Unconventional Behavior of the Noisy Min-Sum Decoder over the Binary Symmetric Channel

Transcription

1 Unconvntional Bhavior of th Noisy Min-Sum Dcodr ovr th Binary Symmtric Channl Christian L. Kamni Ngassa,#, Valntin Savin, David Dclrcq # CEA-LETI, Minatc Campus, Grnobl, Franc, {christian. kamningassa, valntin.savin}@ca.fr # ETIS, ENSEA / CNRS UMR-851 / Univ. Crgy-Pontois, Franc, dclrcq@nsa.fr Abstract This papr invstigats th bhavior of th noisy Min-Sum dcodr ovr binary symmtric channls. A noisy dcodr is a dcodr running on a noisy dvic, which may introduc rrors during th dcoding procss. W show that in som particular cass, th nois introduc by th dvic can hlp th Min-Sum dcodr to scap from fixd points attractors, and may actually rsult in an incrasd corrction capacity with rspct to th noislss dcodr. W also rval th xistnc of a spcific thrshold phnomnon, rfrrd to as functional thrshold. Th bhavior of th noisy dcodr is dmonstratd in th asymptotic limit of th cod-lngth, by using noisy dnsity volution quations, and it is also vrifid in th finit-lngth cas by Mont-Carlo simulation. I. INTRODUCTION In traditional modls of communication or storag systms with rror corrction coding, it is assumd that th oprations of an rror corrction ncodr and dcodr ar dtrministic and that th randomnss xists only in th transmission or storag channl. Howvr, with th advnt of nanolctronics, th rliability of th forthcoming circuits and computation dvics is bcoming qustionabl. It is thn bcoming crucial to dsign and analyz rror corrcting dcodrs abl to provid rliabl rror corrction vn if thy ar mad of unrliabl componnts. Ovr th last yars, th study of rror corrcting dcodrs, spcially Low-Dnsity Parity-Chck (LDPC dcodrs, running on noisy hardwar attractd mor and mor intrst in th coding community. In [1], [2], analytical mthods hav bn proposd to valuat th prformanc of on stp majority logic LDPC dcodrs constructd from faulty gats. In [3] and [4] hardwar rdundancy is usd to dvlop faultcompnsation tchniqus, abl to protct th dcodr against th rrors inducd by th noisy componnts of th circuit. In [5], a class of modifid Turbo and LDPC dcodrs has bn proposd, abl to dal with th nois inducd by th failurs of a low-powr buffring mmory that stors th input soft bits of th dcodr. Vry rcntly, th charactrization of th ffct of noisy procssing on mssag-passing itrativ LDPC dcodrs has bn proposd. In [6], th concntration and convrgnc proprtis wr provd for th asymptotic prformanc of noisy mssag-passing dcodrs, and dnsity volution quations wr drivd for th noisy Gallagr-A and Blif-Propagation (BP dcodrs. In [7] [9], th authors invstigatd th asymptotic bhavior of th noisy Gallagr- B dcodr dfind ovr binary and non-binary alphabts. In This work was supportd by th Svnth Framwork Programm of th Europan Union, undr Grant Agrmnt numbr (i-risc projct. all ths paprs th noisy implmntation of th itrativ mssag passing dcodrs was mulatd by passing ach of th xchangd mssags through a noisy channl. In our prvious works [1], [11], w proposd various rror modls for th arithmtic componnts of th Min-Sum (MS dcodr and drivd dnsity volution quations for th noisy MS. In this papr, w furthr invstigat th asymptotic and finit-lngth bhavior of th noisy MS dcodr ovr th Binary Symmtric Chanl (BSC. W rfin th probabilistic rror modls for th noisy MS dcodr w usd prviously in [1], [11], and discuss thir symmtry proprtis. W conduct a thorough asymptotic analysis of th noisy MS dcodr and highlights a wid varity of mor or lss convntional bhaviors. W also rval th xistnc of a spcific thrshold phnomnon, which is rfrrd to as functional thrshold. Finally, th asymptotic rsults ar also corroboratd through finit lngth simulations. Th papr is organizd as follows. Sction II discusss th probabilistic rror modls and and xplains th motivations bhind thm. Th noisy MS dcodr is also introducd in this sction. Sction III shortly discusss th dnsity volution for th noisy MS dcodr, and provids th notation and dfinitions rquird to undrstand th asymptotic analysis of th noisy MS dcodr, which is thn conductd in Sction IV. Finally, Sction V corroborats th asymptotic analysis through finit-lngth simulations, and Sction VI concluds th papr. II. PROBABILISTIC ERROR MODELS FOR THE NOISY MIN-SUM DECODER A. Noisy Mssag-Passing Dcodrs Th modl for noisy Mssag-Passing (MP dcodrs proposd in [6] incorporats two diffrnt sourcs of nois: computation nois du to noisy logic in th procssing units, and mssag-passing nois du to noisy wirs (or noisy mmoris usd to xchang mssags btwn nighbor nods. Th computation nois is modld as a random variabl, which th variabl-nod or th chck-nod procssing dpnds on. Put diffrntly, an outgoing mssag from a (variabl or chck nod dpnds not only on th incoming mssags to that nod, but also on th ralization of a random variabl, which is assumd to b indpndnt of th incoming mssags. Th mssag-passing nois is simply modld as a noisy channl. Hnc, transmitting a mssag ovr a noisy wir is mulatd by passing that mssag through th corrsponding noisy channl.

