Analysis of Bipartite Graph Codes on the Binary Erasure Channel

Transcription

1 Anayss of Bpartte Graph Codes on the Bnary Erasure Channe Arya Mazumdar Department of ECE Unversty of Maryand, Coege Par ema: Abstract We derve densty evouton equatons for codes on bpartte graphs BG codes for the bnary erasure channe BEC We study the cases of oca codes beng ntroduced on ony one sde of the graph Generazed LDPC codes as we as on both sdes Each oca code s assumed to correct up to umber of erasures one ess than ts dstance We defne and enumerate stoppng sets for BG codes, whch serves an mportant too for anayss of decodng threshods on the BEC I INTRODUCTION Tanner s constructon of codes on graphs 3] assumes that oca constrants codes are mposed on the subsets of edges ncdent to every vertex of the graph A varant of ths constructon was consdered n 3], 7] where the graph was assumed bpartte wth one sde formed of varabe nodes and the other sde wth oca codes such as the one-error-correctng Hammng codes Ths cass of codes was termed Generazed Low Densty Party Chec GLDPC codes Codes on bpartte graphs n whch oca code constrants are mposed on both sdes and the symbos transmtted correspond to graph s edges were studed n 2] and umber of foow-up wors Ths constructon s caed bpartte-graph BG codes Weght dstrbuton of the versons of BG-codes dscussed was computed n 3], 7], 2] Later wors consdered other versons of ths constructon, such as codes constructed from protographs and computed ther weght dstrbutons 5], or anayzed ther performance wth ext charts 4] Let GV V 2, E be a bpartte graph In the standard constructon of LDPC codes, every u V corresponds to a varabe bt and every v V 2 corresponds to a v, v, 2] snge party-chec code C 2, where v s the degree of v Snce the varabe bt s repcated on every edge eavng the correspondng varabe node u V, we can thn of a u, u, ] repetton code C assocated wth the varabe node u where u s ts degree n G Repacng the code C 2 wth genera near bnary codes whch may be dfferent for dfferent vertces v, we obtan GLDPC codes of 3], 7] these papers assumed Hammng codes at every chec vertex If n addton the code C s repaced by nontrva near bnary codes dfferent vertces may have dfferent codes, we obtan the famy of genera BG codes Theoretca anayss of BG codes prevousy was performed usng graph expanson see 2] and ater wors Our purpose n ths paper s to anayze them under message passng decodng Densty evouton equatons aow us to compute the decodng error probabty for teratve decodng These equatons for an ensembe of LDPC codes for the bnary erasure channe BEC are found n ] and descrbed n greater deta n 0] Wth the hep of these equatons t was aso shown that f the channe erasure probabty s beow some threshod, teratve decodng used on an average code n the ensembe defned by random graphs corrects erasures n the receved vector n a but a sma fracton of transmssons Concentraton anayss performed n ] shows that the probabty of devaton from the average approaches zero exponentay wth the code s ength Paper 4] ntroduced stoppng sets as a combnatora too to characterze decodng faures and estmate the threshod of the message-passng agorthm earer these confguratons were consdered n 5] The dstrbuton of stoppng sets for reguar LDPC codes was found n 8] whch aso used t to estmate the boc error probabty of LDPC codes The paper s organzed as foows In Secton II, we fnd densty evouton equatons on the BEC for the ensembe of BG codes We consder separatey the cases when non-trva oca codes are ntroduced n ony one sde and both sdes of the graph To decode the oca codes we