1 Thresholds of absorbing sets in Low-Density-Parity-Check codes Alessandro Tomasoni (alessandro.tomasoni@ieiit.cnr.it) Sandro Bellini (sandro.bellini@polimi.it) Marco Ferrari (marco.ferrari@ieiit.cnr.it) Abstract—In thispaper,weinvestigate absorbingsets, respon- LDPC decoder. Loosely speaking, in this model an AS is 4 sible of error floors in Low Density Parity Check codes. We solved if messages coming from the rest of the graph tend 1 look for a concise, quantitative way to rate the absorbing sets’ to infinity with a rate higher than the wrong messages inside 0 dangerousness.Basedonasimplifiedmodelforiterativedecoding 2 evolution,weshowthatabsorbingsetsexhibitathresholdbehav- the AS. In practical implementations, messages cannot get ior.Anabsorbingsetwithatleastonechannellog-likelihood-ratio arbitrarilylarge.Besides,hypothesesonthegrowthrateofthe n below the threshold can stop the convergence towards the right messagesenteringtheASareneeded.In[9]DensityEvolution a J codeword. Otherwise convergence is guaranteed. We show that (DE) is used, but this is accurate only for LDPC codes with absorbingsetswithnegativethresholdscanbedeactivatedsimply 1 infinite (in practice, very large) block lengths. In [13] the usingpropersaturationlevels.Weproposeanefficientalgorithm 3 saturation is taken into account and the input growth rate is to compute thresholds. evaluated via Discretized DE or empirically via simulation. ] Index Terms—Low Density Parity Check codes, error floor, T In [4] and successive works, the authors rate the trapping absorbing sets. I set dangerousness with the critical number, that is valid for s. harddecodersbutfailstodiscriminatebetweenthesoftentries c I. INTRODUCTION of the iterative decoder. [ In the last years much effort has been spent to identify In this paper, we look for a concise, quantitative way to 1 rate the ASs’ dangerousness with soft decoding. We focus the weak points of Low Density Parity Check (LDPC) code v on Min-Sum (MS) soft decoding that is the basis for any graphs,responsibleforerrorfloorsofiterativedecoders.After 8 LDPCdecoderimplementation,leavingasidemoretheoretical 6 the introduction of the seminal concept of pseudocodewords algorithmssuchasSum-Product(SPA)orLinearProgramming 1 [1],[2] it is now ascertained that these errors are caused by 8 smallsubsetsofnodesoftheTannergraphthatactasattractors (LP). We study the evolution of the messages propagating . insidethe AS, whenthe all-zerocodewordistransmitted.Un- 1 for iterative decoders, even if they are not the support of like [9], we assume a limited dynamic of the Log Likelihood 0 valid codewords. These structures have been named trapping 4 sets [3],[4],[5] or absorbing sets [6],[7],[8] or absorption sets Ratios (LLRs) as in a practical decoder implementation. The 1 AS dangerousnesscan be characterized by a threshold τ. We [9], defined in slightly different ways. In this paper we build v: mainly on [9] and [6],[8],[10]. show that, under certain hypotheses,the decoder convergence Xi Thefirstmeritof[6],[10]hasbeentodefineAbsorbingSets towardstherightcodewordcanfailonlyifthereexistchannel LLRs smaller than or equal to τ. When all channel LLRs are (ASs) from a purely topological point of view. Moreover, the r larger than τ, successful decoding is assured. We also show a authors have analyzed the effects of ASs on finite precision with examplesthat ASs with greater τ are more harmfulthan iterative decoders, on the basis of hardware and Importance ASswith smallerτ. Finally,we provideanefficientalgorithm Sampling simulations [8],[7]. ASs behavior depends on the to find τ. decoder quantization and in [8] they are classified as weak For many ASs, τ < 0. In these cases we can deactivate or strong depending on whether they can be resolved or not ASs simply setting two saturation levels, one for extrinsic by properly tuning the decoder dynamics. In [11] the same messages (in our system model, this level is normalized to research group proposes a postprocessing method to resolve 1), and another level, smaller than |τ|, for channel LLRs. ASs, once the iterative decoder is trapped. This way the code designer can concentrate all efforts on