2 Howvr, in [6] it has bn notd that thr is no ssntial loss of gnrality by combining computation nois and mssag-passing nois into a singl form of nois. Consquntly, th approach adoptd has bn to mrg computation noisy into mssag-passing nois, and to mulat noisy dcodrs by passing th xchangd mssags through diffrnt noisy channl modls. Thus, th noisy Gallagr-A dcodr has bn mulatd by passing th xchangd mssags ovr indpndnt and idntical BSC wirs, whil th noisy BP dcodr has bn mulatd by corrupting th xchangd mssags with boundd and symmtrically distributd additiv nois (.g. uniform nois or truncatd Gaussian nois. Th approach w follow in this work diffrs from th on in [6] in that th computation nois is modld at th lowr lvl of arithmtic and logic oprations that compos th variabl-nod and chck-nod procssing units. This finrgraind nois modling is aimd at dtrmining th lvl of nois that can b tolratd in ach typ of opration. As th main focus of this work is on computation nois, w shall considr that mssags ar xchangd btwn nighbor nods through rror-fr wirs (or mmoris. Howvr, w not that this work can radily b xtndd to includ diffrnt rror modls for th mssag-passing nois (as dfind in [6]. Altrnativly, w may assum that th mssag-passing nois is mrgd into th computation nois, in th sns that adding nois in wirs would modify th probabilistic modl of th noisy logic or arithmtic oprations. B. Probabilistic Error Modls for Noisy Addrs, Comparators and XOR-gats In this sction w dscrib th probabilistic modls for noisy addrs, comparators and xor-oprators, that will b usd in th nxt sction, in ordr to mulat th noisy implmntation of th finit-prcision MS dcodr. 1 Noisy Addr Modl: W considr a θ-bit addr (θ 2. Th inputs and th output of th addr ar assumd to b in V = { Θ,..., 1,, +1,..., +Θ}, whr Θ = 2 θ 1 1. For inputs (x, y V, th output of th noislss θ-bit addr is givn by v = s V (x + y, whr s V : Z V dnots th θ-bit saturation map: s V (z = sgn(z min( z, Θ (1 Th output of th noisy addr will b dfind by injcting rrors in th output of th noislss on. Two main rror injction modls will b usd in this work, both of which ar basd on a bitwis XOR opration btwn th noislss output v and an rror. Th rror is assumd to b drawn from an rror st E V, according to an rror probability distribution p E : E [, 1]. Th two modls diffr in th dfinition of th rror st E, which is chosn such that th bitwis XOR opration (i.. th rror injction may or may not affct th sign of th noislss output. In th first cas th rror injction modl is said to b full-dpth, whil in th scond it is said to b sign-prsrving. W fix a signd numbr binary rprsntation, which can b any of th sign-magnitud, on s complmnt, or two s complmnt rprsntation. Thr ar xactly 2 θ signd numbrs that can b rprsntd by θ bits in any of th abov formats, on of which dos not blong to V (not that V contains only 2Θ + 1 = 2 θ 1 lmnts for symmtry rasons!. W dnot this lmnt by ζ. For instanc, in two s complmnt format, ζ = (Θ + 1, with binary rprsntation 1. Full-dpth rror injction: For this rror modl th rror st is E = V. For symmtry rasons, all rrors ar assumd to occur with th sam probability. It follows that p E ( = 1 and p E ( = 2Θ,, whr > is rfrrd to as th rror injction probability. Finally, th rror injction function is dfind by: ı(v, = { v, if v V, if v = ζ Sign-prsrving rror injction: For this rror modl th rror st is E = {, +1,..., +Θ}. Th rror injction probability is dnotd by, and all rrors ar assumd to occur with th sam probability (for symmtry rasons. It follows that p E ( = 1 and p E ( = Θ,. Finally, th rror injction function is dfind by: ı(v, = v, if v and v V ±, if v =, if v = ζ In th abov dfinition, ı(, is randomly st to ithr or +, with qual probability (this is du onc again to symmtry rasons. Not also that th last two conditions, namly v = and v = ζ, cannot hold simultanously (sinc ζ. Finally, both of th abov rror injction modls satisfy th following symmtry condition: p E ( = p E (, v, w V (4 { ı(v,=w} { ı( v,= w} A particular cas in which th symmtry condition is fulfilld is whn ı( v, = ı(v,, for all v V and E. In this cas, th rror injction modl is said to b highly symmtric. W not that both of th abov modls ar highly symmtric, if on of th sign-magnitud or th on s complmnt rprsntation is usd. In cas that th two s complmnt rprsntation is usd, thy ar both symmtric, but not highly symmtric. Rmark: It is also possibl to dfin a variabl dpth rror injction modl, in which rrors ar injctd in only th λ last significant bits, with λ θ [1]. Hnc, λ = θ corrsponds to th abov full-dpth modl, whil λ = θ 1 corrsponds to th sign-prsrving modl. Howvr, for λ < θ 1 such a modl will not b symmtric, if th th two s complmnt rprsntation is usd! Finally, for any of th abov rror injction modls, th output of th noisy addr is givn by: (2 (3 a pr (x, y = ı (s V (x + y,, (5 whr is drawn randomly from E according to th probability distribution p E. Th rror probability of th noisy addr, i..

3 Pr (a pr (x, y s V (x + y, assuming uniformly distributd inputs, quals th rror injction probability paramtr. Obviously, it would b possibl to dfin mor gnral rror injction modls, in which th injctd rror dpnds on th data (currntly and/or prviously procssd by th addr. Such an rror injction modl would crtainly b mor ralistic, but it would also mak vry difficult to analytically charactriz th bhavior on noisy MP dcodrs. As a sid ffct, th dcoding rror probability would b dpndnt on th transmittd codword, which would prvnt th us of th dnsity volution tchniqu for th analysis of th asymptotic dcoding prformanc (sinc th dnsity volution tchniqu rlis on th all-zro codword assumption. 