Analysis of Absorbing Sets and Fully Absorbing Sets of Array-Based

21 Pages·2009·0.66 MB·English

IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.56,NO.1,JANUARY2010 181
Analysis of Absorbing Sets and Fully Absorbing Sets
of Array-Based LDPC Codes
LaraDolecek, Member, IEEE, ZhengyaZhang, Member, IEEE, Venkat Anantharam, Fellow, IEEE,
MartinJ. Wainwright,Member,IEEE,and Borivoje Nikolic´,SeniorMember,IEEE
Abstract—The class of low-density parity-check (LDPC) codes moderate bit-error rates (BER), say and above [6], [26],
isattractive,sincesuchcodescanbedecodedusingpracticalmes- [28]. As a consequence, LDPC codes have been adopted into
sage-passing algorithms, and their performance is known to ap-
several recent standards including Ethernet [2], digital video
proachtheShannonlimitsforsuitablylargeblocklengths.Forthe
broadcasting[1],[29],andbroadbandwireless[3].
intermediateblocklengthsrelevantinapplications,however,many
However,ithasbeenobservedthatLDPCcodesoftenexhibit
LDPCcodesexhibitaso-called“errorfloor,”correspondingtoa
significantflatteninginthecurvethatrelatessignal-to-noiseratio an error floor, meaning that beyond a certain signal-to-noise
(SNR)tothebit-errorrate(BER)level.Previousworkhaslinked ratio (SNR), there is a significant change in slope in the plot
this behavior to combinatorial substructures within the Tanner of BER versus SNR. For suitably designed codes, these error
graphassociatedwithanLDPCcode,knownas(fully)absorbing
floorsonlyoccuratrelativelylowBERs(e.g.,below ),and
sets. These fully absorbing sets correspond to a particular type
do not pose problems for applications requiring only moder-
ofnear-codewordsortrappingsetsthatarestableunderbit-flip-
pingoperations,andexertthedominanteffectonthelowBERbe- atelylowBER,suchaslow-ratewirelesscommunications.For
haviorofstructuredLDPCcodes.Thispaperprovidesadetailed other applications with low BER requirements, such as mag-
theoreticalanalysisofthese(fully)absorbingsetsfortheclassof neticdiskdrivesandopticalchannels,theseerrorfloorsareex-
(cid:0) array-based LDPC codes, including the characterization of
(cid:0)(cid:2)(cid:3) tremely troublesome. An ongoing line of research has shown
allminimal(fully)absorbingsetsforthearray-basedLDPCcodes
for(cid:2) (cid:0)(cid:2)(cid:3)(cid:3)(cid:3)(cid:4),andmoreover,itprovidesthedevelopmentoftech- thattheseerrorfloorsarecloselyrelatedtothesuboptimalityof
niquestoenumeratethemexactly.Theoreticalresultsofthistype practical message-passing decoders. MacKay and Postol [18]
provide a foundation for predicting and extrapolating the error recognized that certain classes of non-codewords, which they
floorbehaviorofLDPCcodes. referredtoasnear-codewords,cancausethedecodertofail;in
IndexTerms—Absorbing set, bit-flipping,error floor, low-den- particular,an near-codewordisabinarystringofweight
sityparity-check(LDPC)codes,messagepassingdecoding,near- whosesyndromehasweight .Fromsimulationofarate
codeword,trappingset. LDPCcodewithblocklength basedontheMarguliscon-
struction, they found that and near-codewords
are the main contributors to the error floor of this code when
I. INTRODUCTION
usedforthetransmissionoveranadditivewhiteGaussiannoise
L OW-density parity-check (LDPC) codes are a class of (AWGN)channel.Theyalsopostulatedthattheminimumdis-
error-correcting codes based on sparse graphs. Their tance of this code is significantly higher than the weight of
chief appeal is their excellent performance under practical theobservednear-codewords.Dietal.[5]definedacloselyre-
latedconceptofastoppingset,whichgovernstheperformance
decoding algorithms based on message passing, especially for
limitsofiterativedecodingforLDPCcodesoverthebinaryera-
ManuscriptreceivedJanuary31,2008;revisedJanuary19,2009.Currentver- sure channel (BEC). Subsequent work by Orlitsky et al. [21]
sionpublishedDecember23,2009.ThisworkwassupportedinpartbytheNa-
hasprovidedanalyticalcharacterizationofthestoppingsetenu-
tionalScienceFoundation(NSF)underGrantCCF-0635372,andbyMarvell
SemiconductorandIntelCorporationthroughtheUCMICROprogram.The meratorfordifferentensemblesofLDPCcodes.Althoughvery
workofL.DolecekwasalsosupportedbytheUniversityofCaliforniaDisser- useful for determining the performance over a BEC, stopping
tationYearFellowship.Thematerialinthispaperwaspresentedinpartatthe
sets cannot be used directly to determine LDPC performance
IEEEInternationalConferenceonCommunications,Glasgow,Scotland,July
2007. forotherchannels,suchasAWGNchannels,sincethenatureof
L. Dolecek was with the Electrical Engineering and Computer Science errorsinnonerasurechannelsismoresubtle.Formoregeneral