In[9]theauthordefinesabsorptionsets(equivalenttoASs) avoiding only the most dangerous ASs, letting the receiver and identifiesa varietyof ASs for the LDPC codeused in the automaticallydeactivatetheotheroneswithextrinsicmessages IEEE 802.3anstandard. The linear modelof [12], suitable for strong enough to unlock them. Min-Sum (MS) decoding, is refined to meet the behavior of The article is organized as follows. Section II settles the beliefpropagationdecoders.Undersomehypotheses,theerror system model. Section III introduces the notion of equilibria probability level can be computed assuming an unsaturated andthresholds.SectionIVdealswithgeneralizedequilibria,a AlessandroTomasoniandMarcoFerrariarewithConsiglioNazionaledelle tool to study ASs with arbitrary structure. Section V deals Ricerche, Istituto di Elettronica e di Ingegneria dell’Informazione e delle with limit cycles. Section VI studies the message passing Telecomunicazioni, ViaG.Ponzio 34/5, 20133Milano, Italy. SandroBellini behaviorabovethreshold,andprovidesamethodtodeactivate is with Politecnico di Milano, Dipartimento di Elettronica, Informazione e Bioingegneria, Piazza L.daVinci, 20133Milano, Italy. many ASs. Section VII shows practical examples of ASs that 2 +1 x x 1 4 1 2 x 1 2 1 2 1 2 x 2 x 3 x 5 11 5 5 x 12 x +1 x x 4 3 4 3 4 3 10 7 6 4 3 x x 9 8 +1 (a) (b) (c) Fig.2. MessagepropagationwithintheASofFig.1(b).Forsimplicity,CNs inthemiddleofedges arenotshown. Fig.1. Threeabsorbingsets:inFig.1(a),amaximal(4,4)AS;inFig.1(b), a(5,3)AS;inFig.1(c),a(4,0)AS,thatisalsothesupportofacodeword. the iterations, by an N × N routing matrix A. Basically, A = 1 iff there exists an (oriented) path from message i,j behave as predicted by our model during MS decoding on x to message x , going across one VN. For instance, Fig. j i realcompleteLDPCgraphs.SectionVIIIproposesanefficient 2 depicts the LLR exchange within the AS of Fig. 1(b). The algorithmtocomputeτ.SectionIXhighlightsotherinteresting corresponding first row of A is1 properties. Finally, Section X concludes the paper. A =[0 0 0 0 0 0 0 0 0 1 1 0] . (1) {1},{1,...,N} II. SYSTEMMODEL ANDDEFINITIONS To account for saturation we define the scalar function sat(x),sign(x)·min(|x|,1) and we say that x is saturated We recall that a subset D of variable nodes (VNs) in a if |x| ≥ 1, unsaturated otherwise. For vectors, sat(x) is the Tanner graph is an absorbing set (a,b) if [6] element-wise saturation. • every VN in D has strictly more boundaryCheck Nodes For the time being,we considera parallelmessage passing (CNs) in E(D) than in O(D), being E(D) and O(D) decoder,whereallVNsaresimultaneouslyactivatedfirst,then the set of boundary CNs connected to D an even or odd all CNs are simultaneously activated, in turn2. The system number of times, respectively; evolution at the k-th iteration reads • the cardinality of D and O(D) are a and b, respectively. x(k+1) =sat Ax(k)+e+Rλ (2) Besides, D is a fully absorbing set if also all VNs outside D havestrictly less neighborsin O(D) thanoutside.In [10] it is (cid:16) (cid:17) where x(k) are the extrinsic messages within the AS, e is the observedthatapatternofall-onesfortheVNsinD isastable vector of extrinsic messages entering the AS through O(D), concurrentof the all-zerospattern for the iterativebit-flipping and R is a repetition matrix with size N ×a, that constrains decoder, notwithstanding a set O(D) of unsatisfied boundary the a channel LLRs λ to be the same for all messages x j CNs (dark CNs in Fig. 1). ASs behave in a similar manner emanating from the same VN. Referring to Fig. 1(b), also under iterative soft decoding, as shown and discussed in [8], [7]. 1 1 1 0 0 0 0 0 0 0 0 0 IfallCNsareconnectedtoDnomorethantwice,theASis 0 0 0 1 1 0 0 0 0 0 0 0 elementary. Elementary ASs are usually the most dangerous. RT =0 0 0 0 0 1 1 1 0 0 0 0 . (3) Given the code girth, elementary absorbing sets can have 0 0 0 0 0 0 0 0 1 1 0 0 smaller values of a and b than non elementary ones [14]. If   0 0 0 0 0 0 0 0 0 0 1 1 ASs are the support of near-codewords[15], the smaller a is,   the higher the probability of error. Besides, the smaller the Also notethat the row weight of R is unitary, i.e. R1=1. ratio b/a, the more dangerousthe AS is [8]. In this paper we As to the extrinsic messages entering the AS from outside, focus on elementary ASs only, as those in Fig. 1. An AS is we bypass the tricky problem of modeling the dynamical maximalifa=b,asinFig.1(a).Intuitively,maximalASsare behaviorofthedecoderinthewholegraphassumingthateach themildestones,sincetheyhavealargenumberofunsatisfied message enteringthe AS hassaturated to the maximalcorrect CNs. On the opposite, an AS with b = 0 (as in Fig. 1(c)) is LLR(i.e.