2 Noisy Comparator Modl: Lt lt dnot th noislss lss than oprator, dfind by lt(x, y = 1 if x < y, and lt(x, y = othrwis. Th noisy lss than oprator, dnotd by lt pr, is dfind by flipping th output of th noislss lt oprator, with som probability valu that will b dnotd in th squl by p c. Finally, th noisy minimum oprator is dfind by: { x, if ltpr (x, y = 1 m pr (x, y = (6 y, if lt pr (x, y = 3 Noisy XOR Modl: Th noisy XOR oprator, dnotd by x pr, is dfind by flipping th output of th noislss oprator with som probability valu, which will b dnotd in th squl by p x. It follows that: { x y, with probability 1 px x pr (x, y = (7 x y, with probability p x Assumption: W furthr assum that th inputs and th output of th XOR oprator may tak valus in ithr {, 1} or { 1, +1} (using th usual,1 to ±1 convrsion. This assumption will b implicitly mad throughout th papr. C. Nstd Oprators For th Min-Sum dcoding, svral arithmtic/logic oprations must b nstd 1 in ordr to comput th xchangd mssags. Sinc all ths oprations (additions, comparisons, XOR ar commutativ, th way thy ar nstd dos not hav any impact on th infinit-prcision MS dcoding. Howvr, this is no longr tru for finit-prcision dcoding, spcially in cas of noisy oprations, and on nds an assumption about how th abov oprators xtnd from two to mor inputs. Our assumption is th following. For n 2 inputs, w randomly pick any two inputs and apply th oprator on this pair. Thn w rplac th pair by th obtaind output, and rpat th abov procdur until thr is only on output (and no mor inputs lft. Th formal dfinition gos as follows. Lt Ω Z and ω : Ω Ω Ω b a noislss or noisy oprator with two oprands. Lt {x i } i=1:n Ω b an unordrd st of n oprands. W dfin: ω ({x i } i=1:n = ω( (ω(x π(1, x π(2,, x π(n, (8 whr π is a random prmutation of 1,..., n. 1 For instanc, (d n 1 additions whr d n dnots th dgr of th variabl-nod n ar rquird in ordr to comput ach α m,n mssag. Algorithm 1 Noisy Min-Sum (Noisy-MS dcoding Input: y = (y 1,..., y N Y N Output: ˆx = (ˆx 1,..., ˆx N { 1, +1} N rcivd word stimatd codword Initialization for all n = 1,..., N do γ n = q(y n ; for all n = 1,..., N and m H(n do α m,n = γ n; Itration Loop for all m = 1,..(., M and n H(m do CN-procssing β m,n = x pr {sgn(αm,n ( } n H(m\n m pr { αm,n } n H(m\n ; for all n = 1,..(., N and m H(n do VN-procssing α m,n = a pr {γn } {β m,n} m H(n\m ; α m,n = s M (α m,n ; for all n = 1, (..., N do AP-updat γ n = a pr {γn} {β m,n} m H(n ; for all {v n} n=1,...,n do ˆx n = sgn( γ n; if ˆx is a codword thn xit itration loop End Itration Loop D. Noisy Min-Sum Dcodr hard dcision syndrom chck W considr a finit-prcision MS dcodr, in which th a priori information (γ n and th xchangd xtrinsic mssags (α m,n and β m,n ar quantizd on q bits. Th a postriori information ( γ n is quantizd on q bits, with q > q (usually q = q + 1, or q = q + 2. W also dnot by M th alphabt of th a priori information and of th xtrinsic mssags, and by M th alphabt of th a postriori information. Thus: M = { Q,..., 1,, +1,..., Q}, whr Q = 2 q 1 1; M = { Q,..., 1,, +1,..., Q}, whr Q = 2 q 1 1. W furthr considr a quantization map q : Y M, whr Y dnots th channl output alphabt. Th quantization map q dtrmins th q-bit quantization of th dcodr soft input. Th noisy finit-prcision MS dcodr is prsntd in Algorithm 1. W assum that q-bit addrs ar usd to comput both α m,n mssags in th VN-procssing stp, and γ n valus in th AP-updat procssing stp. This is usually th cas in practical implmntations, and allows us to us th sam typ of addr in both procssing stps. This assumption xplains as wll th q-bit saturation of α m,n mssags in th VNprocssing stp. Not also that th saturation of γ n valus is actually don within th addr (s Equation (5. Finally, w not that th hard dcision and th syndrom chck stps in Algorithm 1 ar assumd to b noislss. W not howvr that th syndrom chck stp is optional, and if missing, th dcodr stops whn th maximum numbr of itrations is rachd. E. Sign-Prsrving Proprtis Lt U dnot any of th VN-procssing or CN-procssing units of th noislss MS dcodr. W dnot by U pr th corrsponding unit of th noisy MS dcodr. W say that U pr is sign-prsrving if for any incoming mssags and any nois ralization, th outgoing mssag is of th sam sign as th mssag obtaind whn th sam incoming mssags ar supplid to U.

4 Clarly, CN pr is sign-prsrving if and only if th XORoprator is noislss (p x =. In cas that th noisy XORoprator svrly dgrads th dcodr prformanc, it is possibl to incras its rliability by using classical faulttolrant tchniqus (as for instanc modular rdundancy, or multi-voltag dsign by incrasing th supply voltag of th corrsponding XOR-gat. Th pric to pay, whn compard to th siz or th nrgy consumption of th whol circuit, would b rasonabl. Concrning th VN-procssing, it is worth noting that th VN pr is not sign-prsrving, vn if th noisy addr is so. This is du to th fact that multipl addrs must b nstd in ordr to complt th VN-procssing. Howvr, a sign-prsrving addr might hav svral bnfits. First, th rror probability of th sign of variabl-nod mssags would b lowrd, which would crtainly hlp th dcodr. Scond, if th noisy addr is sign-prsrving and all th variabl-nod incoming mssags hav th sam sign, thn th VN pr dos prsrv th sign of th outgoing mssag. Put diffrntly, in cas that all th incoming mssags agr on th sam hard dcision, th noisy VNprocssing may chang th confidnc lvl, but cannot chang th dcision. This may b particularly usful, spcially during th last dcoding itrations. Finally, th motivation bhind th sign-prsrving noisy addr modl is to invstigat its possibl bnfits on th dcodr prformanc. If th bnfits ar worth it (.g. on can nsur a targt prformanc of th dcodr, th sign-bit of th addr could b protctd by using classical fault-tolrant tchniqus. III. DENSITY EVOLUTION First, w not that our dfinition of symmtry is slightly mor gnral than th on usd in [6]. Indd, vn if th rror injction modls satisfy th symmtry condition from Equation (4, th noisy MS dcodr dos not ncssarily vry th symmtry proprty 2 from [6]. Nvrthlss, th concntration and convrgnc proprtis provd in [6] for symmtric noisy mssag-passing dcodrs, can asily b gnralizd to our dfinition of symmtry. Th dnsity volution tchniqu allows to rcursivly comput th probability mass functions of th xchangd xtrinsic mssags (α m,n and β m,n and of th a postriori information ( γ n, through th itrativ dcoding procss. This is don undr th indpndnc assumption of xchangd mssags, holding in th asymptotic limit of th cod lngth, in which cas th dcoding prformanc convrgs to th cycl-fr cas. Du to th symmtry of th dcodr, th analysis can b furthr simplifid by assuming that th all-zro codword is transmittd through th channl. Du to spac