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

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by  Alessandro Tomasoni

1
Thresholds of absorbing sets in
Low-Density-Parity-Check codes
Alessandro Tomasoni ([email protected])
Sandro Bellini ([email protected])
Marco Ferrari ([email protected])
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<L ≤1. The In Fig. 5(b) we plot the same curves for another AS
ch ch
latter saturation level defines the range of admissible channel embedded in the same LDPC code. This AS is shown in Fig.
LLRs, depicted in Figs. 3(a) and 3(b) as a gray box. The 4 and its τ =−1/9. Once again, reducing L decreases the
ch
7
(a) (b)
VIII. A SEARCH ALGORITHM FORTHRESHOLDS
L = 1
10-10 10-10 ch A. Towards an affordable linear problem
L = 7/15
ch With the aim of deriving an efficient algorithm to compute
L = 7/31
ch the AS threshold,we furthersimplify ProblemIV.1, introduc-
10-12 10-12
ing
Problem VIII.1.
10-14 10-14
R R
BE BE τ˜=max λ
λ,x
10-16 10-16 s.t. −1≤x≤1 (31)
∃j :x <1 (32)
j
10-18 10-18 x≥sat(A(x−1)+(2+λ)1) (33)
wheresat(x),min(x,1).WithrespecttoProblemIV.1,only
f
10-20 10-20 theuppersaturationisstillpresentinsat(·):extrinsicmessages
2.3 2.4 2.5 2.6 2.7 2.3 2.4 2.5 2.6 2.7
SNR [dB] SNR [dB] can nowfassume any negative value. Besides, the constraint
imposed by the equilibrium equalityfhas been relaxed, and
Fig.5. Errorfloorcontributionsofthe(5,3)and(7,3)ASsshowninFigs.2
substituted by an inequality containing only a scalar value λ.
and4,respectively. TheerrorfloorshavebeenobtainedapplyingImportance
SamplingtoarealLDPCcode, underMSsequential decoding. Notwithstanding these modifications, the following theorems
hold:
(a) Theorem VIII.1. As for Problems IV.1 and VIII.1, τ∗ =τ˜.
12
a108 Proof: We show that τ∗ ≤τ˜ and τ∗ ≥τ˜.
6 Assume we are given a solution (x∗,λ∗) of Problem IV.1,
5 10 15 20 25 30 35 40 45 i.e. with min(λ∗) = τ∗. We can exhibit a pair (x˜,λ˜) that
(b)
0 satisfies the constraints of Problem VIII.1. Indeed
-0.1
-0.2 x∗ =sat(A(x∗−1)+2·1+λ∗)
t -0.3
≥sat(A(x∗−1)+(2+min(λ∗))1)
-0.4
-0.5 5 10 15 20 25 30 35 40 45 ≥sat(A(x∗−1)+(2+min(λ∗))1) . (34)
(c)
10-15 Therefore, the pair x˜=x∗,λ˜ =min(λ∗) fulfills the con-
f
10-20 straints of Problem(cid:16)VIII.1, because also ∃j(cid:17): x˜j = x∗j < 1.
R Beingτ˜theresultofamaximization,τ˜≥λ˜ =min(λ∗)=τ∗.
E
B10-25 Focusing on the converse, assume we are given a solution
(x˜,λ˜) of Problem VIII.1, with λ˜ = τ˜. No matter whether
10-30
extrinsic messages are saturated or not, we can always add a
5 10 15 20 25 30 35 40 45
AS topology positive vector ∆≥0 to λ˜·1:
Fig.6. Errorfloorcontributions of48ASs,withvariouschannelsaturation ∆=x˜−sat A(x˜−1)+(2+λ˜)1 . (35)
levels.
(cid:16) (cid:17)
This way, we turn inequality (33) into the equality
f
x˜=sat A(x˜−1)+2·1+λ˜·1+∆ . (36)
error probability, but now L =7/31 does not guarantee an
ch
error-free performance. (cid:16) (cid:17)
The constraints of Problem VIII.1 set x ≥ −1, thus we
f
Fig. 6 refers to a differentcode, with the same blocklength conclude that
and rate. Here we have 48 different AS topologies of size
sat A(x˜−1)+2·1+λ˜·1+∆ ≥−1 (37)
a = 6,8,10,12 (see Fig. 6(a)), b = 4, whose thresholds are
shown in Fig. 6(b). In Fig. 6(c) we plot the BER contribution (cid:16) (cid:17)
and finally
of each topology, with various channel saturation levels. The f
resultsagreewiththepredictionsofourmodel.IfLch =1,all sat A(x˜−1)+2·1+λ˜·1+∆
ASs contribute to the error floor. If L = 7/15 all the (6,4)
ch
ASs,whosethresholdτ =−1/2,aredeactivated.WithLch = f(cid:16) =sat A(x˜−1)+2·1(cid:17)+λ˜·1+∆ . (38)
7/31all(8,4)ASsbelowthresholdaredeactivated.Alsosome
(cid:16) (cid:17)
ASswiththresholdjustabove−7/31gavenoerrors.Besides, To conclude, if a solution (x˜,λ˜) of Problem VIII.1 with
Fig.6(c)showsagoodcorrelationbetweenthethresholdsand λ˜ =τ˜ exists, we can exhibit a generalized equilibrium (x∗ =
the dangerousness of the ASs. x˜,λ∗ = λ˜ ·1+∆) solution of Problem IV.1, with ∆ ≥ 0.