+1,sincewetransmittheall-zerocodeword).Thisis the support of a codeword. areasonablehypothesisaftera sufficientnumberofiterations, For our analysis we assume an MS decoder, that is insen- asobservedin[11]wheretheauthorsbasetheirpostprocessing sitive to scale factors. Thus we can normalize the maximum technique on this assumption. In Section VII we show that extrinsic message amplitude. We recall that, apart from satu- the decoding of a large graph is in good agreement with the ration, the evolution of the messages inside the AS is linear ([9], [12]) since the CNs in E(D) simply forward the input 1Matrixsubscripts indicate subsetsofrowsandcolumns. 2In the following, we will show that the results presented in this paper messages. The relation among the N = 2|E(D)| internal hold for any activation order of VNs and CNs, provided extrinsic messages extrinsicmessages x generatedby VNs can be trackedduring arepropagated justonceperdecoding iteration. 3 predictionsofthismodel.Underthishypothesis,wecanwrite Problem III.1. e=Rd−1−A1 (4) τ′ =max min(λ) λ,x where d is the a×1 vector of the VN degrees. s.t. −1≤x≤1, ∃j :x <1 (7) j From now on, we will consider only left-regular LDPC x=sat(A(x−1)+2·1+Rλ) . codes,with d =3, ∀i.Mostof thetheoremspresentedin the i The constraint (7) restricts the search to bad equilibria, following sections can be extended to a generic VN degree vector d. Luckily, among regular LDPC codes this is also having at least one extrinsic message smaller than 1. We call the case with the most favorable waterfall region. If we set τ′ the threshold, since the AS has no bad equilibria with λ d=3·1, then Rd−1=3·R1−1=3·1−1=2·1, and above that value. In Section VI we will show that the notion (2) becomes of threshold does not pertain only to bad equilibria, but also to any other bad trajectory of (5), not achieving x(∞) =1. x(k+1) =sat A x(k)−1 +2·1+Rλ . (5) In the above optimization problem, for simplicity we did (cid:16) (cid:16) (cid:17) (cid:17) notassign upperand lowerboundsto the channelLLRs λ. In Thisequationismoreexpressivethan(2),asx(k)−1isthegap practice,we can restrictour search in the range−1≤λ≤1. betweenthecurrentstatex(k) andthevaluesthattheextrinsic messagesshouldeventuallyachieve,oncetheAS isunlocked. Theorem III.1. The pair (x = −1,λ = −1) is always an Besides, we will show that −1 ≤ λ ≤ 1. Therefore, 2·1+ equilibrium. Rλ≥1andA x(k)−1 ≤0.Thetwocompetingforcesare Proof:Substituting(x=−1,λ=−1)intheequilibrium nowclearlyvisible.Theformeralwayshelpsconvergence,and (cid:0) (cid:1) equation, we obtain thelattercanamplifynegativeterms(iftheASisnotmaximal, some rows of A have weight larger than 1). The rest of the paper is devoted to unveil the hidden prop- sat A(x−1)+2·1+Rλ′ =sat −2· A1 +1   ertiesof(5),findingsufficientconditionsforcorrectdecoding, ≥1 (cid:0) (cid:1) i.e. x(k) → 1 when k → ∞. We will assume a conservative  =−1=x. (8) |{z} condition to decouple the AS behavior from the rest of the code:we donotstartwithx(0) =0.We takeintoaccountany configuration of extrinsic messages x(0) that may result in a Theorem III.2. The only equilibrium (x,λ) for a system convergencefailure.Westartfromaniterationwiththerestof having λ>1 and −1≤x≤1 is in x=1. the decoder messages saturated to 1. The configuration x(0) insidethe AS, whichis theresultofthe messageevolutionup Proof: Since λ > 1, we can define a strictly positive to that iteration, is unknown. The drawback of this approach quantity ∆ , min(λ)− 1 > 0. Consider parallel message is that we renounce to predict the probability of message passing. Focusing on the evolution of (5), configurations inside the AS leading to decoding errors. On the other hand, if no x(0) can lock the