limitations, dnsity volution quations for th noisy MS dcodr ar not includd hr, but w only provid th notation and dfinitions rquird to undrstand th asymptotic analysis of th noisy MS dcodr conductd 2 Howvr, this proprty is vrifid in cas of highly symmtric fault injction. in Sction IV. W not howvr that th dnsity volution quations w providd in [1] can b rady gnralizd to th rror modls usd in this papr. A. Error Probability, Usful and Targt Error-Rat Rgions 1 Dcoding Error Probability: Th rror probability at dcoding itration l, is dfind as: P (l = 1 z= Q C (l ( z + C (l (, (9 2 whr C (l ( z := Pr( γ (l = z is th probability mass function of a postriori information 3 at dcoding itration l. Hnc, in th asymptotic limit of th cod-lngth, (l givs th probability of th hard bit stimats bing in rror at dcoding itration l. Th following lowr bounds can b drivd from th probability of rror injction within th last of th nstd addrs usd to comput th a postriori information valu. Proposition 1: Th rror probability at dcoding itration l is lowr-boundd as follows: (a For th sign-prsrving noisy addr: (l 1 2 Q. (b For th full-dpth noisy addr: (l Q. Noislss dcodrs xhibit a thrshold phnomnon, sparating th rgion whr th dcoding rror probability gos to zro (as th numbr of dcoding itrations l gos to infinity, from that whr it is boundd abov zro [12]. Things gt mor complicatd in cas of noisy dcodrs. First, th dcoding rror probability has a mor unprdictabl bhavior. It dos not always convrg and it may bcom priodic whn th numbr of itrations gos to infinity (this will b discussd in Sction IV. Scond, th dcoding rror probability is always boundd abov zro if > (Proposition 1, sinc thr is a non-zro probability of fault injction at any dcoding itration. Hnc, a dcoding thrshold, similar to th noislss cas, cannot longr b dfind. Following [6], w dfin blow th notions of usful dcodr and targt rror rat thrshold. W considr a channl modl dpnding on a channl paramtr χ, such that th channl is dgradd by incrasing χ (for xampl, th crossovr probability for th BSC, or th nois varianc for th BI- AWGN channl. W will us subscript χ to indicat a quantity that dpnds on χ. Hnc, in ordr to account for th fact that P (l dpnds also on th valu of th channl paramtr, it will b dnotd in th following by,χ. (l 2 Usful Rgion: Th first stp is to valuat th channl and hardwar paramtrs yilding a final probability of rror (in th asymptotic limit of th numbr of itrations lss than th input rror probability. Th lattr probability is givn by = 1 z= Q C(z + 1 2C(, whr C is th probability mass function of th quantizd a priori information of th dcodr (γ = q(y, s Algorithm 1. P (,χ 3 W drop th variabl-nod indx from th notation, in ordr to indicat th valu of th a postriori information γ n for a random variabl-nod n

5 ( Following [6], th dcodr is said to b usful if is convrgnt, and: P (,χ P (l,χ l> df = lim,χ (l <,χ ( (1 l Th nsmbl of th paramtrs that satisfy this condition constituts th usful rgion of th dcodr. 3 Targt Error Rat Thrshold: For noislss-dcodrs, th dcoding thrshold is dfind as th suprmum channl nois, such that th rror probability convrgs to zro as th numbr of dcoding itrations gos to infinity. Howvr, for noisy dcodrs this rror probability dos not convrg to zro, and an altrnativ dfinition of th dcoding thrshold has bn introducd in [6]. Accordingly, for a targt bit-rror rat η, th η-thrshold is dfind by: { } χ (η = sup χ,χ ( xists and,χ ( < η (11 IV. ASYMPTOTIC ANALYSIS OF THE NOISY MIN-SUM DECODER W considr th nsmbl of rgular LDPC cods with variabl-nod dgr d v = 3 and chck-nod dgr d c = 6. Th following paramtrs will b usd throughout this sction with rgard to th finit-prcision MS dcodr: Th a priori information and xtrinsic mssags ar quantizd on q = 4 bits; hnc, Q = 7 and M = { 7,..., +7}. Th a postriori information is quantizd on q = 5 bits; hnc, Q = 15 and M = { 15,..., +15}. W rstrict our analysis to th BSC channl with crossovr probability p, and furthr assum that th channl input and output alphabt is Y = { 1, +1}. For ach µ {1,..., Q} w dfin th quantization map q µ : Y M by: q µ ( 1 = µ and q µ (+1 = +µ (12 Thus, th a priori information of th dcodr γ n {±µ}. Th paramtr µ will b rfrrd to as th channl-output scal factor, or simply th channl scal factor. Th infinit-prcision MS dcodr is known to b indpndnt of th scal factor µ. This is bcaus µ factors out from all th procssing stps of th dcoding algorithm, and thrfor dos not affct in any way th dcoding procss. This is no longr tru for th finit prcision dcodr (du to saturation ffcts, and w will show shortly that vn in th noislss cas, th scal factor µ may significantly impact th prformanc of th finit prcision MS dcodr. W start by analyzing th prformanc of th MS dcodr with channl scal factor µ = 1, and thn w will analyz its prformanc with an optimizd valu of µ. A. Min-Sum Dcodr with Channl Scal Factor µ = 1 Th cas µ = 1 lads to an unconvntional bhavior, as in som particular cass th nois introducd by th dvic can hlp th MS dcodr to scap from fixd points attractors, and may actually rsult in an incrasd corrction capacity with rspct to th noislss dcodr. This bhavior will b discussd in mor dtails in this sction. W start with th noislss dcodr cas. In this cas, th dcodr xhibits a classical thrshold phnomnon: thr xists a thrshold valu p th, such that ( = for any p < p th. This thrshold valu, which can b computd by dnsity volution, is p th =.39. Now, w considr a p valu slightly gratr than th thrshold of th noislss dcodr, and invstigat th ffct of th noisy addr on th dcodr prformanc. Lt us fix p =.6. Figur 1 shows th dcoding rror probability at itration l, for diffrnt rror probability paramtrs {1 3, 1 15, 1 5 } of th noisy addr. For ach valu, thr ar two suprimposd curvs, corrsponding to th full-dpth ( fd, solid curv and sing-prsrving ( sp, dashd curv rror modls of th noisy addr. Th rror probability of th noislss dcodr is also plottd (solid black curv: it can b sn that it incrass rapidly from th initial valu ( = p and closly approachs th limit valu ( =.323 aftr a fw numbr of itrations. Whn th addr is noisy, th rror probability incrass during th first dcoding