empoy bounded dstance decodng, assumng that each oca code corrects d erasures where d s ts dstance From the densty evouton equatons we obtan the threshod of teratve decodng for both cases We aso fnd an upper bound on the threshod by dervng a stabty condton smar to the one dscussed n 0] In Secton III we defne and enumerate stoppng sets for the reguar ensembe of BG codes Agan we consder the cases of generazed LDPC and douby generazed LDPC codes separatey empoyng the decodng descrbed above We show that the average boc erasure probabty of the ensembe approaches zero at east poynomay n the boc ength n f the erasure probabty of the channe p s ess than some threshod p th, smar to the standard LDPC codes n 8] For p < p th the man contrbuton to the boc erasure probabty s from stoppng sets of sze subnear n n whe the contrbuton from neary-szed stoppng sets decnes exponentay n n Secton IV concudes the paper

2 II DENSITY EVOLUTION EQUATIONS AND THRESHOLD Our anayss w be performed assumng the Tree Channe defned n 0] Asymptotc convergence of the bt error probabty of BG codes to that of the tree channe can be proved smary to that of the standard LDPC codes A Chec-sde generazed BG code Consder transmsson over the BECp wth GLDPC codes Our goa s to derve the densty evouton equatons for ther teratve decodng under whch n every even-numbered teraton a the oca codes at the chec nodes are decoded n parae up to ther mnmum dstance Suppose that the ensembe s characterzed by the degree dstrbutons λ and ρ, where λ s a vector whose th entry λ s the fracton of edges ncdent to varabe nodes of degree and ρ s a matrx n whch ρ,d refers to the fracton of edges ncdent to chec nodes of degree whose oca codes have dstance d We assume that there exst near codes of engths equa to the degrees of the vertces gven by ρ Otherwse the code ensembe can be defned n the same way as standard rreguar LDPC ensembes Ths remar aso appes to genera BG codes beow Denote fp, x p λ,d x x ρ,d d 2 0 Denote by x the average erasure probabty of a bt after teratons Theorem x satsfes the the foowng recurrence: x 0 p, x fp, x,, 2, Proof: The proof s done by nducton wth x 0 p servng as ts base We compute the average erasure probabty of a bt after teratons Consder a chec node of degree and oca code mnmum dstance d Consder the message gong out from ths node It w be an erasure f at east d ncomng messages among the remanng edges are erasures The probabty of ths event equas d 2 x x, 0 as x s the probabty of erasure after the th teraton The probabty that the edge s ncdent to a vertex v V 2 of degree and oca mnmum dstance d equas ρ,d Therefore, the probabty that the message sent aong ths edge to the varabe node s an erasure equas d 2 ] ρ,d x x,d 0 d 2 ρ,d x x,d 0 On the varabe sde consder ode of degree Ths node w send an erasure va an edge f the ncomng messages on a the other edges are erasures and the receved bt tsef was an erasure whch happens wth probabty p d 2 ρ,d x x,d 0 The probabty that an edge comng out from a varabe node of degree s λ Together wth the above ths mpes the theorem Our next goa s to examne some propertes of the functon fp, x Lemma fp, x s ncreasng on both arguments for p, x 0, ] Proof: It s obvous that fp, x s ncreasng on p To show that fp, x s ncreasng on x t suffces to prove that the quantty d 2 ν,d x x x 0 s decreasng on x 0, ] for a, d We have dν d x d 2 x x dx 0 d 2 x x 2 0 d 2 x x d x x d 2 x x ] d + x d 2 x d 0, d 2 whch proves the cam Let P p be the average probabty of erasure after teratons for the ensembe f the channe erasure probabty s p The next two cams foow drecty as a consequence of Lemma Ther proofs are smar to those for LDPC codes 0], Ch 3 Lemma 2 If P p 0 as, and p p, then P p 0 as Moreover, x s monotone n Theorem 2 P p converges to the nearest root of the equaton x fp, x as