decoder, this is true independentlyoftheevolutionofmessagesinside theAS. We x(1) =satA(x(0)−1)+2·1+Rλ will study equilibria, limit cycles and chaotic behaviors (i.e., ≥−2·1 ≥(3+∆)1   aperiodic trajectories) of (5), depending on the channel LLRs   λ and any initial state x(0). ≥sat −2 A|1 +{z(3+}∆)|1 {≥z sat}(−1+∆)·1. (9)   ≤2·1   If sat(−1+∆)=|{1z,}then x(1) =1, and we stop. Otherwise, sat(−1 + ∆) = −1 + ∆ and we go on. For the generic III. EQUILIBRIA, THRESHOLDDEFINITION AND step, assuming x(k) ≥ sat(−1+k∆)·1 and proceeding by PRELIMINARY PROPERTIES recursion, we have InthisSection,westudyequilibriaforthenon-linearsystem x(k+1) =sat A (x(k)−1) +2·1+Rλ (5).   Definition III.1. A pair (x,λ) is an equilibrium iff  ≥(−2+k∆)·1 ≥(3+∆)1    | {z } | {z } x=sat(A(x−1)+2·1+Rλ) . (6) ≥sat −1+k∆ A1 +∆1 ≥sat(−1+(k+1)∆)·1.   ≥1 Equilibriawithx6=1areharmful.Theybehaveasattractors   (10) for the evolution of the extrinsic messages x(k), and can lead |{z} The same inequalitieshold also in case of sequentialmessage touncorrectdecisions.Withtheaimoffindingthemostcritical passing, activatingCNs in arbitraryorder (once per iteration). ASs,thosethatcanleadtoconvergencefailureevenwithlarge values of λ, we would like to solve the following problem: As soon as −1+(k +1)∆ ≥ 1, the recursion ends. We conclude that the message passing algorithm will eventually 4 achieve x=1. Problem IV.1. Being τ′ the result of a maximization, a straight conse- τ∗ =max min(λ) quence of the above two theorems is λ,x Corollary III.1. As for Problem III.1, −1≤τ′ ≤1. s.t. −1≤x≤1, ∃j :xj <1 x=sat(A(x−1)+2·1+λ). The two boundary values τ′ = −1 and τ′ = 1 are the thresholds of maximal ASs and codewords, respectively: The following theorem holds. Theorem III.3. Any support of a codeword has τ′ = 1. Theorem IV.1. As for Problems III.1 and IV.1, τ′ =τ∗. Maximal absorbing sets have τ′ =−1. Proof: We show that τ′ ≤τ∗ and τ′ ≥τ∗. Proof: We start from codewords. (x = −1,λ = 1) is a Every equilibrium is also a generalized equilibrium. Given valid equilibrium for Problem III.1. Indeed: a solution (x′,λ′) of Problem III.1 with min(λ′) = τ′, the solution (x∗,λ∗) with x∗ = x′ and λ∗ = Rλ′ satisfies the constraints of Problem III.1. Being τ∗ the result of the sat(A(x−1)+2·1+Rλ)=sat −2 A1 +3·1 maximization in Problem IV.1, we conclude that τ∗ ≥τ′.   =2·1 On the converse, generalized equilibria may not be equi- =sat(−4·1+3·1)=−1=x.(11) libria. Indeed, λ∗ could not be compatible with the repetition |{z} forced by matrix R. Notwithstanding this, if a generalized By Corollary III.1 we conclude that τ′ =1. equilibrium (x∗,λ∗) exists, then also an equilibrium (x′,λ′) ReferringtomaximalASs,forany−1≤x(0) ≤1andλ> exists, with x′ ≤x∗ and λ′ =min(λ∗)·1. Consider channel −1, we can define a strictly positive quantity ∆,min(λ)− LLRsλ′ =min(λ∗)·1.Weexplicitlyprovideaninitialization (−1)>0. Focusing on the evolution of (5), x(0) for (5) that makes the extrinsic messages achieve an equilibrium (x′,λ′), with x′ ≤x∗. First, note that x(1) =satA(x(0)−1)+2·1+Rλ x(k+1) =sat A x(k)−1 +2·1+Rλ′  ≥−2·1 ≥(1+∆)1  (cid:16) (cid:16) (cid:17) (cid:17)  | {z } | {z } =sat A x(k)−1 +2·1+min(λ∗) R1 . (15) ≥sat −2 A1 +(1+∆)1 ≥sat(−1+∆)·1. (12)     (cid:16) (cid:17) =1 =1     If we set x(0) =x∗, we obtain the inequality |{z} If sat(−1+∆)|={z1}, then x(1) =1, and we stop. Otherwise, sat(−1 + ∆) = −1 + ∆ and we go on. For the generic x(1) =sat(A(x∗−1)+2·1+min(λ∗)1) step, assuming x(k) ≥ sat(−1+k∆)·1 and proceeding by ≤sat(A(x∗−1)+2·1+λ∗)=x∗ =x(0). (16) recursion, we obtain Proceeding by induction, x(k+1) =sat A (x(k)−1) +2·1+Rλ x(k+1) =sat A x(k)−1 +2·1+min(λ∗)1    ≥(−2+k∆)·1 ≥(1+∆)1  ≤sat A x(k(cid:16)−1)(cid:16)−1 +2(cid:17)·1+min(λ∗)1 =x(cid:17)(k) (17)  | {z≥s}at(−|1+{z(k+}1)∆)·1. (13) (cid:16) (cid:16) (cid:17) (cid:17) since A ≥ 0, ∀i,j. The above equation states that the i,j As soon as −1 + (k + 1)∆ ≥ 1, the recursion ends. The sequence {x(k)} is monotonically decreasing. Yet, it cannot message passing algorithm will eventually achieve x = 1, assume arbitrarilysmall values, since extrinsicmessages have that is not a valid equilibriumfor Problem III.1. We conclude a lower saturation to −1. We conclude that {x(k)} must