itrations, bhaving similarly to th noislss cas. It may approach th limit valu from th noislss cas, but starts dcrasing aftr som numbr of dcoding itrations. Howvr, not that it rmains boundd abov zro (although not apparnt in th figur, according to th lowr bounds from Proposition 1, and it can actually b numrically vrifid that ths bounds ar narly tight. Th abov bhavior of th MS dcodr can b xplaind by xamining th volution of th probability mass function of th a postriori information, dnotd by C (l (s Sction III, for l. In th noislss cas, C(l rachs a fixd point of th dnsity volution for l 2. Not that sinc all variablnods ar of dgr d v = 3, it can b asily sn that for l 1, C (l is supportd only on vn valus of M. Ths gaps in th support of th probability mass function sm to lad to favorabl conditions for th occurrnc of dnsity-volution fixd points. In th noisy cas, C (l volvs virtually th sam as in th noislss cas during th first itrations. Howvr, th nois prsnt in th addr progrssivly fills th gaps in th support of C (l, which allows th dcodr to scap from th fixd point attractor. It is worth noting that nithr th noisy comparator nor th XOR-oprator can hlp th dcodr to scap from ths fixdpoint attractors, as thy do not allow filling th gaps in th support of C (l. W focus now on th usful rgion of th noisy MS dcodr. W assum that only th addr is noisy, whil th comparator and th XOR-oprator ar noislss. Th usful usful rgion for th sign-protctd noisy addr modl is shown in Figur 2. Th usful rgion is shadd in gray and dlimitd by ithr a solid black curv or a dashd rd curv. Although on would xpct that ( = p on th bordr of th usful rgion, this quality only holds on th solid black bordr. On th dashd rd bordr, on has ( < p. Th rason why th usful rgion dos not xtnd byond th dashd rd bordr is that for points locatd on th othr sid of this bordr th squnc ( (l l> is priodic, and hnc it dos

6 (l (dcoding rror probability at itration l (3,6 rgular LDPC, (4,5 quantization, noisy addr = 1E 5 = 1E 15 = 1E 3 Noislss MS [fd, =1 3] [sp, =1 3] [fd, =1 15] [sp, =1 15] [fd, =1 5] [sp, =1 5] Itration numbr (l BSC Crossovr Probability (p Usful Rgion (3,6 rgular LDPC, (4,5 quantization, µ = 1 P < p B A D C Non Convrgnc Rgion Usful Rg Bordr: = p Usful Rg Bordr: < p P > p non convrgnc rgion Addr Error Probability ( Thrshold Valu (3,6 rgular LDPC, (4,5 quantization Noislss MS add[ sp, = 1 4] add[ fd, = 1 5] comp[p c = 5 3] XOR[p x = 3 4] XOR[p x = 2 4] Channl Scal Factor (µ Figur 1. Effct of th noisy addr on th asymptotic prformanc of th MS dcodr (p =.6 Figur 2. Usful and non-convrgnc rgions of th MS dcodr with sign-prsrving noisy addr Figur 3. Thrshold valus of noislss and noisy MS dcodrs with various channl scal factors (l (rror probability at itration l (3,6 rgular LDPC, (4,5 quantization, sign protctd noisy addr P ( = 9.11 E Itration numbr (l (a A(p =.3, =.27 Figur 4. (l (rror probability at itration l (3,6 rgular LDPC, (4,5 quantization, sign protctd noisy addr Dcoding rror probability P (l Itration numbr (l (b B(p =.3, =.3 (l (rror probability at itration l (3,6 rgular LDPC, (4,5 quantization, sign protctd noisy addr Itration numbr (l (c C(p =.3, =.39 (l (rror probability at itration l (3,6 rgular LDPC, (4,5 quantization, sign protctd noisy addr P ( = 6.5 E Itration numbr (l (d D(p =.3, =.42 of th noisy MS dcodr, for p =.3 and sign-prsrving noisy addr with various valus not convrg. Th rgion shadd in brown in Figur 2 is th non-convrgnc rgion of th dcodr. Not that th nonconvrgnc rgion gradually narrows in th uppr part, and thr is a small portion of th usful rgion dlimitd by th non-convrgnc rgion on th lft and th black bordr on th right. Finally, w not that points with = (noislss dcodr and p >.39 (thrshold of th noislss dcodr rprsntd by th solid rd lin suprimposd on th vrtical axis in Figur 2 ar xcludd from th usful rgion. Indd, for such points P ( clos to zro, w hav P ( > p ; howvr, for gratr than but pa 2 Q. W xmplify th dcodr bhavior on four points locatd on on sid and th othr of th lft and right boundaris of th non-convrgnc rgion. Ths points ar indicatd in Figur 2 by A, B, C, and D. For all th four points p =.3, whil =.27,.3,.39, and.42, rspctivly. Th rror probability ( (l l> is plottd for ach on of ths points in Figur 4. Th point A blongs to th usful rgion, and it can b sn from Figur 4(a that ( (l l> convrgs to ( = < p. For th point B, locatd just on th othr sid of th dashd rd bordr of th usful rgion, ( (l l> xhibits a priodic bhavior (although w only plottd th first 5 itrations, w vrifid th priodic bhavior on th first itrations. Crossing th non-convrgnc rgion from lft to th right, th amplitud btwn th infrior and suprior limits of ( (l l> dcrass (point C, until it rachs again a convrgnt bhavior (point D. Not that D is outsid th usful rgion, as ( (l l> convrgs to ( =.65 > p. Th non-convrgnc rgion gradually narrows in th uppr part, and for <.1 it taks th form of a discontinuity lin: ( taks valus clos to 1 4 just blow this lin, and valus gratr than.5 abov this lin. Not that points (, p with p < pa pa = 2 Q 3 cannot blong to th usful rgion, sinc from Proposition 1 w hav ( 2 Q > p. Morovr, w not that th bottom bordr of th usful rgion (solid black curv is virtually idntical to, but slightly abov, th lin dfind by p = 2 Q. B. Optimization of th Channl Scal Factor In this sction w show that th dcodr prformanc can b significantly improvd by using an appropriat choic of th channl scal factor µ. Figur 3 shows th thrshold valus for th noislss and svral noisy dcodrs with channl scal factors µ {1, 2,..., 7}. For th noisy dcodrs, th thrshold valus ar computd for a targt rror probability η = 1 5 (s Equation (11. Th solid black curv in Figur 3 corrspond to th noislss dcodr. Th solid rd curv and th dottd blu curv corrspond to th MS dcodr with sign-prsrving noisy addr and full-dpth noisy addr, rspctivly. Th addr rror probability 4 is = 1 4 for th sign-prsrving noisy addr, and = 1 5 for th full-dpth addr. Th two curvs ar 4 According to Proposition 1, a ncssary condition to achiv a targt rror probability ( η = 1 5 is 2 Qη = for th signdprotctd addr, and 2η 2 Q+1 2 Q = for th full-dpth addr.