3 Usng Lemma 2 we can defne the threshod of teratve decodng for the erasure channe Defnton Consder the ensembe of GLDPC codes characterzed by the degree dstrbuton λ, ρ The threshod probabty p λ, ρ s defned as p λ, ρ sup {p 0, ] : P p 0 as } Theorem 2 provdes the foowng characterzaton of the threshod p λ, ρ sup {p 0, ] : x fp, x has no soutons n 0, ]} nf {p 0, ] : x fp, x has a souton n 0, ]} To determne the threshod numercay we pot fp, x x The argest vaue of p for whch the entre curve s beow the x-axs gves us the threshod Stabty Condton: We have x fp, x fp, 0 + x fp, 0x + Ox 2 Assume that there are no nodes of degree, so fp, 0 0 By a cacuaton smar to the proof of Lemma, we have fp, 0 x p λ,d pλ 2 ρ,2 whch gves ρ,d d + x d 2 x d x0 d 2 x pλ 2 ρ,2 x + Ox 2 2 Notce that the ony contrbuton to the near term comes from chec nodes whose oca codes have dstance 2 Ths gves a condton for the fxed pont at 0 beng stabe Formay speang, we have Theorem 3 If pλ 2 ρ,2 > then m P p > 0 On the other hand f pλ 2 ρ,2 <, then there exsts ζ > 0 such that m P p 0 for a p 0, ζ Coroary p λ, ρ λ 2 ρ,2 3 B Genera BG code We now consder the ensembe of BG codes wth oca constrants ntroduced on both sdes of the graph G used on the BECp In contrast to the LDPC code constructon, the bts of the transmsson are assocated wth the edges of G Our purpose s to derve densty evouton equatons for message passng decodng assumng that the oca codes are decoded up to ther dstance The ensembe s characterzed by two matrces λ and ρ The, dth entry of λ ρ denotes the fracton of edges ncdent to a eft resp, rght node of degree whose oca code has dstance d Let f λ p, x p d 2 λ,d x x,d 0 and f ρ p, x p,d x x ρ,d d 2 0 We w assume that one teraton of decodng conssts of decodng a the rght codes n parae, updatng the bt vaues on the eft and decodng a the eft codes n parae Theorem 4 The erasure probabty x of a bt after teratons of message passng decodng satsfes the foowng recurrence: x 0 p, y f ρ p, x, x f λ p, y Proof: The proof agan goes by nducton Its base hods true by defnton Suppose that the statement s true up to teratons Let y be the probabty of erasure beng sent to the eft sde aong a randomy chosen edge n the th teraton Ceary, an edge w carry an erasure to eft f t was erased n transmsson and the rght decodng nvovng t dd not recover ts vaue Thus, y p d 2 ρ,d x x,d 0 The expresson for x foows mmedatey Lemma 3 The functon gp, x f λ p, f ρ p, x s ncreasng n both arguments for p, x 0, ] Proof: From Lemma t foows that both f λ p, x and f ρ p, x are ncreasng functons of p and x 0, ] Let P 2 p be the ensembe-average probabty of erasure after teratons We have the foowng: Lemma 4 If P 2 p 0 as, and p p, then P 2 p 0 as Moreover, x s monotonc n Theorem 5 P 2 p converges to the nearest root of the equaton x gp, x, as Defnton 2 Consder a BG code characterzed by the dstrbutons λ, ρ The threshod probabty p λ, ρ equas p λ, ρ sup {p 0, ] : P 2 p 0 as } 4 The threshod probabty can be characterzed as foows: p λ, ρ sup {p 0, ] : x gp, x has no souton n 0, ]} nf {p 0, ] : x gp, x has a souton n 0, ]}

