Performance study of Non-binary LDPC Codes over GF(q)

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1 Proeedigs 585 CSNDSP08 Performae study of Nobiary DPC Codes over GF().S. Gaepola, R.A. Carraso, I. J. Wassell ad S. e Goff Shool of Eletrial, Eletroi ad Computer Egieerig, iversity of Newastle {v.s.gaepola, r.arraso, stephae.legoff}@l.a.uk Computer aboratory, iversity of Cambridge ijw4@am.a.uk Abstrat owdesity Parity Chek (DPC) odes are kow to perform well i the presee of Additive White Gaussia Noise (AWGN) but for very large blok legths. It has bee proposed to defie the odes over high order Galois fields to overome this limitatio. I this paper we ostrut ew uasiyli obiary DPC odes with moderate ode legths from Reed Solomo odes with two message symbols proposed by i et al defied over large fiite fields. We evaluate the performae of these odes o the AWGN hael by omputer simulatio ad show that they outperform biary DPC odes of the same legth i biary bits. Idex Terms obiary, DPC odes, AWGN, FFTBP deodig, fiite fields. I. INTRODCTION Biary ow Desity Parity Chek (DPC) odes over GF () redisovered by Makay ad Neal [3], [4] have bee observed to display ear Shao limit performae whe deoded usig probabilisti soft deisio deodig algorithms. owever these ear Shao s limit performaes are obtaied for radomly ostruted odes of very large blok legths. It has bee show that this limitatio a be overome by defiig the ode over higher order Galois Fields [5], [6], [8]. It is reasoable to assume that o biary DPC odes perform better tha biary DPC odes o haels with oise bursts, give the fat that oseutive bits are grouped together formig symbols i the o biary alphabet GF ( ). The lassial BP algorithm used i deodig o biary DPC odes has a omputatioal omplexity domiated by O ( ) makig the deodig over higher order fields omputatioally ifeasible. owever it has bee GF a show that the belief propagatio over ( ) be oveietly trasferred ito freuey domai. log [8]. O salig dow the omplexity to ( ) This paper shows that o biary odes with oly moderate ode legths outperform biary DPC odes with large blok legths by osiderable margis. We also demostrate by simulatio that with workig i very high order fields we a approah hael apaity for shorter blok legths. II. CONSTRCTION OF NONBINARY DPC CODES No biary DPC odes over Galois fields GF (), where is a prime umber, a be see as a geeralizatio of biary DPC odes over GF (). A vetor spae projeted over a fiite field GF () is used to deote the elemets i GF (). We selet to be of the form b () where b is a positive iteger suh that b >. The ode is defied i terms of a ultra sparse parity hek matrix. K is deoted as the legth of the message while N is used to deote the odeword legth. We defie the rate of the DPC ode R, where M R ( N M) N () N K. The retagular [ M N] ultra sparse parity hek matrix is ostruted havig a mea olum weight at least eual to two while the row weight ρ is made as uiform as possible. Suh algebrai ostrutio methods esure that ) eah row has exatly ρumber of elemets; ) eah olum has exatly umber of elemets; 3) ay two rows or two olums have more tha oe plae where they both have o zero ompoets. The first two oditios esure that the parity hek matrix has uiform row ad olum weights formig a (,ρ ) regular DPC ode while the third oditio esures that the miimum distae of the ode geerated is at least ad the Taer graph of the ode is free of yles of legth four. Algebrai /08/$ IEEE