7 suprimposd for 1 µ 6, and diffr only for µ = 7. Th corrsponding thrshold valus ar qual to thos obtaind in th noislss cas for µ {2, 4, 6}. For µ {1, 3, 5}, th MS dcodrs with noisy-addrs xhibit bttr thrsholds than th noislss dcodr. This is du to th fact that th mssags alphabt M is undrusd by th noislss dcodr, sinc all th xchangd mssags ar ncssarily odd (rcall that all variabl-nods ar of dgr d v = 3. For th MS dcodrs with noisy addrs, th nois prsnt in th addrs lads to a mor fficint us of th mssags alphabt, which allows th dcodr to scap from fixd-point attractors and hnc rsults in bttr thrsholds (Sction IV-A. Figur 3 also shows a curv corrsponding to th MS dcodr with a noisy comparator having p c =.5, and two curvs for th MS dcodr with noisy XOR-oprators, having rspctivly p x = and p x = Concrning th noisy XOR-oprator, it can b sn that th thrshold valus corrsponding to p x = ar vry clos to thos obtaind in th noislss cas, xcpt for µ = 7 (th sam holds for valus p x < Howvr, a significant dgradation of th thrshold can b obsrvd whn slightly incrasing th XOR rror probability to p x = Morovr, although not shown in th figur, it is worth mntioning that for p x 5 1 4, th targt rror probability η = 1 5 can no longr b rachd (thus, all thrshold valus ar qual to zro. Finally, w not that xcpt for th noisy XOR-oprator with p x = 3 1 4, th bst choic of th channl scal factor is µ = 6. For th noisy XOR-oprator with p x = 3 1 4, th bst choic of th channl scal factor is µ = 3. This is rathr surprising, as in this cas th mssags alphabt is undrusd by th dcodr: all th xchangd mssags ar odd, and th fact that th XOR-oprator is noisy dos not chang thir parity. Assumption: In th following sctions, w will invstigat th impact of th noisy addr, comparator and XOR-oprator on th MS dcodr prformanc, assuming that th channl scal factor is µ = 6. C. Study of th Impact of th Noisy Addr (µ = 6 In ordr to valuat th impact of th noisy addr on th MS dcodr prformanc, th usful rgion and th η-thrshold rgions hav bn computd, assuming that only th addrs within th VN-procssing str noisy ( >, whil th CN-procssing stp is noislss (p x =. This rgions ar rprsntd in Figur 5 and Figur 6, for th signprsrving and th full-dpth noisy addr modls, rspctivly. Th usful rgion is dlimitd by th solid black curv. Th vrtical lins dlimit th η-thrshold rgions, for η = 1 3, 1 4, 1 5, 1 6 (from right to th lft. Not that unlik th cas µ = 1 (Sction IV-A, thr is no non-convrgnc rgion whn th channl scal factor is st to µ = 6. Hnc, th bordr of th usful rgion corrsponds to points (, p for which ( = p. Howvr, it can b obsrvd that thr is still a discontinuity lin (dashd rd curv insid th usful rgion. This discontinuity lin dos not hid a priodic (non-convrgnt bhavior, but it is du to th occurrnc of an arly platau phnomnon in th convrgnc of ( (l l. This phnomnon is illustratd in Figur 8, whr th rror probability ( (l l is plottd as a function of th itration numbr l, for th two points A and B from Figur 5. For th point A, it can b obsrvd that th rror probability (l rachs a first platau for l 5, and thn drops to for l 25. For th point B, (l bhavs in a similar mannr during th first itrations, but it dos not dcras blow th platau valu as l gos to infinity. Although w hav no analytic proof of this fact, it was numrically vrifid for l In Figur 1, w plottd th asymptotic rror probability ( as a function of p, for th noislss dcodr ( =, and for th sign-prsrving noisy addr with rror probability valus = 1 4 and =.5. In ach plot w hav also rprsntd two points p (U and p (DL, corrsponding rspctivly to th valus of p on th uppr-bordr of th usful rgion, and on th discontinuity lin. Hnc, p (DL coincids with th classical thrshold of th MS dcodr in th noislss cas, and it can b sn as an appropriat gnralization of th classical thrshold to th cas of noisy dcodrs. In th following, p (DL will b rfrrd to as th functional thrshold of th noisy dcodr, and th sub-rgion of th usful rgion locatd blow th discontinuity lin will b rfrrd to as th functional rgion. Within this rgion, if th addr rror probability is small nough, it can b obsrvd that th lowr-bounds providd in Proposition 1 ar tight. D. Study of th Impact of th Noisy XOR-oprator (µ = 6 Th usful rgion and th η-thrshold rgions of th dcodr, assuming that only th XOR-oprator usd within th CN-procssing stp is noisy, ar plottd in Figur 7. Similar to th noisy-addr cas, a discontinuity lin can b obsrvd insid th usful rgion, which dlimits th functional rgion of th dcodr. Comparing th η-thrshold rgions from Figurs 5, 6 and 7, it can b obsrvd that in ordr to achiv a targt rror 1 6, th rror probability paramtrs of th noisy addr and of th noisy XOR-oprator must satisfy: < , for th full-dpth noisy-addr; < 3 1 5, for th sign-protctd noisy-addr; p x < 7 1 5, for th noisy XOR-oprator. (morovr, valus of p x up to ar tolrabl if p is sufficintly small Th most stringnt rquirmnt concrns th rror probability of th full-dpth noisy-addr, thus w may considr that it has th most ngativ impact on th dcodr prformanc. On th othr hand, th lss stringnt rquirmnt concrns th rror probability of th noisy XOR-oprator. Finally, it is worth noting that in practical cass th valu of p x should b significantly lowr than th valu of (givn th high numbr of lmntary gats containd in th addr. Morovr, sinc th XOR-oprators usd to comput th signs of CN mssags rprsnt only a small part of th dcodr, this part of th circuit could b mad rliabl by using classical probability P (