4 Stabty Condton: Wrtng out a quadratc Tayor poynoma for g we obtan x gp, x gp, 0 + x gp, 0x + Ox 2 We get gp, 0 f λ p, f ρ p, 0 0 Moreover, whch gves x p 2 x gp, 0 f λp, f ρ p, 0f ρp, 0 f λp, 0f ρp, 0 p λ,2 λ 2 p ρ 2 ρ,2 x + Ox 2 5 Agan ony oca codes wth mnmum dstance 2 contrbute to the near term We have the foowng stabty condton for the fxed pont at 0, Theorem 6 If p 2 λ,2 ρ,2 x > then m P 2 p > 0 On the other hand f p2 λ,2 ρ,2 x <, then there s ζ > 0 such that m P 2 p 0 for a p 0, ζ An upper bound on the threshod probabty s gven n the next coroary Coroary 2 p λ, ρ C Ext Charts λ,2 ρ,2 An anayss of threshod probabty can aso be done usng the ext charts For the defnton of ext functon we refer the reader to Chapter 3 of 0] In the foowng anayss we consder b-reguar bpartte graph codes wth constant eft degree be and rght degree The anayss though can be easy generazed to rreguar bpartte graph wth a gven degree dstrbuton Let us assume that we have the codes C, ] and C, 2 ] on the eft and rght nodes respectvey Moreover we assume that, these codes are chosen from an ensembe of random near codes of the same parameters An anayss smar to the one beow appears n 9] The ext functon of a near code wth parameters n, ] n an erasure channe s gven by Equaton 40 of ], H n, x n n x g x n g g gẽ g,n, n g + ẽ g,n, ] 6 where, ẽ h,n, s the unnormazed nformaton functon defned n 6] for a code wth parameters n, ] For random near codes wth parameters n, ], the expected nformaton functon s gven by, Eẽ h,n, n h h h n r ] r r0 ] n h r ] 2 rn h +r where a r] s the Gaussan bnoma coeffcent defned by ] r a 2 a 2 r 2 r 2 Consder transmsson over BECp In ths case the ext functon from the eft hand sde of the bpartte graph to an edge s H E x H, x 7 The ext functon from the rght hand vertces, H E2,p s H E2,p x ph, 2 px 8 Ext Chart of BG Code wth 24,8 random near oca codes on BEC H E x H x E2,035 H E2,042 x H E2,045 x x Fg The Ext Chart for a Bpartte Graph Code wth random near component codes In Fg we have potted the ext chart for a bpartte graph code wth parameters 24 and 2 8 The ext functons are average of a near codes of parameters 24, 8] The fgure shows the chart for a channe wth erasure probabtes p 035, p 042 and p 045 From the ext chart anayss, the threshod probabty for the above mentoned code s gven by p 045 The code has rate > III STOPPING SETS AND THEIR ENUMERATION Let us defne stoppng sets n the context of teratve decodng on the BEC Defnton 3 A subset of edges s caed a stoppng set f erasures n these edges submtted to the next teraton

5 of teratve decodng are not recovered n ths teraton By defnton, empty set s a stoppng set For GLDPC codes stoppng sets can be equvaenty defned as subsets of varabe nodes The set of stoppng sets w be denoted by Γ Consder an ensembe of BG codes of ength n Defne the normazed ensembe-average stoppng set dstrbuton γα, α 0, ] as foows Let γα m n E {S Γ : S αn} ] 9 n n p th sup{p : max γα α 0,] p α + αh hp] < 0} 0 α where, h s the bnary entropy functon n nats Let P B C n, p be the boc erasure probabty of a code C n from the ensembe on the BECp Theorem 7 Let p < p th Then such that EP B C n, p] E {S Γ : S } ]p +exp Θn where the expectaton s over the ensembe of BG codes of ength n and m n n 0 For standard LDPC codes ths resut appears n 8] The proof for the generazed case s competey anaogous Moreover t s nown that for reguar LDPC codes, the rght-hand sde of goes to 0 poynomay wth n We w show that the same hods for genera BG codes and w aso fnd an expresson for γα These resuts mpy a computabe bound on the threshod for boc error rate of BG codes The quanttatve defnton of stoppng sets depends on the oca decodng empoyed adopted n the teratve agorthm In our resuts beow we agan assume that a oca code can correct up to d erasures, where d s ts dstance In the foowng we consder reguar BG codes wth eft degree and rght degree Moreover we