8
Beingτ∗ theresultofa maximization,we concludethatτ∗ ≥ to this solution. As already highlighted, S˙unsat could con-
min(λ∗)=min(λ˜·1+∆)≥λ˜ =τ˜. tain indices referring to saturated variables. If at least one
Once again, Problems IV.1 and VIII.1 are not equivalent, element of xS˙unsat is not saturated, this does not impair
as the solutions are different (the second one is not even a the outer maximization, since the same solution (x˙,λ˙) is
generalized equilibrium). Anyway, the two thresholds are the a maximizer with another pattern of saturations Sunsat =
same. j ∈S˙unsat :x˙ <1 6= ∅. In this case, (32) is satisfied.
j
Problem VIII.1 is still non-linear and multimodal. Besides, Onn the contrary,if allomessages in S˙unsat were saturated, we
equations are still not differentiable. We further elaborate,
could achieve the maximum λ not respecting (32), obtaining
rewriting Problem VIII.1 in another form that does not rely
τ˙ ≥ τ˜. We now prove that this cannot happen. Indeed, if
on the sat(·) function:we define a partition of {1,...,N} in
we set x = 1, (42) becomes λ ≤ −1. Yet, similarly
the two subsets Sunsat and Ssat of unsaturated and saturated Sunsat
to Theorem III.1, there are always other legitimate solutions
messagefs, respectively4. We also introduce a permutation having λ=−1 that do not violate the constraints of Problem
matrix Π, that reorganizes extrinsic messages, putting the
VIII.2,e.g.(x=−1,λ=−1),forwhich(43)hasnomeaning
unsaturated ones on top:
and (42) becomes −2(A − I)1 ≤ 0, that is true because
x A1 ≥ 1. We conclude that τ˙ ≥ −1 and that substituting
Sunsat =Πx. (39)
1 xSunsat < 1 with xSunsat ≤ 1 does not harm the threshold
(cid:20) (cid:21)
computation.
Accordingly, we permute the routing matrix A, and divide it
Once again, we do not distinguish any more between τ′,
in four submatrices, having inputs/outputs saturated or not:
τ∗, τ˜ and τ˙ since they match, and simply use τ.
A A In principle, we could solve the inner maximization of
ΠAΠ−1 = Sunsat,Sunsat Sunsat,Ssat . (40)
A A Problem VIII.2, repeatedly runningan optimizationalgorithm
Ssat,Sunsat Ssat,Ssat
(cid:20) (cid:21)
suited to linear equality and inequality constraints (e.g., the
We are now ready to introduce
simplex algorithm), and retaining only the largest value of τ
Problem VIII.2. amongall possible configurationsof saturated messages. This
is practically unfeasible for two reasons. First, optimization
τ˙ = max max λ
algorithms are time-consuming and we should resort to them
Sunsat⊆{1,...,N}λ,xSunsat
Sunsat6=∅ with caution. Besides, the number of configurations to test
s.t. −1≤x ≤1 (41) grows exponentially with N. Solving Problem VIII.2 with a
Sunsat
brute-force search becomes unpracticable even for moderate
(A −I)(x −1)+(1+λ)1≤0
Sunsat,Sunsat Sunsat valuesof N. In the following,we developmethodsto discard
(42)
most Sunsat configurations.
A (x −1)+(1+λ)1≥0. (43)
Ssat,Sunsat Sunsat
B. Pruning tests
NotethattheuseofSunsat intheaboveproblemisslightly
Test 1 exploits the following two theorems:
misleading:evenif theouter(leftmost)maximizationsets j ∈
Sunsat, thanks to (41) the inner maximization could achieve Theorem VIII.3. If A containsat least one row
Sunsat,Sunsat
itsmaximumeveninxj =1,andnotinxj <1.Weshallshow with all-zero elements, there are no solutions satisfying the
thatthisrelaxationdoesnotimpairthethresholdcomputation. constrains of Problem VIII.2.
The following theorem holds.
Proof: The proof is a reductio ad absurdum. Consider
Theorem VIII.2. As for Problems VIII.1 and VIII.2, τ˜=τ˙. any row of ASunsat,Sunsat with null weight, say the one
corresponding to the j-th in x. Then, by (42),
Proof: We only give a sketch of the proof, since it is
simple but quite long. We show that Problem VIII.1 implies xj ≥2+λ≥1 (44)
Problem VIII.2, and vice-versa.
where the second inequality holds since λ ≥ −1. The above
First, (32) means that Sunsat 6= ∅. Rewriting
result x ≥1 contradicts the hypothesis j ∈Sunsat .
(33) in the modified order, we obtain x ≥ j
Sunsat
ASunsat,Sunsat(xSunsat −1) + (2 + λ)1, since in the Theorem VIII.4. If ASunsat,Sunsat containsat least one row
first block of inequalities the sat(·) operator is useless. with exactly one element equal to 1, say Aj,h, and if the
Therefore (42) must be true. As for the second block of column vector ASsat,h has weight larger than 0, then there
inequalities, they hold only if thfe argument of the sat(·) are no solutions satisfying the constraints of Problem VIII.2.
exceeds 1, i.e., A (x −1)+(2+λ)1 ≥ 1,
Ssat,Sunsat Sunsat Proof:Theproofisareductioadabsurdum.Considerany
that immediately leads to (43). f nonnullelementofA ,sayA .Notethatthemaximum
Ssat,h i,h
Analogous arguments hold for the converse. The only
weightofanyrowandcolumnofAis2,beingtheVNdegree
tricky point is the following. Let (x˙,λ˙) be a maximizer for d = 3. Thus, either A is the only non-null element of the
ProblemVIII.2,withλ˙ =τ˙,andS˙unsat thesetcorresponding row A , or at mio,hst another element A =1 exists in
i,Sunsat i,k
A . If A has weight 1, by (43)
i,Sunsat i,Sunsat
4The adoption of both Sunsat and Ssat is redundant, since Ssat =
{1,...,N}\Sunsat,butsometimesweusebothofthemforcompactness. 1≤xh−1+2+λ=xh+1+λ. (45)
9
Column h (case 3) Column m (case 1,2,3)
If A has weight 2, by (43)
i,Sunsat
1≤x −1+x −1+2+λ≤x +1+λ (46)
h k h
Row j (case 1)
wherethesecondinequalityholdssincex ≤1.Ineithercase,
k
x ≥x −1+2+λ≥1 (47) Row j (case 2)
where the first ineqjualityhcomes from (42). The above result deleet ASunsat,Sunsat
x ≥1 contradicts the hypothesis j ∈Sunsat .
j
TheoremsVIII.3andVIII.4suggestsufficientconditionsto Row m (case 1,2,3)
discard configurations of Sunsat. The advantage of Test 1 is Row j (case 3)
simplicity. The weakness of Test 1 is that it does not take ad-
m
vantage of previous maximizations, with other configurations ove
of Sunsat.
A
Ssat,Sunsat
Test 2 exploits the threshold τ discovered up to that time.
It starts initializing lower and upper bounds (l and u, respec-
tively) for the minimum channel LLR λ and for xSunsat: Fig.7. Row-columnerasurepossibilities inASunsat,Sunsat.
λ
l≤ ≤u (48)
x
Sunsat C. Tree based, efficient search of the AS threshold
(cid:20) (cid:21)
where Test2istypicallymoreeffectivethanTest1,asitcandetect
τ λ
l= , u= max (49) a large number of configurations of Sunsat not improving
−1 1
(cid:20) M×1(cid:21) (cid:20) M×1(cid:21) the threshold τ. Yet, it can be applied only when Sunsat is
and M =card(Sunsat). In the mostgeneralcase, λ =1. formed.Onthecontrary,Test1canalsobeappliedduringthe
max
Yet we are mainly interested in the negative semi-axis, since construction of Sunsat:
in Section VI we have shown that we can deactivate an AS
TheoremVIII.5. LetSviolation ⊆Sunsat bethesetofindices
only if the threshold is negative. Therefore, in the threshold
satisfying Theorems VIII.3 or VIII.4. Erasing only a subset of
computation algorithm we can lower λ (of course, we
max Sviolation from Sunsat, the other elements in Sviolation still
must keep λ ≥ 0), exchanging some information loss
max satisfy conditions of Theorems VIII.3 or VIII.4.
(we return min(τ,λ ) = λ when τ > 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