8 BSC Crossovr Probability (p (3,6 rgular LDPC, (4,5 quantization, MS/ sign protctd noisy addr.1 ( Usful Rg Bordr (P Discontinuity Lin = p.9 (Functional Thrshold ( < 1 6 B A 1 6 < ( < 1 5 ( < p 1 5 < ( < Addr Error Probability ( 1 4 < ( < 1 3 BSC Crossovr Probability (p (3,6 rgular LDPC, (4,5 quantization, MS/ sign protctd noisy addr.1 ( Usful Rg Bordr (P = p Discontinuity Lin.9 (Functional Thrshold ( < < ( < 1 5 ( < p 1 5 < ( < Addr Error Probability ( 1 4 < ( < 1 3 BSC Crossovr Probability (p (3,6 rgular LDPC, (4,5 quantization, MS/ noisy XOR oprator.1 ( Usful Rg Bordr (P = p Discontinuity Lin.9 (Functional Thrshold ( < 1 6 ( < p 1 6 < ( < < ( < < ( < XOR Error Probability (p x Figur 5. Usful and η-thrshold rgions of th MS dcodr with sing-prsrving noisy addr Figur 6. Usful and η-thrshold rgions of th MS dcodr with full-dpth noisy addr Figur 7. Usful and η-thrshold rgions of th MS dcodr with noisy XOR-oprator (l (rror probability at itration l (3,6 rgular LDPC, (4,5 quantization, sign protctd noisy addr ( = 3.33 E Itration numbr (l (a Point A(p =.77, = 1 4 (l (rror probability at itration l (3,6 rgular LDPC, (4,5 quantization, sign protctd noisy addr Itration numbr (l P ( = 4.74 E-2 (b Point B(p =.772, = 1 4 Figur 8. Illustration of th arly platau phnomnon (points A and B from Figur 5 BSC Crossovr Probability (p (3,6 rgular LDPC, (4,5 quantization, MS / noisy comparator.1 Usful Rg Bordr (TH Thrshold (p < ( < p ( > p ( = Comparator Error Probability (p c Figur 9. Usful rgion and thrshold curv of th MS dcodr with noisy comparator ( (dcoding rror probability (3,6 LDPC, (4,5 quantization, MS/ noislss, µ = 6.1 ( (.1 = p (BSC crossovr probability p (DL =.77 (a = (noislss dcodr p (U =.85 ( (dcoding rror probability (3,6 LDPC, (4,5 quantization, MS/ sign protct. noisy addr, µ = 6.1 ( (.1 P 3.33 E p (BSC crossovr probability (b = 1 4 p (DL =.77 p (U =.85 ( (dcoding rror probability (3,6 LDPC, (4,5 quantization, MS/ sign protct. noisy addr, µ = (.33 ( p (BSC crossovr probability (c =.5 p (DL =.65 p (U =.72 Figur 1. Asymptotic rror probability P ( as a function of p ; noislss and noisy MS dcodr with sign-protctd noisy addr fault-tolrant mthods, with a limitd impact on th ovrall dcodr dsign. E. Study of th Impact of th Noisy Comparator (µ = 6 This sction invstigats th cas whn comparators usd within th CN-procssing str noisy (p c >, but = p x =. Contrary to th prvious cass, this cas xhibits a classical thrshold phnomnon, similar to th noislss cas: for a givn p c dnotd by p (TH, such that ( >, th xists a p -thrshold valu, = for any p < p (TH. Th thrshold valu p (TH is plottd as a function of p c in Figur 9. Th functional rgion of th dcodr is locatd blow th thrshold curv, and ( = for any point within this rgion. In particular, it can b sn that ( = for any p.39 and any p c >. Although such a thrshold phnomnon might sm surprising for a noisy dcodr, it can b asily xplaind. Th ida bhind is that in this cas th crossovr probability of th channl is small nough, so that in th CN-procssing stp only th sign of chck-to-variabl

9 mssags is important, but not thir amplituds. In othr words a dcodr that only computs (rliably th signs of chcknod mssags and randomly chooss thir amplituds, would b abl to prfctly dcod th rcivd word. Finally, w not that th usful rgion of th dcodr xtnds slightly abov th thrshold curv: for p c clos to, thr xists a small rgion abov th thrshold curv, within which < ( < p. V. FINITE LENGTH PERFORMANCE OF THE NOISY MIN-SUM DECODER Th goal of this sction is to corroborat th asymptotic analysis from th prvious sction, through finit-lngth simulations. Unlss othrwis statd, th (3, 6-rgular LDPC cod with lngth N = 18 bits from [13] will b usd throughout this sction. A. Early stopping critrion As dscribd in Algorithm 1, ach dcoding itration also compriss a hard dcision stp, in which ach transmittd bit is stimatd according to th sign of th a postriori information, and a syndrom chck stp, in which th syndrom of th stimatd word is computd. Both stps ar assumd to b noislss, and th syndrom chck stcts as an arly stopping critrion: th dcodr stops whn whthr th syndrom is +1 (th stimatd word is a codword or a maximum numbr of itrations is rachd. W not howvr that th syndrom chck stp is optional and, if missing, th dcodr stops whn th maximum numbr of itrations is rachd. Th rason why w strss th diffrnc btwn th MS dcodr with and without th syndrom chck stp is bcaus, as w will s shortly, th noislss arly stopping critrion may significantly improv th bit rror rat prformanc of th noisy dcodr in th rror floor rgion. Unlss othrwis statd, th MS dcodr is assumd to implmnt th noislss stopping critrion (syndrom chck stp. Th maximum numbr of dcoding itrations is fixd to 1 throughout this sction. B. Finit-lngth prformanc for various channl scal factors Figur 11 shows th bit rror rat (BER prformanc of th finit-prcision MS dcodr (both noislss and noisy with various channl scal factors. For comparison purposs, w also includd th BER prformanc of th Blif-Propagation dcodr (solid black curv, no markrs and of th infinitprcision MS dcodr (dashd blu curv, no markrs. It can b obsrvd that th worst prformanc is achivd by th infinit-prcision MS dcodr (! and th finit-prcision noislss MS dcodr with channl scal factor µ = 1 (both curvs ar virtually indistinguishabl. Th BER prformanc of th lattr improvs significantly whn using a sign-prsrving noisy addr with rror probability =.1 (dashd rd curv with mpty circls. For a channl scal factor µ = 6, both noislss and noisy dcodrs hav almost th sam prformanc (solid and dashd grn curvs, with triangular markrs. Rmarkably, th achivd BER is vry clos to th on achivd by th Blif-Propagation dcodr! Ths rsults corroborat th asymptotic analysis from Sction IV-B concrning th channl scal factor optimization. C. Error floor prformanc Surprisingly, th BER curvs of th noisy dcodrs from Figur 11 do not show any rror floor down to 1 7. Howvr, according to Proposition 1, th dcoding rror probability should b lowr-boundd by (l 1 2 Q = (s also th η-thrshold rgions in Figur 5. Th fact that th obsrvd dcoding rror probability may dcras blow th abov lowr-bound is du to th arly stopping critrion (syndrom chck stp implmntd within th MS dcodr. Indd, as w obsrvd in th prvious sction, th abov lowr-bound is tight, whn l (th itration numbr is sufficintly larg. Thrfor, as th itration numbr incrass, th xpctd numbr of rronous bits gts closr and closr to 1 2 Q N =.34, and th probability of not ( having any rronous bit within on itration approachs N Q =.967. As th dcodr prforms mor and mor itrations, it will vntually rach an rror fr itration. Th absnc of rrors is at onc dtctd