that at least one element in λ must be equal to −1, therefore achieve a new equilibrium x′ ≤x∗. τ′ =−1. The equilibrium (x′,λ′) satisfies all the constraints of ProblemIII.1. Beingτ′ theresultofamaximization,τ′ ≥τ∗. IV. GENERALIZEDEQUILIBRIA The above statements do not claim that the two problems Most of the effort of this Section is in the reformulationof are equivalent. Indeed, they can be maximized by different Problem III.1, to make it manageable. First, in place of equi- pairs (x,λ). Anyway, as long as we are interested in the AS libria, we considera slightly moregeneralcase, removingthe threshold, we can deal with Problem IV.1 instead of Problem repetition matrix R and assuming N unconstrained channel III.1 and with generalized equilibria instead of equilibria. LLRs λ. Definition IV.1. A pair (x,λ) is a generalizedequilibriumiff V. LIMIT CYCLES In this Section, we focus on limit cycles, i.e. on extrinsic x=sat(A(x−1)+2·1+λ) . (14) messages that periodically take the same values. We show that they have thresholds smaller than or equal to equilibria. Accordingly, we write the following optimization problem. Therefore, we will neglect them. 5 Definition V.1. The sequence({x′′(0),...,x′′(L−1)},λ′′)is a L binary matrices, with size N ×N: limit cycle with period L iff ∀k, L−1 x′′(k+1modL) =sat A x′′(kmodL)−1 +2·1+Rλ′′ . Wi(,lj) ∈{0,1}, W(l) =I. (22) l=0 (cid:16) (cid:16) (cid:17) (1(cid:17)8) X Then Limit cycles can be interpreted as equilibria of the aug- L−1 L−1 mented AS, described by an augmented matrix A′′ of size W(l)x′′(l) = W(l)sat A¯(x′′(l−1 modL)−1) (NL)×(NL). While the VN and CN activation order does Xl=0 Xl=0 (cid:16) not matter in case of equilibria (at the equilibrium, extrinsic +A(x′′(l)−1)+2·1+Rλ′′ messages do not change if we update them all together, or one by one in arbitrary order), this is not true in case of limit L−1 (cid:17) =sat W(l)A¯(x′′(l−1 modL)−1) cycles. Indeed,the associated set of equationsdependson the decoding order. Xl=0 L−1 In case of parallelmessage passing,onecan write a system + W(l)A(x′′(l)−1) (23) of equations with NL rows, where the l-th horizontal stripe l=0 of N equationsrepresents the evolutionof extrinsic messages X L−1 L−1 from state x′′(l−1 modL) to x′′(l) +2· W(l)1+ W(l)Rλ′′ x′′(0) x′′(0) Rλ′′ l=0 l=0 ! X X x′′(1) x′′(1) Rλ′′ =I =I  . =satA′′ . −1+2·1+ .  . .. .. .. whereinthesecondequat|ion{Wz (l)}enter|sin{tozthe}sat(·)func-         x′′(L−1)  x′′(L−1)  Rλ′′tionsince1·sat(x)=sat(1·x)and0·sat(x)=sat(0·x), ∀x.                (19) Choose a vector x∗ of extrinsic messages as Instead, in sequential (or serial-C [16]) decoding CNs are x∗ =min x′′(l) , ∀j. (24) j j l activated one by one, in turn, immediately updating the a- (cid:16) (cid:17) posterioriLLRsoftheVNsconnectedthereto.Theaugmented As a consequence, we have x∗ ≤x′′(l), ∀l and matrixchanges,sinceonlythefirstCNsuseextrinsicmessages producedatthepreviousiteration,whileallothersexploitmes- L−1 L−1 sages generated during the same iteration. We can represent W(l)x′′(l) =sat W(l)A¯(x∗−1) thisbehaviordefiningtwomatricesA¯ andA,binarypartitions l=0 l=0 of A, X X  A¯ ,A ∈{0,1}, A¯ +A=A (20)  =I i,j i,j  | {z } and writing an augmented matrix as L−1 + W(l)A(x∗−1)+2·1+Rλ′′+∆ AA¯ A0 ······ 00 A0¯ Xl=0 ,λ∗  A′′ =0 A¯ 0 0 . (21) =sat(A=(Ix∗−1)+2·1+λ∗) | {z(25}) ... ... ... ... ... | {z } 0 0 ··· A¯ A with ∆ = lL=−01W(l)A¯(x′′(l−1 modL) − x∗) +   L−1W(l)A(x′′(l) − x∗) ≥ 0, being A ≥ 0,∀i,j.   l=0 P i,j A¯ and A have upper and lower triangular shapes, due to Finally, choose the partition W(l) that implements the P the sequential update order. Note that (21) is valid not only min(·) function, i.e., (cid:8) (cid:9) for sequentialCN message passing decoding,but also for any L−1 arbitraryorder3,aslongasallextrinsicmessagesareactivated W(l)x′′(l) =x∗ (26) in turn, once per decoding iteration. Parallel message passing l=0 is a special case of (21), with A¯ =A and A=0. Therefore X thus achieving a generalized equilibrium (x∗,λ∗) with λ∗ = we provide the following theorem only for the most general Rλ′′+∆≥Rλ′′. case. A straight consequence of Theorem V.1 is that limit cycles Theorem V.1. If there exists a limit cycle can be neglected, when we compute the AS threshold. ({x′′(0),x′′(1),...,x′′(L−1)},λ′′), a generalized equilibrium (x∗,λ∗) with λ∗ ≥Rλ′′ exists, too. VI. BEHAVIOR ANALYSIS ABOVETHRESHOLD Wealsohavetotakeintoaccountpotentialchaoticbehaviors Proof: Consider any partition of the identity matrix I in of the extrinsic messages in (5). In principle, x(k) could even evolve without achieving any equilibrium or limit cycle. Yet, 3Inthiscase,theloweranduppertriangular shapeislost. above the threshold τ the extrinsic messages x(k) achieve 1. 6 TheoremVI.1. Letτ bethesolutionofProblemIII.1orIV.1. (a) τ≥Lch If λ>τ ·1, for any starting x(0) with −1≤x(0) ≤1, for a -1 0 1 = equilibrium sufficiently large K ≥0 --LL LL cchh cchh = limit cycle x(k) =1, ∀k ≥K. (27) (b) τ<-Lch = aperiodic Proof: For the time being, consider channel messages trajectory λˆ that can assume only quantized values between −1 and -1 0 1 --LL LL 1, with uniform step δ = 1, Q ∈ N. Assume that also cchh cchh Q extrinsic messages xˆ are quantized numbers, with the same Fig.3. Trajectories ofanAS,orderedw.r.t.theirminimumchannel LLR. step δ. Therefore, xˆ(0) can only assume (2Q)N different values. Letting the system 2 xˆ(k+1) =sat A xˆ(k)−1 +2·1+Rλˆ (28) (cid:16) (cid:16) (cid:17) (cid:17) evolve, it is clear that extrinsic messages at every time k >0 mustbelongtothesamesetof(2Q)N values.Whenλˆ >τ·1, 1 3 4 the analysis presented in previous Sections assures that the only remaining equilibrium is x=1. Indeed, other equilibria cannotexistsincetheywouldneedmin λˆ ≤τ.ByTheorem 7 6 5 V.1, also limit cycles do not exist, both in case of parallel (cid:16) (cid:17) and sequential decoding.Therefore, the only value that xˆ can Fig.4. A(7,3)elementary absorbingset. assume more than once is 1 (otherwise we would incur in equilibria or cycles). We can conclude that x = 1 will be reached in at most K = (2Q)N iterations. After that, the decodingtrajectories within ASs can be very differentin case extrinsic messages will remain constant and the absorbing set of positive or negative thresholds: will be defused. If x(0) and λ are not quantized, we can always identify • ifτ ≥0,thesaturationofchannelLLRsto±Lchdoesnot destroybadtrajectories.Thisisgraphicallyrepresentedin a sufficiently small quantization step δ and a quantized pair Fig. 3(a); xˆ(0),λˆ s.t. • if τ < 0, we can set Lch < −τ = |τ|. With this (cid:16) (cid:17) choice, channelLLRs can neverlead to bad trajectories, xˆ(0) ≤x(0), τ ·1<λˆ ≤λ (29) as depicted in Fig. 3(b). Therefore,by TheoremVI.1 the since Q is dense in R. Finally, writing the inequality AS is defused. xˆ(k+1) =sat A xˆ(k)−1 +2·1+Rλˆ VII. SIMULATION RESULTS ≤sat A(cid:16) x(cid:16)(k)−1 (cid:17)+2·1+Rλ (cid:17)=x(k+1) (30) The behavior of bad structures under iterative decoding in a large code graph is in good agreement with the theory (cid:16) (cid:16) (cid:17) (cid:17) and recalling that xˆ(k) achieves 1 in at most K = (2Q)N developed so far. For instance, consider the AS (5,3) with iterations, we conclude that also x(k) must achieve 1 in at topology shown in Fig. 2. For this AS, τ =−1/3 (a method mostK iterations,bythe SqueezeTheoremappliedtoxˆ(k) ≤ to compute thresholds will be presented in the next Section). x(k) ≤1. In Fig. 5(a) we plot its contribution to the error-floor Theorem VI.1 states that there cannot exist bad equilibria, of an LDPC code having block size 30000 and rate 4/5. limit cyclesor chaotic behaviors(in short, bad trajectories) if The simulations are run using Importance Sampling over a the minimumchannelLLRexceedsthe solutionτ ofProblem Gaussian channel, with SNR around 2.5 dB. We always let III.1 or IV.1. This reinforces the name threshold assigned to the quantized channel LLRs vary in the range [−7,7], while τ (we do not distinguish any more between τ′ and τ∗), that the extrinsic LLRs are quantized with a varying number q e is not limited to equilibria, but pertains to all bad trajectories. of bits. Therefore extrinsic messages belong to the interval In Fig. 3(a) we represent bad trajectories, ordering them −2qe−1+1,2qe−1−1 , and Lch = 2qe−71−1. Decisions are w.r.t. their minimum channel LLR, in the range −1≤λ≤1. taken after 20 iterations of MS sequential decoding. (cid:2) (cid:3) By TheoremsV.1 and VI.1, the rightmostbad trajectory is an FromFig.5(a)itisapparentthattheprobabilitythattheMS equilibrium. decoderbe locked by the AS is lowered when L is reduced ch Theresultsfoundsofarcanbeexploitedtodeactivatemany from 1 to 7/15. However, this is larger than |τ| = 1/3 and ASs during the decoding process, using two different satura- an error floor still appears. In agreement with the predictions tion levels. Without loss of generality we set the saturation of our theory, if we set L = 7/31 < |τ| the AS is always ch level of extrinsic messages equal to ±1, and the saturation unlocked and the error probability is zero. levelof channelLLRs equalto ±L , with0 λ ), for an max max max increasedcapability to discard saturationpatterns,resulting in Proof: Assume that only one element involved in some an execution speedup. violation, say the m-th, is erased by Sunsat. The proof for a generic subset of violations, erased all together, can Test2analyzestheinequalityconstraintsofProblemVIII.2 be achieved repeating the following argument many times, in turn, rewritten as discarding one element after the other. C λ ≤f (50) When element m passes from Sunsat to Ssat, not only the x Sunsat rowA mustbeerased,butalsothecolumnA (cid:20) (cid:21) m,Sunsat Sunsat,m with mustbecanceled.LookingatanyotherrowofASunsat,Sunsat leading to a violation, say the one corresponding to element 1 A −I C = M×1 Sunsat,Sunsat (51) j, three events can happen (see Fig. 7): (cid:20)−1(N−M)×1 −ASsat,Sunsat (cid:21) 1) A had weight 0: after column deletion, it still A 1 −2·1 j,Sunsat f = Sunsat,Sunsat M×1 M×1 . (52) has weight 0 and the hypothesis of Theorem VIII.3 is 1 −A 1 (cid:20) (N−M)×1 Ssat,Sunsat M×1(cid:21) still valid; For every variable involved,the test tries to tighten the gap 2) Aj,Sunsat had weight 1, and its element equal to 1 betweenthecorrespondinglowerandupperbounds,exploiting was exactly in the m-th column (the erased one): after bounds(upperorlower,dependingonthecoefficientsigns)on deletion, the row assumes weight 0, thereforesatisfying the other variables. The process can terminate in two ways: the hypothesis of Theorem VIII.3; 3) A had weight 1, and the element equal to 1, 1) bounds l and u cannot be further improved, and l≤u: j,Sunsat say A did not lie in the m-th column: after deletion, τ andSunsat arecompatiblewiththe existenceof other j,h the row of A still has weight 1. Since the Sunsat,Sunsat equilibria, having thresholds larger than τ; weightofthecorrespondingcolumnA isstill1(it Ssat,h 2) for at least one index j, we achieve u < l : equilibria j j had weight1 by hypothesis),TheoremVIII.4 still holds having thresholdslargerthan the currentlydiscoveredτ for the j-th element. cannot exist, for that Sunsat. Either Theorem VIII.3 or VIII.4 are still valid, and the other The initialization in (49) influences the algorithm effective- violations do not disappear. ness: the more the discovered threshold τ gets large, the The aboveTheoremgivesusthe freedomto erase elements more Test 2 will effectively detect impossible configurations inSviolation alltogetherfromSunsat,andsimultaneouslyadd of Sunsat, speeding up the solution of Problem VIII.2. themto Ssat. Therefore,we can imaginea tree searchamong 10 all possible configurations of saturated messages. is not punctured, otherwise nothing is left to optimize and At the root node, Ssat = ∅. At successive steps, some λp =0>τ ·1. extrinsicmessagesaremarkedasalreadyvisited(“fixed”,from Consider now the case τ ≥ 0. Since at least one entry of here on). In addition, fixed messages are labeled as saturated λp is equal to 0, we have min(λp) ≤ 0. Thus min(λp) ≤ τ or not. Extrinsic messages not fixed (say “free”) are always and the threshold of the same AS without puncturing is not unsaturated. This implicitly defines Ssat and Sunsat. For the exceeded. current configuration, Test 1 is performed. Three things can Therefore, ASs having τ <−L cannot become harmful. ch happen: • Case 1: Test 1 claims that Problem VIII.2 may have B. Thresholds are rational numbers solutions for that Ssat; A final, not trivial property of thresholds is the following. • Case 2: Test 1 claims that Ssat is