2 CSNDSP Proeedigs ostrutio methods of DPC odes geerally ivolve first ostrutig irulat permutatio matries usig o biary elemets i GF () ad dispersig them ito a base matrix ostrutig the overall parity hek matrix. The sparse parity hek matries osidered i this paper are regular parity hek matries havig uiform row weights ρ ad olum weights geerated based o Reed Solomo (RS) odes over GF () with two iformatio symbols [], []. The ozero elemets of the parity hek matrix are defied from the RS ode to maximize the etropy of the orrespodig symbol of the sydrome vetor suh that T z (3) where ( K,, K ),, N suh that GF() deotes a valid ode word ad T deotes the traspose of the parity hek matrix. Gaussia elimiatio a be used o the parity hek matrix i order to obtai the systemati geerator matrix G. III. DECODING OF DPC CODES OER FINITE FIEDS The deodig problem i obiary DPC odes is to iteratively proess hek odes ad variable odes ad determie the most probable reeived ode word ĉ suh that z ˆ 0, where the likelihoods of is determied aordig to the hael model. The deodig of o biary DPC odes usig lassial belief propagatio (BP) algorithm was first proposed i [5] ad [6] with BCJR algorithm for hek ode proessig. The lassial BP algorithm yielded a omputatioal omplexity domiated by ( ) O maily owig to the BCJR blok i the hek ode proessig step, makig the deodig over higher order fields omputatioally ifeasible. Evidetly BP deodig of DPC odes over higher order fields reuire prohibitively large umber of omputatios rulig them out for pratial implemetatio. The idea of trasferrig the hek ode proessig ito the freuey domai ad salig dow the deodig omplexity was first proposed i [8] ad followed up by [3]. This paper otais a simple desriptio of the FTTBP algorithm ad shows how the Fast adamard Trasforms a be used i hek ode proessig. Fator graphs used to deode o biary DPC odes reuire two additioal bloks ) permutatio blok; ) reorderig blok; ompared to biary fator graphs. We a modify the biary fator as show i Figure to represet the iterative deodig of DPC ode. The fator graph is a bipartite graph osistig of set of variable odes (irular bloks at the top) ad hek odes (suare bloks at the bottom). Iferee over fator graph a aomplished by meas of passig messages betwee hek odes ad variable odes alteratively. I o biary DPC odes over GF (), the messages passed alog the edges of the fator graph orrespods to poit disrete probability set rather tha a sigle message. These probability distributios are exhaged iteratively i fidig a valid ode word. h h h h ( ) h Figure. Fator graph of o biary DPC odes deodig h N The fator graph show i Figure oets variable odes to hek odes through itermediate stages; permutatio ad re orderig. We a see from the fator graph that there are umber of permutatio bloks oeted to eah variable ode ad they eah orrespod to ozero etries foud uder orrespodig olums i the parity hek matrix. The permuted likelihood values are the set through a re orderig blok before oetig them to the hek odes. The algorithm is iitialized with the likelihood values of the reeived odeword. The likelihood value of odeword over GF() orrespods to the probabilities of th reeived odeword symbol beig eual to eah o biary elemet i GF (). If we defie the th reeived symbol by y k ad the likelihood value of the th symbol beig eual to a, a GF() by p ( y a), we a defie the th symbol likelihood values at the output. Eah likelihood value is oeted to umber of ozero etries i eah olum. The symbol likelihood values are used to iitialize the messages set from eah variable ode towards hek odes durig the first ru of the deoder. We a see that this is just a opy of the symbol likelihood values alog eah edge oeted to the variable ode. If we use ad

3 Proeedigs 587 CSNDSP08 to deote the iomig ad out goig messages relative to the variable ode respetively, we a deote the pdf message set alog the j th edge oeted to the th variable ode by thus iitialize the deoder by settig j j. We a (4) It is uite oveiet to referee the symbol likelihood values set from variable odes to the hek odes usig the biary represetatio of the elemets i GF (). This refereig system plays a pivotal role i the Fast Fourier trasformig likelihood values as explaied shortly. Eah j otais umber of disrete probabilities, ad effetively beomes a probability distributio. The iitialized symbol likelihood values are the set from variable odes to orrespodig permutatio bloks. It a be see from the Figure that umber of permutatios bloks are oeted to eah variable ode, relatig to eah ozero etry i the orrespodig olum. I the ase of (, ρ) regular o biary ode, there are ρ umber of ozero etries i eah row. Therefore, if h, K h,, h are the ozero etries i the j ji K jρ j th row of the parity hek matrix, we a write the j th parity hek euatio as, ρ h (5) ji i It is evidet from the euatio (5) that ulike i the ase of biary DPC, the parity hek euatios of o biary DPC odes otai ozero elemets i its parity hek euatios. This implies that the probability distributios otaied i the messages eeds to be permuted aordigly, takig the o biary elemets i the parity hek euatio ito aout. Due the struture of the Galois fields, the permutatio blok beomes a yli shift of the probabilities exept for the probability of the reeived ode word beig eual to 0. We a see that the it reuires i umber of yli shifts i the diretio of asedig order of filed elemets are eeded to permute the likelihood values i the ase that ozero etry h i ji α. The permuted pdfs are the subjeted to hek ode proessig, progressig the belief values further. The lassial BP uses well kow BCJR algorithm with all possible forward ad bak partial sums [5], [6] O. yieldig a deodig omplexity of ( ) It a be see from the trellis that this method yields a omputatioal omplexity domiated by O ( ) rulig out the deodig over high order Galois fields. That this limitatio a be oveietly overome by trasferrig the hek ode proessig i to freuey domai by usig FFT ad overtig the ovolutioal ode ito a produt ode as show i Figure. The FFT over Galois fields has a speial struture that a be effiietly represeted by a radix butterfly diagram as show i Figure. The Fourier trasforms over fiite sets, iludig Galois b fields of GF() where, a be deomposed i to set of d order Fourier trasforms applied alog eah dimesio of field [8], [3]. The fast Fourier trasform over fiite fields, groups is redued i to a reursive set of sums ad differees of the values haged by its bit loatios i the referee. We also defie the dimesio of the field elemets as the bit loatio uder osideratio; ragig from to b FFT Figure. Radix butterfly of FFT over (8) GF The radix butterfly show i Figure allows us to ompute the sums ad differees of the likelihood values reursively alog eah dimesio with eah reursio represetig the appliatio of d order FFT, whih is a sum ad a differee, alog a separate dimesio. Further aalysis ofirms that the FFT over Galois field redues i to Fast adamard Trasform () whih a be performed usig Walshadamrd matrix of the order eual to the field order. We a use the elemetal Walsh adamard matrix whih also orrespods to the d order FFT over Galois fields i (6) (6) ad apply the observatio (7) repeatedly i order to ostrut Walshadamard matrix mathig the field order. (7) This ojeture a be aommodated i our alulatios by reorderig the likelihood values projeted ) i the asedig order before the GF(

4 CSNDSP Proeedigs Fast adamard Trasformatio step. The reordered poit likelihood sets a the be trasferred ito freuey domai simply by multiplyig the likelihood values by order Walsh adamard Matrix as show i euatio (8). The Fast adamard Trasform geerates umber of disrete probability values. F T (8) ( ) W. As desribed i Figure, the deoder osists of total M umber of hek odes whih were simplified ito simple produt odes usig. Every hek ode oets ρ umber of Fourier trasformed poit probability sets usig termbyterm produt operatio i freuey domai. The termbyterm produt of two poit pdfs a be defied by, K ). (, K, ) ). (), K, ( ). ( ( ( ( )) (9) The M umber of rows of is represeted by M umber of produt odes i the deoder ad the ρ umber of ozero etries i eah row is represeted by ρ umber of edges oeted to eah of the produt ode. The belief propagatio aross horizotal plae a the be omputed simply by multiplyig termbyterm all the Fourier trasformed probability values oeted ito a produt ode with eah other, exept with the message alog the edge ρ pdfs. If uder osideratio totallig up to ( ) we use ad to deote the iomig ad out goig pdfs relative to the variable ode, we a deote the output of the i th edge oeted to m th produt ode by ρ mi mj j j i (0) The pdf messages set from the hek odes to the variable odes take the same path i the opposite way. Every operatioal blok o its path towards the hek odes implies the iverse of operatio it performed o its way towards the hek ode. The pdf messages at the output of the hek odes are i the freuey domai ad eeds to be overted bak to its dual. The blok i the opposite diretio implies the iverse Fast adamard Trasform applied to the poit pdf. We a observe that the iverse Fast adamard Trasform is exatly the same as the Fast adamrd Trasform ad we a multiply eah poit pdf set, agai by the order Walsh adamard matrix i order to obtai the iverse Fast adamard Trasform as show below. I ( ) T. () The messages are the subjeted to the iverse of the reorderig, dispersig the likelihood values i to their origial loatios followed up by the iverse of the yli shift operatio, exatly the same umber of yli shifts of the likelihood values it was subjeted ow i the opposite diretio. The likelihood values are the proessed at the variable odes, propagatig the belief values i the vertial plae. ariable ode proessig implies the likelihood termbyterm multipliatio of ( ) values alog the edges oeted to variable ode, exept for the message alog the edge uder osideratio. We a defie the output of the j th edge oeted to th variable ode by j i i j () ard deisio is the take o the likelihood values at the variable odes by determiig the symbols from the o biary alphabet GF() as show below arg max GF( ) ˆ (3) i The freuey domai implemetatio of the hek ode proessig redues the omplexity to. log i ompariso with the omplexity ( ) O domiated by O ( ) i. It is iterestig to ote that this is exatly the same omplexity redutio foud i CooleyTukey FFT algorithm. The omplexity redutio allows fast proessig of vast amouts of data eablig deodig obiary DPC odes defied over very large order Galois fields. Deodig over higher order fields failitates ear asymptoti performae, drawig loser to the hael apaity. PERFORMANCE COMPARISON The optimum fast belief propagatio deodig usig Fourier trasforms is osidered with respet to differet field orders. I order to make a fair ompariso, the ode words with respet to differet fiite fields are ostruted to have the same amout of biary iformatio. I Figure 3 the odes osidered are () (4 GF, N 040 ), ( GF ), N 06 ), ( GF (6), N 504 ), ( GF (64), N 378 ), ( GF (56), N 48 ). All the odes ρ leadig to are ostruted havig the same row weight 8 ad the same olum weight 4 exatly the same ode rates i all five ases R. These odes are simulated i Additive White Gaussia Noise (AWGN) Chael. Also represeted i the Figure 3 is the Bit Error Rate

5 Proeedigs 589 CSNDSP08 BER.00E00.00E0.00E0.00E03.00E04.00E05.00E06 (BER) performae urve for the uoded bits uder the ifluee of the same hael oditios. It ould be observed that there is ideed a performae gai i movig ito higher order field reahig towards the Shao s limit. I. CONCSION I this paper we establish that workig i a higher order Galois field, sigifiatly improve the performae of the DPC ode with moderate ode legths. This a be uite oveiet i implemetatio of DPC odes as it yields reasoable performae eve with relatively smaller frame sizes. The fast deodig algorithm based o fast Fourier trasforms redues the omputatioal omplexity of the belief propagatio algorithm sigifiatly ad workig i higher order Galois fields are made omputatioally feasible. It is demostrated by simulatio that there is a sigifiat performae gai betwee the odes over () GF, GF (4), GF (6), GF(64) ad GF (56). It is evidet from the simulatio results that usig very high order DPC odes we ould draw ear the hael apaity realizig earasymptoti performae levels (Eb/N0) SNR db BER oded AWGN (0000 trials) GF(56), N5 GF(), N040 GF(4), N06 GF(6), N504 GF(64), N378 Figure 3. Performae ompariso, DPC ode, R, GF (), N 040 ), ( GF (4), N 06 ), ( GF (6), N 504 ), ( GF (64), N 378 ), ( GF (56), N 48 ). ACKNOWEDGEMENT This researh is supported by the Egieerig ad Physial Siees Researh Couil (EPSRC) K, The K govermet s leadig fudig agey for researh ad traiig i egieerig ad physial siees. REFERENCES []. Rathi ad R. rbake, "Desity Evolutio, Thresholds ad the Stability Coditio for Nobiary DPC Codes." [] R. Gallager ad. Codes, "Cambridge," i MA: MIT Press Moograph, 963. [3] D. MaKay ad R. Neal, "Good Codes based o ery Sparse Matries," Cryptography ad Codig: 5th IMA Coferee, Cireester, K, Deember 80, 995: Proeedigs, 995. [4] D. MaKay ad R. Neal, "Near Shao limit performae of low desity parity hek odes," Eletrois etters, vol. 33, pp , 997. [5] M. Davey ad D. MaKay, "ow desity parity hek odes over GF ()," Iformatio Theory Workshop, 998, pp. 707, 998. [6] M. Davey, "Errororretio usig owdesity Parity Chek Codes," iv. of Cambridge PhD dissertatio, 999. [7] D. Sridhara ad T. Fuja, "ow desity parity hek odes over groups ad rigs," Iformatio Theory Workshop, 00. Proeedigs of the 00 IEEE, pp. 6366, 00. [8]. Barault ad D. Deler, "Fast Deodig Algorithm for DPC over GF ( )," The Pro. 003 Iform. Theory Workshop, pp , 003. [9] C. Poulliat, M. Fossorier, ad D. Deler, "Desig of o biary DPC odes usig their biary image: algebrai properties," Iformatio Theory, 006 IEEE Iteratioal Symposium o, pp. 9397, 006. [0] S. i, S. Sog, Y. Tai,. a, ad. Zeg, "Algebrai Costrutios of Nobiary QuasiCyli DPC Codes," Commuiatios, Ciruits ad Systems Proeedigs, 006 Iteratioal Coferee o, vol., 006. [] Y. Tai,. a,. Zeg, S. i, ad K. AbdelGhaffar, "Algebrai Costrutio of QuasiCyli DPC Codes for the AWGN ad Erasure Chaels," Commuiatios, IEEE Trasatios o, vol. 54, pp , 006. [] B. Zhou, Y. Tai,. a, S. Sog,. Zeg, ad S. i, "CT08: Costrutio of igh Performae ad Effiietly Eodable Nobiary QuasiCyli DPC Codes," Global Teleommuiatios Coferee, 006. GOBECOM'06. IEEE, pp. 6, 006. [3] D. Deler ad M. Fossorier, "Deodig Algorithms for Nobiary DPC Codes Over GF()," Commuiatios, IEEE Trasatios o, vol. 55, pp , 007.

University of Arizona ECE/OPTI 500C: Photonic Communications Engineering I C Fall Forward Error Correction (FEC) By Ivan B.

University of Arizona ECE/OPTI 500C: Photonic Communications Engineering I C Fall Forward Error Correction (FEC) By Ivan B. Uiversity of Arizoa ECE/OPTI 500C: Photoi Commuiatios Egieerig I C Fall 00 Forward Error Corretio (FEC) By Iva B. Djordjevi j Codig for Optial Chaels Importat lasses of odes: Liear lok odes Cyli odes Covolutioal