by th noislss syndrom chck stp, and th dcodr stops. To illustrat this bhavior, w plottd th Figur 12 th BER prformanc of th noisy MS dcodr, with and without arly stopping critrion. Th noisy MS dcodr compriss a sign-prsrving noisy addr with =.1, whil th comparator and th XOR-oprator ar assumd to b noislss (p c = p x =. Two cods ar simulatd, th first with lngth N = 18 bits, and th scond with lngth N = 1 bits. In cas that th noislss arly stopping critrion is implmntd (solid curvs, it can b sn that non of th BER curvs show any rror floor down to 1 8. Howvr, if th arly stopping critrion is not implmntd (dashd curvs, corrsponding BER curvs xhibit an rror floor at , as prdictd by Proposition 1. D. Finit-lngth prformanc for various paramtrs of th probabilistic rror modls In this sction w invstigat th finit-lngth prformanc whn all th MS componnts (addr, comparator, and XORoprator ar noisy. In ordr to rduc th numbr of simulations, w assum that p x. Concrning th noisy addr, w valuat th BER prformanc for both th signprsrving and th full-dpth rror modls. Simulation rsults ar prsntd in Figurs In cas th noisy-addr is sign-prsrving, it can b sn that th MS dcodr can provid rliabl rror protction for all th nois paramtrs that hav bn simulatd. Of cours, dpnding on th rror probability paramtrs of th noisy componnts, thr is a mor or lss important dgradation of th achivd BER with rspct to th noislss cas. But in all cass th noisy dcodr can achiv a BER lss than 1 7. This is no longr tru for th full-dpth noisy addr: it can

10 (3,6 rgular LDPC, N = 18; (4,5 quantization (3,6 rgular LDPC, (4,5 quantization, sign protctd noisy addr, =.1 N = 18, with syndrom chck N = 18, wout syndrom chck N = 1, with syndrom chck N = 1, wout syndrom chck (3,6 rgular LDPC, (4,5 quantization, add[sign protctd], p c =, p x =.1 Bit Error Rat Crossovr probability (p BP, float point (noislss MS, float point (noislss MS, noislss, µ = 1 MS, add[ sp, =.1], µ = 1 MS, noislss, µ = 6 MS, add[ sp, =.1], µ = 6 Figur 11. BER prformanc of noislss and noisy MS dcodrs with various channl scal factors Bit Error Rat N = Crossovr probability (p Figur 12. BER prformanc with and without arly stopping critrion (MS dcodr with signprsrving noisy addr, =.1 N = 1 Func. Thrshold =.77 Bit Error Rat p = p =.1, p =.1 a c x Crossovr probability (p Noislss Min Sum =.1, p x =.1 =.1, p x =.1 =.5, p x =.1 Figur 13. BER prformanc, noisy MS, signprsrving noisy addr, p c =, p x =.1 (3,6 rgular LDPC, (4,5 quantization, add[full dpth], p c =, p x =.1 (3,6 rgular LDPC, (4,5 quantization, add[full dpth], p c =, p x =.1 (3,6 rgular LDPC, (4,5 quantization, add[sign protctd], p c =, p x = Bit Error Rat p = p =.1, p =.1 a c x Crossovr probability (p Noislss Min Sum =.1, p x =.1 =.1, p x =.1 =.5, p x =.1 Bit Error Rat p = p =.1, p =.1 a c x Crossovr probability (p Noislss Min Sum =.1, p x =.1 =.5, p x =.1 Bit Error Rat p = p =.1, p =.1 a c x Crossovr probability (p Noislss Min Sum =.1, p x =.1 =.5, p x =.1 Figur 14. BER prformanc, noisy MS, fulldpth noisy addr, p c =, p x =.1 Figur 15. BER prformanc, noisy MS, fulldpth noisy addr, p c =, p x =.1 Figur 16. BER prformanc, noisy MS, signprsrving noisy addr, p c =, p x =.1 b sn that for p c =.5, th noisy dcodr cannot achiv bit rror rats blow 1 2. VI. CONCLUSION This papr invstigatd th asymptotic and finit lngth bhavior of th noisy MS ovr th BSC channl. W dmonstratd th impact of th channl scal factor on th dcodr prformanc, both for th noislss and for th noisy dcodr. W also highlightd th fact that an inappropriat choic th channl scal factor may lad to an unconvntional bhavior, in th sns that th nois introduc by th dvic may actually rsult in an incrasd corrction capacity with rspct to th noislss dcodr. W analyzd th asymptotic prformanc of th noisy MS dcodr in trms of usful rgions and targt-ber thrsholds, and furthr rvald th xistnc of a diffrnt thrshold phnomnon, which was rfrrd to as functional thrshold. Finally, w also corroboratd th asymptotic analysis through finit-lngth simulations. REFERENCES [1] S. K. Chilappagari, M. Ivkovic, and B. Vasic, Analysis of on stp majority logic dcodrs constructd from faulty gats, in Proc. of IEEE Int. Symp. on Information Thory, 26, pp [2] B. Vasic and S. K. Chilappagari, An information thortical framwork for analysis and dsign of nanoscal fault-tolrant mmoris basd on low-dnsity parity-chck cods, IEEE Trans. on Circuits and Systms I: Rgular Paprs, vol. 54, no. 11, pp , 27. [3] C. Winstad and S. Howard, A probabilistic LDPC-codd fault compnsation tchniqu for rliabl nanoscal computing, IEEE Trans. on Circuits and Systms II: Exprss Brifs, vol. 56, no. 6, pp , 29. [4] Y. Tang, C. Winstad, E. Boutillon, C. Jgo, and M. Jzqul, An LDPC dcoding mthod for fault-tolrant digital logic, in IEEE Int. Symp. on Circuits and Systms (ISCAS, 212, pp [5] A. M. Hussin, M. S. Khairy, A. Khajh, A. M. Eltawil, and F. J. Kurdahi, A class of low powr rror compnsation itrativ dcodrs, in IEEE Global Tlcom. Conf. (GLOBECOM, 211, pp [6] L. R. Varshny, Prformanc of LDPC cods undr faulty itrativ dcoding, IEEE Trans. Inf. Thory, vol. 57, no. 7, pp , 211. [7] S. Yazdi, H. Cho, Y. Sun, S. Mitra, and L. Dolck, Probabilistic analysis of Gallagr B faulty dcodr, in IEEE Int. Conf. on Communications (ICC, 212, pp [8] S. Yazdi, C. Huang, and L. Dolck, Optimal dsign of a Gallagr B noisy dcodr for irrgular LDPC cods, IEEE Comm. Lttrs, vol. 16, no. 12, pp , 212. [9] S. Yazdi, H. Cho, and L. Dolck, Gallagr B dcodr on noisy hardwar, IEEE Trans. on Comm., vol. 66, no. 5, pp , 213. [1] C. L. Kamni Ngassa, V. Savin, and D. Dclrcq, Min-Sum-basd dcodrs running on noisy hardwar, in Proc. of IEEE Global Communications Confrnc (GLOBECOM, 213. [11], Analysis of Min-Sum basd dcodrs implmntd on noisy hardwar, in Proc. of Asilomar Confrnc on Signals, Systms and Computrs, 213. [12] T. J. Richardson and R. L. Urbank, Th capacity of low-dnsity paritychck cods undr mssag-passing dcoding, IEEE Transactions on Information Thory, vol. 47, no. 2, pp , 21. [13] D. J. MacKay. Encyclopdia of spars graph cods. [Onlin]. Availabl: