDPC odes ae entitled to existene in the theoy and patie of antinoise oding. Foming of egula LDPC odes is defined in onseutive ode. Regula LDPC ode with a ode length n foms based on the hek matix H. Сhek matix H has a fixed value of ones in the matix ow W and a fixed value of zeos in the olumn W [2]. It s onsideed that hek matix H has a low density of ones when density of ones in hek matix H is less than 50% of all the hek matix elements. he LDPC ode eo-oeting ability is speified based on speifi paametes of hek matix H: n, W, W. At the same time, positions of ones in the hek matix Н ae based on andom pemutations the basi sub matix H 1 olumns. Eah olumn of basi sub matix H 1 inludes only solus one. he egula LDPC ode ate is defined as a funtion of hek matix H paametes (3): W n n ( W 1) W W W 1 R = = 1 + (3) n W n Withal, LDPC odes hek maties Н with the same matix paametes, but diffeent positions of ones in hek matix, an geneate antinoise odes with diffeent ode distanes and espetively diffeent eo-oeting abilities. Hene the task aises to seah the best hek matix H with known paametes n, W, W by the iteion of maximal eo-oeting ability of LDPC ode: t ( d ) max max 2 / 2. LDPC ode hek matix H an be epesented as: H 1 π 1( H1) H = M π WC 1( H1), (4) Whee Н 1 basi submatix, π i ( H1) submaties ae geneated by andom eaangement of basi submatix olumns Н 1, i=1,2,,w 1. Chek matix H an be tansfomed into the matix fom: H = [ A In k ], (5)

3 URYVSKY L., OSYPCHUK S.: COMPARAIVE ANALYSIS OF LDPC AND BCH CODES ERROR-CORRECING CAPABILIIES 7 Whee A some non-spase fixed matix with ze- n k k ; In k identi- os, ones and dimensions (( ) ) ty matix with dimensions ( n k) ( n k ). he geneation matix G an be epesented as: G = I k A (6) If the hek matix Н is pesented as (5), then the geneation matix G (6) an be simply given fom the matix Н by tansfomation. he matix G is also named as geneative matix so fa as ode wods that an be epesented as linea ombinations of matix G ows. he maties Н and G ae elated as [2]: GH = 0, HG = 0 (7) Code distane d fo egula LDPC ode is defined as the least olumns numbe of hek matix H that oveall gives 0. he analytial desiption fo LDPC ode eo-oeting abilities doesn t exist so fa; howeve, the fowad and bakwad theoems exist fo LDPC ode distane [10]. heoem 1. If any l d 1 olumns of linea ode hek matix Н ae linealy independent, then a minimal ode distane will be at least d. If d linealy independent olumns ae found, then minimal ode distane is equal d. heoem 2. If minimal ode distane is equal d, then any l d 1 olumns of hek matix H ae linealy independent and exatly d linealy independent olumns exist. hus, it s possible to onlude fom theoems 1 and 2 that LDPC ode distane d an be identified fom maties H and G as the next: the d value equals the least olumns numbe of matix Н that sum up to 0; the d value equals the least ow weight (the numbe of ones in the ow) in matix G. he LDPC odes eo-oeting ability is eseahed in uent wok based on the desibed popeties of hek and geneating maties. he same LDPC ode wod length n=1000 and diffeent hek matix H paametes esult in diffeent antinoise ode ates R and diffeent numbes of oeted eos pe ode wods espetively. Given esults in expeiments ae ompaed with eo-oeting abilities of BCH odes with ode wod length n=1023 bits. LDPC odes eo-oeting ability Known methods fo LDPC ode distane d seah omplexity gows exponentially as is shown in [2]. Known seah methods give a possibility to define the ode distane value fo odes with ode length less that n<1000 bits in tems of spending sensible time esoues fo a seah. he LDPC ode eo-oeting value an be found fom maties H and G by seah, but the given ode distane value an be not the best fo used LDPC matix paametes. he study of LDPC ode eo-oeting abilities with ode length n=1000 bits is pefomed in this wok based on the theoems 1 and 2. his idea is shown on Fig. 2. he line #1 (Fig. 2) indiates the ode distane d seah omplexity when use only theoem 1 (by using matix Н); the line #2 designates a ode distane d seah omplexity when use only the theoem 2 (by using matix G). he omplexity tem means in this ontext the numbe of elementay opeations to exeute in the speifi time point and save the same pogess of the end esults eeiving. Fig. 2. A gaphial epesentation of appliation the theoems 1, 2 fo ode distane seah poess he point A (Fig. 2) indiates the moment when it s bette to use theoem 1 befoe that, but it s bette to use theoem 2 afte point A. he point A oesponds the ase when ode distane d = 6 (t = 2) is found by theoem 1. As soon as the matix Н has d = 6, then seah by theoem 1 stops and futhe seah of ode distane d ontinues by edution the matix H to matix G in anonial fom (5). hus, the minimal time fo seah LDPC ode distane spends if ombine theoems 1, 2 fo the seah poess. he desibed algoithm above fo LDPC ode distane seah is implemented on Java language. LDPC ode distane esults fo n=1000 ae obtained fom set of numeial expeiments pefomed on high pefomane omputing luste in NUU KPI. Matix H and found LDPC ode eo-oeting paametes ae pesented in able II.

4 8 INFORMAION AND ELECOMMUNICAION SCIENCES VOLUME 5 NUMBER 1 JANUARY JUNE 2014 ABLE II. GIVEN PARAMEERS OF LDPC CODES n W W k R d t LDPC and BCH odes ompaison he method fo seah the best antinoise blok ode by iteion of maximal appoah to the Shannon limit is desibed in [9]. his method implies the blok ode seletion with using up-to-date theoy of antinoise oding. his method is used fo alulation and eo-oeting abilities ompaison of LDPC and BCH odes. Regading the method [9] and poblem statement (Fig. 1), the next fatos an be noted: 1) if the manipulation QPSK is used, then an initial bit eo pobability p bit =10-2 is eahed when the signal to noise atio in the hannel is h 2 =7.3 db; 2) if antinoise blok ode with ode length n=1000 bits is used and equied eliability is p bit_eq =10-6, then it s needed to oet up to t=28 eos [9]. Only in this ase the equied eliability p bit_eq =10-6 an be ahieved on the eeive side; 3) to satisfy the equied eliability p bit_eq =10-6, the value d/2n fo antinoise ode should meet he limit onditions of antinoise odes existene with some eo-oeting abilities ae desibed by Plotkin limit [10]. A suffiient ondition fo antinoise odes with speifi eo-oeting abilities is defined by the Vashamov-Gilbet limit (VG). heeby, the Plotkin and VG iteia povides an oppotunity to ompae diffeent eo-oeting blok odes in the same elative oodinates R = f (d/2n) fo assessment the eo-oeting apabilities. he dependeny R = f(d/2n) is shown on the Fig. 3. he Plotkin and VG limits ae shown in these oodinates. he positions of some BCH (able I) and given expeimentally LDPC odes (able II) ae plotted on Fig. 3 with ode ates 0, If take into aount the ode length n ~ 1000 bits and set the value d/2n=0.03, then the tiangle between Plotkin and VG limits shows the shaded aea (Fig. 3) whih outlines the egion of antinoise odes paametes that an povide expeted eliability p bit_eq =10-6 [9]. If ompae the haateistis of BCH odes with n=1023 bits and LDPC odes with n=1000 bits, then it s possible to say that the next ode ates fit the atio d/2n=0.03: R BCH =0.7 and R LDPC =0.73. If R BCH =R LDPC =0,55, then d/2n BCH =0,05 a nd d/2n LDPC =0,07. he above two instanes demonstate a bette oeting apabilities of LDPC odes in ompaison with BCH odes (R LDPC >R BCH if d/2n LDPC =d/2n BCH =onst; o d/2n LDPC >d/2n BCH when R LDPC =R BCH ; o d LDPC >d BCH when n=onst). If ontinuously hange the paamete h 2, then it s possible to get dependeny R=f(h 2 ) (Fig. 4). Both Plotkin and VG limits ae stood below the Shannon limit in oodinates R=f(h 2 ). Consequently, if hannel paamete h 2, uent bit eo pobability p bit, equied eliability p bit_eq ae known, then it s not possible to ome to Shannon limit neae, than it s defined by Plotkin limit. Fig. 3. BCH (n=1023) and LDPC (n=1000) Speified above onditions give an oppotunity to hoose the antinoise ode that lies to Shannon limit losely as muh as possible. he LDPC and BCH odes positions in oodinates R = f (h 2 ) ae shown on Fig. 4. R BCH 0.4 LDPC 0.2 Shannon limit Plotkin limit Vashmov-Gilbet limit SNR, db Fig. 4. LDPC (n=1000) and BCH (n=1023) odes

5 URYVSKY L., OSYPCHUK S.: COMPARAIVE ANALYSIS OF LDPC AND BCH CODES ERROR-CORRECING CAPABILIIES 9 As shown on Fig. 4, LDPC odes stand a little bit losely to Shannon limit than BCH odes. his behavio takes a plae moe and moe if LDPC ode ate dops down: R<0.7. If R<0.7, then LDPC ode is pefeable when hoose between LDPC and BCH. he indisputable advantage of LDPC ode is a possibility to inease the ode wod length n ight up to tens of thousands bits. his explained by elatively simple methods of oding and deoding. ogethe with that, the advantage of BCH ode is the oppotunity to define ode paametes analytially and hoose appopiate ode with needed paametes (e.g., it s possible to hoose the BCH ode ate with a step 0.01 fo ode length n=1023 bits) to meet the equiements of eos oeting egading the method desibed in [9]. Conlusions he poedue omplexity of the LDPC hek matix H seahing with good eo-oeting ability gows exponentially with ineasing a ode wod length. hee is no need fo BCH odes to pefom the seah matix poedue beause of the natue of enoding/deoding poesses. his is an advantage of BCH odes. Reseah showed that LDPC odes an be haateized as antinoise odes with good eo-oetion popeties. Relative numbe of oeted eos pe ode wod is almost the same fo LDPC (n=1000) and BCH (n=1023) odes. LDPC odes have a little bit bette eo-oeting abilities than BCH odes have if ode ate R<0.7 (it s implied that othe paametes like ode length, signal to noise atio, manipulation method, equied eliability ae the same fo LDPC and BCH). he oded ates ae obtained fo LDPC (n=1000) and BCH odes (n=1023) fo gaussian hannel when signal to noise value is known. Given ode ates R BCH =0.7 и R LDPC =0.73 noties that it s possible to use both LDPC and BCH odes with a speifi manipulation type to satisfy equied infomation eliability. Aoding to numeial LDPC and BCH ode ates values, LDPC ode an be eommended as moe effetive if signal to noise values ae bigge than 7 db. If signal to noise values ae smalle than 7 db, then LDPC and BCH odes an be used pai passu. hese eommendations ae easonable fo manipulation QPSK, ode wod length 1000 bits and equied infomation eliability he LDPC ode wod length an eah tens of thousands. his is possible beause of elatively simple ode wods enoding/deoding poedues, and this is an advantage of LDPC odes. As opposed to LDPC, the BCH odes have moe omplex enoding/deoding poedues. As a esult of this, BCH odes with long ode wods n>1000 ae less patial than LDPC odes, o they ae tehnially ompliated with ealization. Refeenes 1.. Ohtsuki, LDPC odes in ommuniations and boadasting, IEIC ans. Commun., vol. 90-B, no. 3, pp , Mah D. MaKay, Infomation heoy, Infeene, and Leaning Algoithms. Cambidge Univesity Pess, D. MaKay, Good eo-oeting odes based on vey spase maties, IEEE ans. Inf. heoy, vol. 45, no. 2, pp , Mah D. MaKay, R. Neal, Nea Shannon limit pefomane of low density paity hek odes, Eleton. Lett., vol. 32, no. 18, pp , August ian, C. Jones, J. Villaseno, R. Wesel, Constution of iegula LDPC odes with low eo floos, Communiations, ICC 03, IEEE Intenational Confeene, vol. 5, pp , May R. Gallage, Low-Density Paity-Chek Codes. MI Pess, V.Vagauzin, Neaby the Shannon limit, elemultimedia Jounal, pp. 3 10, [Online]. Available: shennon.pdf 8. M. Luby, M. Mitzenmahe, A. Shokollahi, D. Spielman, Impoved low-density paity-hek odes using iegula gaphs and belief popagation, SRC ehnial Note, 9 p., L. Uyvsky, K. Pokopenko, A. Pieshkin, Noise ombating odes with maximal appoximation to the Shannon limit, eleommuniation Sienes, vol. 2, no. 1, pp , Januay-June [Online] Available: eleommuniation%20sienes%20n.1% p df 10. N. Sloane, F. MaWilliams, he heoy Of Eo- Coeting Codes. Bell Laboatoies, Amstedam, Noth-Holland, S. Miyamoto, K. Kasai, K. Sakaniva, Suffiient onditions fo a egula LDPC ode bette than an iegula LDPC ode, IEICE ans. Fundamentals, vol. E90 A, no. 2, pp , Febuay Reeived in final fom Apil 19, 2014