assume that a the oca codes on one sde are the same t suffces to assume that they have the same mnmum dstance or are decoded up to the same number of erasure Extenson of the resuts to the case of rreguar BG codes wth gven eft and rght degree dstrbutons as we as dfferent code dstances s straghtforward athough the resuts become more cumbersome A GLDPC codes Consder reguar GLDPC codes wth eft degree and rght degree Let V 2 and V be the sets of chec nodes and varabe nodes respectvey Let d denote the mnmum dstance of the oca codes at the vertces v V In each teraton these codes are decoded up to d erasures Denote by N R S the neghborhood n V 2 of a subset of vertces S V From Defnton 3 we have the foowng Defnton 4 A stoppng set S V s a subset of vertces such that any vertex v N R S s connected to at east d vertces n S Accordng to the above defnton, the set of stoppng sets s cosed under the unon operaton So every subset of varabe nodes has a unque maxma stoppng set whch can be an empty set If Ω V s the set of bts varabe nodes erased n transmsson, then the set of erasures whch remans when the decoder stops s equa to the unque maxma stoppng set contaned n Ω Theorem 8 The expected number of stoppng sets of sze s n the reguar ensembe of GLDPC codes s gven by, E {S Γ : S s} ] coef n s + x d x n s n, x s ] where coeffx, x denotes the coeffcent of x n fx 2 Proof: Let E be the set of edges of G Gven a chec node c the number of ways of choosng of ts socets s coef+x, x The number of ways of choosng of ts socets such that 0 or d equas coef + x d x, x So the number of ways of choosng e chec node socets from a the socets of V 2, such that every chec node connected to the socets s connected to at east d of them, s gven by coef+x d x ] V2, x e Let U V, U s The number of edges ncdent to U s e s The probabty that these e edges are such that U becomes a stoppng set equas P ru Γ coef + x d coef E e + x d x ] V2, x e ] x n, x s n s Fnay there are n s ways of choosng U, whch proves the theorem Theorem 9 The normazed average stoppng set dstrbuton s gven by x 0 γα n + d x α 0 where x 0 s the postve souton of the equaton α + d hα, 3 α x 0 0 Proof: From the defnton of γα we have

6 n γα m n n n αn coef + x d n α n ] x n, x α ] n hα + m n coef n n d ] x n, x α n hα + m n n + x x d,x d+,,x : d xα n n x d, x d+,, x, n d x d x Let y x n Notcng that the above sum contans umber of terms poynoma n n and usng asymptotc propertes of mutnoma coeffcents, we have γα hα + hy d, y d+,, y, y + d max y d,y d+,,y : d yα d y n where hz,, z t t z n/z Evauatng the maxmum, we obtan the cam of the theorem Next et us use the above resuts to estmate the threshod of boc error rate for teratve decodng of GLDPC codes We need the foowng emma Lemma 5 Let < m and d >, then d ] coef + x 2 x m, x d m + d 2 3 Proof: The proof proceeds n the same way of countng as n Lemma 8 of 8], whch gves a smar statement for d 2 Except generazng ths, we use the foowng bound, d m d + m d Theorem 0 Let p 0, ], then E {S Γ : S } ]p O n /d such that m n n 0 4 Proof: We use the bound of the above emma as foows, s s s E {S Γ : S s} ]p s n s n s coef + x d s d n s ] x n, x s s + n s 2 3 s d n s whch can be bounded above by decreasng geometrc sequences, for a on Therefore, E {S Γ : S s} ]p s O n /d s whch proves the theorem The above theorem aong wth Equaton estabshes p th as the threshod of boc erasure probabty beow whch the expected boc erasure probabty goes to 0 wth n Numerca vaues of p th can be easy computed from Equatons 0 and 3 B Genera BG codes Consder reguar BG codes wth eft degree and rght degree Let E be the set of edges of G, E n V V 2 Consder the ensembe of graphs obtaned by connectng the vertces n V wth the vertces n V 2 usng a permutatons on the set of n edges Denote by d and d 2 the oca dstances at the vertces of V and V 2, respectvey In teratons, the oca codes w be decoded to correct d erasures, where d d or d 2 as approprate Defnton 5 Let S E be a subset of edges and et V S and V 2 S be the sets of eft and rght nodes to whch they are ncdent Then S s caed a stoppng set f every u V S has S-degree at east d n S and every v V 2 S has S- degree at east d 2 As before, the set of stoppng sets s cosed under tang the unon Therefore, every subset of edges has a unque maxma stoppng set whch can be an empty set If Ω E s the subset of bts erased n transmsson, then the set of erasures whch remans when the decoder stops s equa to the unque maxma stoppng set of Ω Theorem The expected number of stoppng sets of sze s n the, -breguar ensembe of BG codes s gven by E {S Γ : S s} ] 2 coef + x d x n, x s ] n s Proof: Consder an s-subset U E From Theorem 8, the number of ways of choosng s rght node socets from a the socets of V 2, such that every rght node connected to the socets s connected to at east d 2 of them, s gven by p s p s

7 coef+x d 2 x ] V2, x s A smar expresson can be found for the eft sde If the graph s chosen randomy then the events of U satsfyng the constrants on the eft sde and the rght sde are ndependent So we have P ru Γ] 2 coef + x d n s 2 ] x n, x s because V n and V 2 n Snce there are n s ways of choosng U, ths competes the proof Theorem 2 The normazed average stoppng set dstrbuton s gven by γα 2 n + ] d x x α hα where x,, 2 s the postve souton of the equaton α + α x 0 d Proof: From the defnton of γα we have, γα m n n 2 coef + x d n n αn hα + m n 2 n coef + x n d ] x n, x αn x n, x αn ] We proceed the same way as Theorem 9, obtanng γα hα + 2 hy d, y d +,, y, max y d,y d +,,y : d y α d y + d y n Evauatng the maxmum, we obtan the statement of the theorem Theorem 3 Let p 0, ], then E {S Γ : S } ]p O n /d /d 2 such that m n n 0 5 Proof: We use the bound of the Lemma 5, to have, s E {S Γ : S s} ]p s p s s 2 coef + x d s n s x n, x s ] 2 s d s + n 2 3 s s d n s p s whch can be bounded from above by decreasng geometrc sequences for on Therefore, a E {S Γ : S s} ]p s O n /d /d2 s From, f the erasure probabty satsfes p < p th, the expected boc error probabty goes to 0 wth n We can compute p th numercay usng Equaton 0 and Theorem 2 IV CONCLUDING REMARKS We notce from both equatons 2 and 5, that for an ensembe of BG codes wth oca code mnmum dstance at east 3, the near terms dsappear, mang x Ox 2 Ths means that the fxed pont of densty evouton equatons at 0 s aways stabe n these cases In case of conventona LDPC codes, the stabty condton mposes an upper bound on the threshod probabty, but n an ensembe of BG codes wth oca code mnmum dstance at east 3 no such constrant s mposed on the threshod If we tae an ensembe of graphs wth gven degree dstrbuton and consder GLDPC codes on t, we obtan a better threshod from the densty evoutons compared to that of LDPC codes on the same ensembe The same appes for the threshod p th for the average boc error probabty Consder a breguar ensembe of bpartte graphs wth eft and rght degrees and respectvey For GLDPC codes, f the rate of the oca codes at the chec node s fxed at R L, then the overa rate of the code s R L, whch s ower than LDPC codes on the same graph Thus the better decodng threshod of these codes s at the expense of decreasng the rate For a genera BG code on a breguar graph, the overa rate s R +R 2, where codes of rate R and R 2 are used on the eft and rght nodes respectvey These codes have better rate and threshods compared to GLDPC codes 2 The stoppng set dstrbuton above s derved n the context of bounded dstance decodng If we now the ran dstrbuton of the oca codes, then t s aso possbe to fnd the stoppng set dstrbuton under ML decodng at the vertces For a gven vertex v et C v be the oca code assocated wth t Denote by Ev and N L v, respectvey the set of edges ncdent to v and the set of varabe nodes connected to v The

8 defntons of stoppng sets n ths case shoud be modfed as foows Defnton 6 Stoppng sets for GLDPC codes wth oca ML decodng: Let S V and et N R S be the set of chec nodes connected to S S s caed a stoppng set f for every v N R S the set of edges N L v \ S does not contan an nformaton set of C v Defnton 7 Stoppng sets for genera BG codes wth oca ML decodng: A subset S E s caed a stoppng set f for any u V S, the set Eu \ S does not contan an nformaton set of C u, and for any v V 2 S, the set Ev\S does not contan an nformaton set of C v 5] V V Zyabov and M S Pnser, Compexty of Decodng Low-Densty Codes n Transmsson Over a Channe wth Erasures, Prob Inform Trans, Vo 0, no, 974, 5 28 In a future wor, we pan to study the performance of BG codes under message passng decodng on the bnary symmetrc channe and other dscrete channes Acnowedgement The author s gratefu to Aexander Barg for numerous hepfu dscussons Ths research s supported n part by NSF grant CCF05524 REFERENCES ] A Ashhmn, G Kramer and S ten Brn, Extrnsc Informaton Transfer Functons: Mode and Erasure Channe Propertes, IEEE Transactons on Informaton Theory, Vo 50, No, November ] A Barg and G Zémor, Dstance Propertes of Expander Codes, IEEE Transactons on Informaton Theory, Vo 52, No, January ] J Boutros, O Pother and G Zémor, Generazed Low Densty Tanner Codes, Proceedngs of IEEE Internatona Conference on Communcatons ICC, Vo, Vancouver, BC, Canada, 999, pp ] C D, D Proett, I E Teatar, T J Rchardson and R L Urbane, Fnte-Length Anayss of Low Densty Party-Chec Codes on Bnary Erasure Channe, IEEE Transactons on Informaton Theory, Vo 48, No 6, June ] D Dvsaar, Ensembe Weght Enumerators for Protograph LDPC codes, Proceedngs of IEEE Internatona Symposum on Informaton Theory ISIT, Seatte, USA, Juy, 2006, pp ] T Heeseth, T Kove and V I Levenshten, On the Informaton Functon of an Error-Correctng Code, IEEE Transactons on Informaton Theory, Vo 43, No 2, March 997 7] M Lentmaer and K Sh Zgangrov, On Generazed Low Densty Party Chec Codes Based on Hammng Component Codes, IEEE Communcaton Letters, Vo 3, No 8, 999 8] A Ortsy, K Vswanathan and J Zhang, Stoppng Set Dstrbuton of LDPC code ensembes, IEEE Transactons on Informaton Theory, Vo 5, No 3, March ] E Paon, M Fossorer and M Chan Anayss of Generazed LDPC Codes wth Random Component Codes for the Bnary Erasure Channe, Internatona Symposum on Informaton Theory and ts Appcatons ISITA, Seou, Korea, October 29 - November, ] T J Rchardson and R L Urbane, Modern Codng Theory, avaabe onne at ] T J Rchardson, M A Shoroah and R L Urbane, Desgn of Capacty-Approachng Irreguar Low-Densty Party-Chec Codes, IEEE Transactons on Informaton Theory, Vo 47, No 2, February ] M Spser and D A Speman, Expander Codes, IEEE Transactons on Informaton Theory, Vo 42, No 6, November 996 3] R M Tanner, A Recursve Approach to Low Compexty Codes, IEEE Transactons on Informaton Theory, Vo IT-26, No 5, September 98 4] Y Wang and M Fossorer, Douby Generazed LDPC codes, Proceedngs of IEEE Internatona Symposum on Informaton Theory ISIT, Seatte, USA, Juy, 2006, pp