incompatible with Theorem IX.2. Thresholds τ ∈Q. any solution of Problem VIII.2, and all the elements generating a test violation are free; Proof: We focus on Problem VIII.2. We will prove • Case 3: Test 1 claims that Ssat is incompatiblewith any the theorem for any constrained saturation pattern Sunsat. solutionofProblemVIII.2,butsomeelementsgenerating Therefore, the result will hold for the maximum across all a test violation have been previously fixed (and marked possible Sunsat. The proof is slightly cumbersome, and in- as unsaturated). volves standard concepts of linear programming theory. Depending on the answer of Test 1, we expand the tree in First, note that constraints −1≤x≤1 bound the feasible different manners: space of extrinsic messages. By Theorem III.1, the constraint −1 ≤ λ ≤ 1 can be added without modifying the result. • in Case 1, in turn we fix one of the free messages, Therefore, the above constraints and the others in Problem and branch the tree, labeling the last element as either VIII.2 define a polytope P in M +1 dimensions saturated or not, calling the algorithm recursively; • in Case 2, we fix and mark as saturated all elements of I(M+1)×(M+1) 1(M+1)×1 Sunsat that generate violations, and call the algorithm y∈P ⇐⇒ −I y≤ 1 (53) (M+1)×(M+1) (M+1)×1     recursively; C f • in Case 3, or if all variables have been already fixed, we     take no action. ,C′ ,f′ After Test 1, before the tree branching, we either perform where | {z } | {z } λ optimization or not: y, (54) x Sunsat • in Case 1or 2,Test 2 isexecuted.In case ofnegativere- (cid:20) (cid:21) sult,we returnτ =−1;otherwise,the simplexalgorithm Geometrically, the above inequality constraints reported in canonicalformrepresenthalf-spacesthat“shave”thepolytope. is eventually performed to solve Problem VIII.2 for the current Sunsat; The polytope P is convex, since it results from the intersec- tion of half-spaces, that are affine and therefore convex. To • in Case 3, we return the partial result τ =−1. conclude, our optimization problem can be re-stated as This way, Test 1 speeds up the construction of Ssat and prunesmanybranches.Test2avoidstheexecutionofthesim- max wTy, s.t. y∈P (55) plex algorithm for many useless configurations, not detected y by Test 1. T with w = 1 0 ... 0 . Our feasible region cannot be empty, since we already know that a solution (λ = (cid:2) (cid:3) IX. ADDITIONAL PROPERTIES −1,xSunsat =−1), i.e. y=−1 always exists. From Linear Programming Theory [18], we know that the A. Punctured LDPC codes numberofindependentconstraintsatanyvertexis M+1 and Puncturingisapopularmeanstoadaptthecoderateoreven that at least one vertex is an optimizer in linear programming achieveratecompatibility[17].Aninterestingextensionofour problems(thelatterpartistheenunciationoftheFundamental theory is that harmless ASs having τ <−L are deactivated ch Theorem of Linear Programming). even in case of puncturing. Focus on a vertex v that is also a maximizer, and on the Theorem IX.1. Puncturing at most a−1 VNs of an AS does M+1 linearly independentconstraints satisfied with equality not increase the threshold. in that point. Let A be the set of these constraints. We can write C′ v = f′ . Being C′ full-rank, Proof: Assume that an AS (a,b) of threshold τ is A,{1,...,M+1} A A,{1,...,M+1} wecanachieveafullQR-likedecompositionC′ = punctured in less than a VNs. Let λp be the set of channel A,{1,...,M+1} QL, being Q orthogonal, and L lower triangular. Note that, LLRs, with null messages for the punctured VNs. First, beingthe entriesof C′ rational(actually,integer), consider the case τ < 0. Assume that after puncturing, a A,{1,...,M+1} we can always keep the elements of Q and L rational, e.g. bad trajectory {x(k)} exists, with min(λp) > τ. This is an performinga Gram-Schmidtdecomposition.Multiplyingboth absurdum,sinceλpisalegitimatesolutionwithoutpuncturing, sides of the above equation by QT, we obtain and the definition of threshold given e.g. in Problem III.1 is contradicted. This holds as long as at least one variable Lv=QTf′ . (56) A

Thresholds of absorbing sets in Low-Density-Parity-Check codes

Similar Thresholds of absorbing sets in Low-Density-Parity-Check codes