Department, Massachusetts Institute of Technology, Cambridge, MA 02139
channels, pioneering work by Richardson [23] introduced the
USA. She is now with the Electrical Engineering Department, University
of California, Los Angeles (UCLA), Los Angeles, CA 90095 USA (e-mail: operationallydefinednotionofatrappingsetinordertoaddress
[email protected]). theerrorfloorofLDPCcodes,anddevelopedafastnumerical
Z.ZhangwaswiththeDepartmentofElectricalEngineeringandComputer
methodforestimatingtheerrorprobabilityinthelowBERre-
Sciences,UniversityofCalifornia,Berkeley.HeisnowwiththeDepartment
ofElectricalEngineeringandComputerScience,UniversityofMichigan,Ann gion. Follow-up work by Chilappagari et al. [4] used trapping
Arbor,MI48109USA(e-mail:[email protected]). setstostudyerrorfloorsofLDPCcodesonabinarysymmetric
V. Anantharam, B. Nikolic´, and M. J. Wainwright are with the Electrical channel.Otherresearchershavestudiedcloselyrelatednotions
Engineering and Computer Science Department, University of California,
ofelementarytrappingsets[17],pseudocodewordsforiterative
Berkeley, Berkeley, CA 94720 USA (email: [email protected];
[email protected];[email protected]). decoding[14],[15],andpseudocodewordsforlinear-program-
CommunicatedbyT.Etzion,AssociateEditorforCodingTheory. mingdecoding[12].
ColorversionsofFigures1and2inthispaperareavailableonlineathttp://
Inpreviousexperimentalwork[32],wedesignedahardware
ieeexplore.ieee.org.
DigitalObjectIdentifier10.1109/TIT.2009.2034781 emulator to explore the low BER regime of various classes of
0018-9448/$26.00©2009IEEE
182 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.56,NO.1,JANUARY2010
Fig.1. Hardware-basedBER/FERperformanceofseveralarray-basedLDPCcodeswithbitdegreefour.
structuredLDPCcodes.Onthebasisoftheseexperiments,we TABLEI
isolated a set of graphical substructures in the Tanner graph ERRORSTATISTICSFORTHE(cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:0)(cid:2)(cid:6)(cid:4)(cid:4)(cid:7)ARRAY-BASEDLDPCCODE
[27] representation of the parity check matrix, referred to as (CHECKDEGREE(cid:8)(cid:9)(cid:10),BITDEGREE(cid:8)(cid:11))
absorbingsets,thatcausevariousmessage-passingdecodersto
failbyconvergingtothesenon-codewordstates.Likestopping
sets, absorbing sets have a purely combinatorial definition in
terms of the parity check matrix of a given LDPC code. They
canbeviewedasaspecialsubclassofanear-codewordoratrap-
pingset,inparticularonethatisguaranteedtobestableunder
a bit-flipping decoder. With this property, and in contrast to
previousworks,absorbingsetsdescribethedominantdecoding TABLEII
failures of various message passing algorithms and provide a ERRORSTATISTICSFORTHE(cid:0)(cid:2)(cid:2)(cid:4)(cid:5)(cid:0)(cid:2)(cid:4)(cid:2)(cid:11)(cid:7)ARRAY-BASEDLDPCCODE
(CHECKDEGREE(cid:8)(cid:11)(cid:12),BITDEGREE(cid:8)(cid:11))
newanalyticalfoundationforstudyingerrorfloors.
We also point out that the insights on the structure of the
absorbingsetsasthedominantcontributorstotheLDPCcode
performance in the low BER region are not only useful from
the theoretical viewpoint, but they also play an extremely im-
portant role in the systematic development of an efficient de-
coder design required by practical very-large-scale integration
(VLSI)implementations.Thepropertiesofabsorbingsetsarea
criticalfactorinthedesignofahigh-throughputhardwareem- codeword lengths, and
ulatorthatoperatesinverylowBERregions[33],[34].Inad- ,withrates and ,respectively.These
dition, an ongoing work has demonstrated that the absorbing codes have check degrees and , respectively, and all
sets are also a fundamental component in an importance-sam- three have bit degrees equal to . While the performance re-
pling-basedmethodforestimatingerrorfloorprobabilities[9]; portedinFig.1ofthesethreecodesisquitesimilar,itisimpor-
inthedevelopmentofexplicittheoreticalboundsondecoder’s tanttonoticethatthelowBERperformanceofallthreecodes
performance [7]; and for the systematic improvementof prac- is dominated by the (fully) absorbing sets (this object
ticaldecodingalgorithms[25],[34]. will be precisely defined in Section II-B), as indicated in Ta-
In order to motivate the theoretical results in this paper, we bles I, II, and III which list the statistics (counts) of the error
present several experimental results obtained on our hardware eventscapturedatseveralSNRpoints.Eachrowcorrespondsto
emulator [32]. This hardware platform is capable of reaching thenumberofcaptureddecodingerrorswhenthedecodercon-
remarkableBERlevelsbelow forstructuredfinite-length vergedtoan absorbingset,fordifferent pairs.
LDPC codes, decoded using a sum–product algorithm. Fig. 1 While inthis paper wefocus ondescribing indetail the ab-
showstheperformanceofseveralarray-basedLDPCcodes[11], sorbingsetsforthefamilyofhigh-ratearray-basedLDPCcodes,
alongwiththeuncodedtransmissioncurve,forboththebit-and itisworthreiteratingthattheabsorbingsetsareageneralprop-
frame- error rate (FER). The results are for three different ertyofthefactorgraphdescribingtheparitycheckmatrixofan
DOLECEKetal.:ANALYSISOFABSORBINGSETSANDFULLYABSORBINGSETSOFARRAY-BASEDLDPCCODES 183
Fig.2. Hardware-basedBER/FERperformanceofthe(cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:0)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)Reed–SolomonbasedLDPCcode.
TABLEIII absorbing sets. For this class of structured LDPC codes, we
ERRORSTATISTICSFORTHE(cid:0)(cid:6)(cid:5)(cid:4)(cid:10)(cid:0)(cid:6)(cid:11)(cid:5)(cid:3)(cid:9)ARRAY-BASEDLDPCCODE provethenonexistenceofvariouspossiblecandidateabsorbing
(CHECKDEGREE(cid:12)(cid:4)(cid:8),BITDEGREE(cid:12)(cid:4)) sets, and having thereby explicitly constructed minimal ab-
sorbingsets, we characterize theircombinatorial structure and
cardinalities.
Compared to the results in [8], in this work we provide the
complete and explicit characterization of all (fully) ab-
sorbingsets(versuspureexistenceandasymptoticproperties),
and a detailed theoretical analysis of all candidate configura-
tionsofminimalabsorbingsetsforcolumnweightsofinterest.
Additionally,theexperimentalevidenceprovidedearlierinthis
LDPC code, not just of a specific class of codes on which sectionmotivatestheusefulnessofstudyingabsorbingsets.
we focus in depth in the remainder of the paper. As an il- Theremainderofthispaperisorganizedasfollows.Webegin
lustration, Fig. 2 shows the BER/FER versus SNR curve in Section II with a brief overview of the class of array-based
of another high-performance LDPC code, the LDPCcodes[11],andthenformallyintroducethedefinitionof
Reed–SolomonLDPCcode[6].Whilethiscodealsohasavery absorbingsets.InSectionIII,weprovideadetailedstudyofthe
absorbingsetsforcolumnweights and forthestan-
good performance in the moderate BER region, it also expe-
dardparitycheckmatrices ofsuchcodes,andenumerate
riences a significant flattening of the BER/FER versus SNR
all such sets of smallest size. All of the theoretical results are
curve at lower BER levels. For this code, the dominant errors
stated in this section, with some of the more technical proofs
are the (fully) absorbing sets. In fact, almost all errors
deferredtotheAppendix.SectionIVconcludesthepaper.
capturedinthe5.0–5.5dBrangeare fullyabsorbingsets.
Itisimportanttopointoutthatinalloftheaboveexamples, II. BACKGROUND
none of the recorded decoding errors in the low BER region
Webeginwithbackgroundonarray-basedLDPCcodes,and
wereduetotheminimumdistanceoranyothernontransmitted
codewords. thenprovideaprecisedefinitionofabsorbingsets.
Under maximum-likelihood decoding, it is well known that
the minimum distance and the weight enumerator of a code A. Array-BasedLDPCCodes
arekeyfactorsthatdetermineitserror-correctingperformance.
Array-based LDPC codes [11] are regular LDPC codes pa-
Given that absorbing sets (as opposed to neighboring code-
rameterizedbyapairofintegers ,suchthat ,and
words) are the dominant error event for iterative decoders, it
isaprime.Givena permutationmatrix oftheform
is natural to consider the analogs of minimum distance and
weight enumerator for absorbing sets. With the above moti-
vating examples in mind, we now turn to the in-depth study
of the minimal absorbing sets of high-rate array-based LDPC
codes.ThisclassofcodesisanexemplarofastructuredLDPC (1)
. . . .
. . . .
code with excellentperformance inthe moderate BER region, . . . .
but whose low BER performance is governed by the minimal
184 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.56,NO.1,JANUARY2010
shouldalsohavestrictlymoreemptysquaresthanfullsquares
asneighbors.
Note that is a fully absorbing set if and only if for
all , where de-
notes the symmetric difference between and , is
the Hamming weight of a binary string , and is a binary
Fig.3. Anexampleofa(cid:0)(cid:2)(cid:0)(cid:2)(cid:3)absorbingset.
stringwithsupport .
The name “absorbing set” indicates the absorbing nature of
weformthe paritycheckmatrix these combinatorial objects in the sense that they act as local
minimumstatesforthedecodingalgorithm:uponenteringthis
configurationthedecodingalgorithmis“absorbed”init.Forthe
special case of a bit flipping algorithm [24], the configuration
describedasafullyabsorbingsetisstable,sinceeachbitnode
receives strictly more messages from the neighboring checks
(2)
.. .. .. .. thatreinforceitsvaluethanmessagesthatsuggesttheopposite
. . . .
bit value. However, as illustrated in the examples in the Intro-
duction(furtherimplementationdetailsandadditionalexamples
areinourconcurrentwork[34]),absorbingsetsalsocontrolthe
We use to denote the binary linear code defined by this
errorfloorbehaviorofmoresophisticatedmessage-passingde-
paritycheckmatrix(2).Therate1ofthiscodeis
coders,suchasthesum–productalgorithm.
[20].Fan[11]firstdemonstratedthatarray-basedLDPCcodes
have very good performance in the low SNR region, and they III. THEORETICALRESULTS
havesubsequentlybeenproposedforanumberofapplications,
Ourgoalistodescribeminimalabsorbingsetsandminimal
including digital subscriber lines [10] and magnetic recording
fullyabsorbingsets ofthefactorgraphoftheparitycheck
[30].
matrix ,for ,wheretheminimalityreferstothe
smallestpossible ,andwhere isthesmallestpossibleforthe
B. AbsorbingSets
given .
Aconvenientrepresentationofan paritycheckmatrix We use the following notation throughout the paper. Recall
ofabinarylinearcodeisintermsofitsfactororTannergraph that is a matrix of ’s and ’s. It is convenient
[13],[16],[27].Inparticular,let denoteabi- to view as a two-dimensional array of component
partitegraph,inwhichthesetofvertices isassociatedwith submatriceswiththerows intherange (also
bitsinthecode(columnsof ),andtheset isassociatedwith referredtoasrowgroups)andthecolumns intherange
checksofthecode(rowsof ).Theedgeset isdefinedby (also referred to as column groups). Each column
thestructureof :inparticular,thereexistsanedge of isuniquelydescribedbyapair where denotes
ifandonlyif and .Elementsof arecalled“bit thecolumnindexofthesubmatrixthiscolumnbelongsto,and
nodes”andelementsof arecalled“checknodes.” ,denotestheindexofthiscolumnwithinthe
Forthearray-basedLDPCcodesdefinedinSectionII-A,the submatrix.
factor graphassociated with doesnothaveanycyclesof Our main results are summarized in Theorems 1 and 2. Let
lengthfour,andthusthegirthisatleastsix(see[11]).Forany bethefactorgraphassociatedwiththeparitycheckmatrix
subset of we let denote the subset of check nodes ofthearray-basedLDPCcode .Thefirstresultchar-
neighboringtheelementsof .Foranysubset of ,let acterizestheminimal(fully)absorbingsets.
(resp., )bethesetofneighboringverticesof in inthe
Theorem1(Minimality):
graph witheven(resp.,odd)degreewithrespectto .With
(a) Forthe family,allminimalabsorbingsetsaremin-
thissetup,wehavethefollowing.
imalfullyabsorbingsets,andareofsize .
Definition1: Givenanintegerpair ,an absorbing (b) Forthe family,theminimalabsorbingsetsareofsize
setisasubset ofsize ,with ofsize andwith , and the minimal fully absorbing sets are of size
thepropertythateachelementof hasstrictlyfewerneighbors .
in thanin .Wesaythatan absorbingset (c) For the family, and for , the minimal ab-
isan fullyabsorbingset,ifinaddition,allbitnodesin sorbing sets and the minimal fully absorbing sets are of
havestrictlymoreneighborsin thanin . size .
An example of a absorbing set is given in Fig. 3, Our second result deals with the scaling behavior of the
where dark circles represent bits in the set , dark squares number of absorbing sets. Recall the standard asymptotic
constitutetheset ,whitesquaresconstitutetheset notation : we say that some positive function grows as
isgivenbysolidlines,and isgiven if there exist constants such that
by dashed lines. Observe that each element in has more ,for sufficientlylarge.
neighbors with even degree than odd degree. All check nodes
Theorem2(Scaling): Recallingthattheblocklength
not in the picture are denoted by empty squares. For this set
ofthe codecorrespondstothenumberofcolumnsinthe
to be a fully absorbing set, every bit node not in the figure
paritycheckmatrix ,wehavethefollowing.
(a) For ,thenumberofminimal(fully)absorbingsets
1Notethattheparitycheck matrix(cid:0) isnot fullrank,hencetheslight
increaseinrateover(cid:4)(cid:0)(cid:2)(cid:3)(cid:4). in growsas .
DOLECEKetal.:ANALYSISOFABSORBINGSETSANDFULLYABSORBINGSETSOFARRAY-BASEDLDPCCODES 185
(b) For ,thenumberofminimalabsorbingsetsaswell Webeginwithelementarystructuralconditionsandlemmas
as the number of minimal fully absorbing sets grows as that play a central role throughout the paper, summarized in
. SectionIII-A.
(c) For andforallblocklengths thenumberof
minimalabsorbingsetsaswellasthenumberofminimal
A. Preliminaries
fullyabsorbingsetsgrowsas .
Although Theorem 2 states the result in terms of the Let be the factor graph associated with , so bit
-scaling behavior, our techniques in fact provide an exact nodesandchecknodesin representcolumnsandrowsin
count of the number of minimal (fully) absorbing sets. Note ,respectively.Inthegraph ,bitnodeshavedegree
that Theorem 1(a) implies that for , the smallest (fully) andchecknodeshavedegree .Thereisatotalof bitnodes
absorbing sets are codewords; in fact, for this code, these and checknodes.Eachbitnodein receivestheunique
absorbing sets are the minimum distance codewords. This label that describes the corresponding column of .
resultshouldbecontrastedwiththeassertionsofTheorem1(b) Eachchecknodein receivesalabel ifthecorresponding
and (c), for and , respectively, which establish rowof belongstotherowgroup .Multiplebitnodescan
the existence of (fully) absorbing sets strictly smaller than havethesame or label,butnotboth.Multiplechecknodes
the minimum distance of the code. In particular, for , canhavethesame label.
the minimum distance is six [20], [31], whereas for Wenotethatthestructureoftheparitycheckmatriximposes
and , the minimum distance is between eight and ten thefollowingconditionsontheneighboringbitnodesandcheck
[20],[31].Therefore,for both and ,the minimal nodes.
absorbing sets and minimal fully absorbing sets are strictly BitConsistency:Forabitnode,allitsincidentchecknodes,
smallerthantheminimumdistanceofthecode. labeled through , must have distinct labels, i.e., these
Subsequent sections contain detailed proofs of the above checknodesareindistinctrowgroups.
statements. In particular, the proofs of Theorems 1 and 2 are Check Consistency: All bit nodes, say through
presentedinthefollowingorder. , participating in the same check node must have dis-
(cid:129) In Section III-A, we summarize the principal structural tinct values,i.e.,theyareallindistinctcolumngroups.
propertiesoftheparitycheckmatrix ,thatarereferred Bothconditionsfollowfromthefactthattheparitycheckma-
toasthebit-,check-,cycle-,andpattern-consistencycon- trix of consistsofatwo-dimensionalarrayofpermu-
straints,respectively.Thesestructuralpropertieswillbere- tationmatricesofequalsize.
peatedlyexploitedinestablishingthemainresults.
Lemma1: (PatternConsistency)Thepermutationsubmatrix
(cid:129) SectionIII-BcontainstheproofofstatementsgiveninThe-
hasthefollowingproperties.
orems1(a)and2(a).Inparticular,afterhavingshownthat
(a) The entryof is ifandonlyif .
thenumberofbitsinaminimalabsorbingsethastobeat
(b) Let and be in the same row group of . If
least forthecodedescribedby ,Lemma3provides
entry of isnonzero,thensoisentry of
an explicitcountof (fully)absorbing sets,thusim-
where .
mediatelyimplyingTheorem1(a)andTheorem2(a).
Proof: Assumethatthecolumnsandrowsof areindexed
(cid:129) SectionIII-CcontainstheproofofstatementsgiveninThe-
with through .Recallthat has“ ”inthetopmostrowin
orems 1(b) and 2(b). We first show that for the code de-
thecolumnindexed (lastcolumn),andthateachsubsequent
scribedby ,thenumberofbitsinaminimalabsorbing
row has “ ” in a position that is a cyclic shift to the right of
sethastobeatleast .Wethenprovethat absorbing
thepositionof“ ”inthepreviousrow.Multiplying with
sets exist, and that these are not fully absorbing sets. We
cyclicallyshiftstheentriesbyonepositiontotheleft.Thus,
thenshowthat absorbingsetsaretheminimalfully
has“ ”inthetopmostrowincolumn ,andhas
absorbingsets,thusestablishingTheorem1(b).Lemma4
“ ”insomerow inthecolumn ,from
providestheexplicitcountsof absorbingsets,andof
whichthestatement(a)follows.Thus, has“ ”inrow in
fullyabsorbingsets,therebyimplyingthestatement
thecolumnindexed and has“ ”in
inTheorem2(b).
row inthecolumnindexed .Equating
(cid:129) ThelengthiestSectionIII-Ddealswithprovingthestate-
theseexpressionsintermsof ,thestatementin(b)follows.
mentsgiveninTheorems1(c)and2(c).Toprovethestate-
mentsregardingtheminimalityof (fully)absorbing We will refer to the constraints of the type described in
setsforthecodedescribedby ,weproceedwithase- Lemma1aspatternconsistencyconstraints.
riesofauxiliarylemmasthatestablishthenonexistenceof
Lemma 2: (Cycle consistency) Consider a cycle in of
certain smaller candidate absorbing sets, which holds for
length , involving bit nodes, with labels through
sufficiently large code parameter (specifically
and checknodes, withlabels through ,suchthat
will be sufficient for all auxiliary results). The sequence
bit nodes and participate in the check labeled
of the intermediate results consists of: 1) Lemma 5 that
and participateinthechecklabeled ,and
proves that absorbing sets do not exist, 2) Lemma
soon,untilchecklabeled inwhich and par-
6 that proves that absorbing sets do not exist, and
ticipate.Then
3) Lemma 7 that proves that absorbing sets do not
exist. Finally, Lemma 8 provides an in-depth analysis of
the (fully)absorbingsets,whereweprovidetheex-
plicitenumerationandstructuraldescriptionofthesesets. (3)
186 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.56,NO.1,JANUARY2010
whereasthepatternconsistencyconstraintsofLemma1(b)give
(7)
Thereare waysofchoosing ,whichalsodetermines .
Since ,wemusthave ,sowehave ways
Fig.4. (Labeled)candidate(cid:0)(cid:2)(cid:0)(cid:3)(cid:4)absorbingset. ofchoosing ,whichalsodetermines .Wethenhave ways
of choosing , which also determines . Since , we
have waysofchoosing ,whichalsodetermines .To
Proof: ThepatternconsistencyconstraintsofLemma1(b)
verifythateveryoneofthesechoicessatisfiesalltheequations
give
itonlyremainstoverifythat .Thisholdsbecause
(8)
.
. (4)
. Now,foranychoiceofrowgrouplabelsforthechecks,and
column labels for the bits that satisfy the bit and check con-
sistency constraints and the pattern consistency constraints of
Expand into Lemma1(b),thereisauniquewaytochoosetherowindexin
the individual row groups so that the pattern consistency con-
straint of Lemma 1 are satisfied. This completes the proof of
Hence
Lemma3.
FromLemma3(andrecallingthattheblocklength ),
(5)
weconcludethatthenumberof (fully)absorbingsetsfor
Byrearrangingtheterms,relation(3)follows. thecodedescribedby is ,therebyestablishingThe-
orem2(a).
Constraints of the type (3) will subsequently be referred to
as cycle consistency constraints. Note that the cycle consis-
C. ProofofTheorems1(b)and2(b)
tency constraints are a consequence of the pattern consistency
constraints. In our preceding analysis with , note that ab-
sorbingsetsareactuallycodewords,sotheperformanceofthe
B. ProofofTheorems1(a)and2(a)
cyclecodeunderiterativedecodingisdominatedbylowweight
WestartbyprovingTheorem1(a).Thecode hasuniform codewords.Wenowturntothecase ,whichleadstomore
bitdegreetwo,andisthusacyclecode.Eventhoughsuchcodes interestingresults.Inparticular,ourproofofTheorem1(b)es-
areknowntobepoor[22],weincludetheanalysisforthesake tablishesthe existenceofminimalabsorbing setsandminimal
ofcompleteness. fully absorbing sets, for which the number of bit nodes is
Let denote the factor graph of . Let strictlysmallerthantheminimumdistance ofthecode.
be an absorbing set in . Each bit node in has Let denote the factor graph of . Let
degree in andisrequiredtohavestrictlymoreneighbors be an absorbing set in . Each bit node in has
in than in . This implies that is empty. The degree in andisrequiredtohavestrictlymoreneighbors
absorbing set is of type . It is thus a fully absorbing set, in thanin .
andisinfactacodeword. Suppose .Inthegraph ,anevennumberofedges
Sincethematrix hasthetoprowconsistingofidentity from terminatesin .Thus,either or corre-
matrices,thecodewordsof areofevenweight.Moreover, spondingtothesituationsinFig.5.Ineithercase,therewould
since the bottom row of consists of distinct component beacycleoflength in ,whichcannothold[11],implying
submatrices, no two columns of sum to zero. Therefore, that .
andevenandtherearenocyclesoflength inthiscode. Suppose .Inthegraph ,anevennumberofedges
Wenowconsider .Let and from terminates in . Thus, either or .
bethebitnodesparticipatinginacandidate ab- Suppose .ThismustcorrespondtothefirstforminFig.6,
sorbing set. These nodes must necessarily be arranged as in or the second form in Fig. 6, which again involves a cycle of
Fig. 4. length in ,acontradiction[11].
ThefollowingresultprovesTheorem1(a). Still with , the remaining case to consider is .
Lemma 3: There is a total of (fully) ab- Inthiscase,eachbitnodein wouldthenconnecttoexactly
onechecknodein implyingtheunlabeledformofFig.7.
sorbingsetsinthecodedescribedby .
Notethatthereisacycleoflength .Supposethatthesethree
Proof: The bit consistency conditions are automatically
bitnodesareindexedas and ,respec-
satisfiedbythenumberingoftherowgroupsinFig.4.Thecheck
tively,where and aredistinct(bythecheckconsistency)
consistencyconstraintsgive
and . Without loss of generality, as-
and (6) sumethat and shareacheckintherowgroup
DOLECEKetal.:ANALYSISOFABSORBINGSETSANDFULLYABSORBINGSETSOFARRAY-BASEDLDPCCODES 187
Fig.5. Candidate(cid:0)(cid:2)(cid:0)(cid:0)(cid:3)absorbingsets.(a)(cid:0)(cid:4)(cid:5).(b)(cid:0)(cid:4)(cid:2).
Fig.8. Candidate(cid:0)(cid:6)(cid:2)(cid:6)(cid:3)absorbingset(solidcircles),withanadjacentbitnode
(emptycircle).
Eventhougha fullyabsorbingsetseemsplausible,care
mustbetakenwithrespecttobitnodesoutsideacandidatefully
absorbingset,thatalsoparticipateintheunsatisfiedchecks.As
wenowshow,a fullyabsorbingsetcannot exist,though
theexistenceofa absorbingsetimpliesa fullyab-
sorbingset.
Suppose first that a fully absorbing set were to exist.
Since ,itisthennecessarythatnobitnodeoutsideofthe
absorbing set participates in more than one unsatisfied check
adjacent to a absorbing set. Since and
share a check, . Consider the bit node labeled
thatconnects to , asinFig. 8.Since ,itfollowsfrom
Fig. 6. Candidate (cid:0)(cid:6)(cid:2)(cid:7)(cid:3) absorbing sets. (a) First candidate. (b) Second Lemma1(b)that .Equations(9)–(11)implythat
candidate. so that bit node
alsoconnectstothechecklabeled ,asshowninFig.8.This
eliminatesthepossibilityofa fullyabsorbingset.
A absorbingset(i.e.,acodewordofweight )cannot
exist since the minimum distance of the code is [31]. The
nextcandidatesizeforthesmallestfullyabsorbingsetis .
Eachoftheunsatisfiedchecksinanysuchconfigurationwould
necessarilyconnecttoonlyoneofthebitnodes,elsewewould
have a cycle of length , a contradiction [11]. Given this, no
satisfied checknode can connectto all four bit nodes, else we
wouldhaveacycleoflength ,acontradiction[11].Sincethere
Fig.7. (Labeled)candidate(cid:0)(cid:6)(cid:2)(cid:6)(cid:3)absorbingset. aretenedgesfromthebitnodesthatgotosatisfiedcheckswe
nowseethattheremustbefivesatisfiedchecksinanycandidate
fullyabsorbingset.Thetwobitnodesthateachhaveall
and shareacheckintherowgroup ,andthat
theirthreeedgesgoingtosatisfiedchecknodesmustthenshare
and share a check in the row group , where
exactlyonesatisfiedcheck(theyhavetoshareatleastone,and
andaredistinctbythebitconsistencycon-
cannotsharemorethanone[11]).Wehavethereforeconcluded
dition.Wemayassumewithoutlossofgeneralitythat
thatanycandidate fullyabsorbingsetmustlooklike(an
and . Note that the bit consistency constraints
unlabeledversionof)Fig.8.Theexistenceofsuch fully
forcethevaluesof and tobeasgiveninFig.7.
absorbing sets is proved in Lemma 4, which also counts the
In the remainder of the discussion, we first prove the exis-
numberofsuchsets.
tence of a absorbing set. We then show that these
absorbing sets are not fully absorbing sets. This result will in Lemma 4: The total number of absorbing sets and
turn imply the existence of fully absorbing sets, which fullyabsorbingsetsinthefactorgraph is ,
arethusminimalfullyabsorbingsetsfor . and ,respectively.
Thebitconsistencyconstraintsareautomaticallysatisfiedby Proof: Referring to Fig. 7, the bit consistency and the
ourlabelingoftherowgroupsinFig.7.Thecheckconsistency check consistency constraints are satisfied for the given labels
constraintsreducetothedistinctnessof and .Thepat- ofrowgroupsandsince and aredistinct.Then and
ternconsistencyconstraintsofLemma1(b)give can each be chosen in ways, and then can be chosen
in ways. This fixes by (9), by (10), and then
(9)
by(11).Thereisthen a uniqueway tochoosethe rowindices
(10) in the individual row groups so that the pattern consistency
(11) conditionsofLemma1aresatisfied.Thus,thetotalnumberof
absorbingsetsis .
Theexistenceofasolutionandhenceofa absorbingsetis Turning to counting fully absorbing sets, every such
givenintheproofofLemma4below,whichcountsthenumber setmustlooklikeanunlabeledversionofFig.8,andsoitcon-
ofsuch sets. tainsexactlytwodistinct absorbingsets(corresponding,
188 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.56,NO.1,JANUARY2010
Lemma5: For ,thefactorgraphfamily doesnot
containany absorbingsets.
Proof: Without loss of generality, we may let
and , where and dis-
tinct by the bit consistency conditions. Then, by propagating
thebitconsistencyconditionsateachremainingvertex,andex-
ploitingthesymmetry,itsufficestoconsider
either or where
andaredistinct.
For the case , we es-
tablishthefollowingcycleconsistencyconditionsbasedonthe
cycleswithinthegraphinFig.9:
and
Fig.9. Depictionofthecandidate(cid:0)(cid:2)(cid:0)(cid:2)(cid:3)absorbingset. (12)
Byaddingandsubtractingtheconditionsin(12),itfollowsthat
respectively, to removing one of the bit nodes that connects
to an unsatisfied check). From Fig. 7 one can see that every
and
absorbingsetiscontainedinthreedistinct fullyab-
sorbing sets (for each pair of unsatisfied checks in Fig. 7 one (13)
canfindabitnodethatthesechecksconnectto,whichwhenap-
Since are distinct, relation (13) implies that ’s would
pendedtothe absorbingsetgivesa fullyabsorbing
havetobeallthesame,whichcontradictsthecheckconsistency
set).Thetotalnumberof fullyabsorbingsetsistherefore
constraint.
.
For the case , again
ObservethatLemma4immediatelyimpliesTheorem1(b). basedonthecyclestructureinFig.9,weobtainthecyclecon-
sistencyconditions
D. ProofofTheorems1(c)and2(c)
Inordertoestablishthat (fully)absorbingsetsaremin-
imal for and , we will first show that ab-
(14)
sorbing sets for do not exist. This section contains the
following auxiliary results on the nonexistence of certain can-
Welet and .Bythe
didateabsorbingsets,whichholdforsufficientlylargecodepa-
checkconsistencycondition,allof and arenonzero.
rameter (specifically willbesufficientforallauxiliary
Substituting and in(14)andthenexpressing and
results).Inparticular:
intermsof ,onearrivesatthecondition
—Lemma5provesthat absorbingsetsdonotexist;
—Lemma6provesthat absorbingsetsdonotexist,and (15)
—Lemma7provesthat absorbingsetsdonotexist.
Finally, Lemma 8 provides an in-depth analysis of the Itcanbeverifiedthatthisconditioncannotholdforanychoice
absorbingsets. of , where and are distinct
Let denote an absorbing set in , for . There are ways of assigning numerical
thefactorgraphof .If (respectively, )thenatleast values to . Substituting each numerical assignment
six(respectively,nine)edgesfrom in terminatein , yieldspossiblechoicesofprime forwhichtheex-
whichimpliestheexistenceofacycleoflength in ,which pressionin(15)becomeszero .Thecondition(15)holds
isfalse[11].Thus, . for .Therefore,for doesnotcontain
Suppose andnotethat mustbeeven.Wecannothave absorbingsets.
, since this would imply the existence of a codeword of
Wenextshowthat absorbingsetsdonotexistforthepa-
weight , which is false [31]. If , one can conclude that
rameter largeenough.Inparticularweestablishacongruential
theremustbeacycleoflength inthecode(whetherthenumber
constraintinvolvingthelabelsoftheedgesemanatingfromthe
ofedgesgoingintounsatisfiedchecksistwoorfour),andthisis
bitsintheabsorbingsetthatcannotholdfor largeenough.
false,[11].Thus,wemusthave and,sinceeachbitnode
musthaveatleastthreeedgesgoingtosatisfiedchecks,theim- Lemma6: For ,thefactorgraphfamily doesnot
possibilityofacycleoflength [11]impliesthattheabsorbing containany absorbingsets.
setcanbedescribedasinFig.9.Inthisfigure,eachvertexrep- Proof: Sinceeachbitnodeintheabsorbingsethasatmost
resentsadistinctbitnodeofthecandidate absorbingset oneneighboringunsatisfiedchecknode,itfollowsthat .
and each edge represents a satisfied check node that connects Bycountingthenumberofedgesemanatingfromthebitnodes
tothebitnodesintheabsorbingset,thatcorrespondtoitsend- in the candidate absorbing set (some of which have three sat-
pointsinthefigure.Thefollowinglemmaestablishesthatsuch isfied and one unsatisfied check, and the rest having all four
setsdonotexistiftheprime islargeenough. checkssatisfied) andending at satisfiedchecks,itfollowsthat
DOLECEKetal.:ANALYSISOFABSORBINGSETSANDFULLYABSORBINGSETSOFARRAY-BASEDLDPCCODES 189
that . We let and , where
anddistinct.Bypropagatingpossiblevalues
of the labels for remaining edges, while maintaining bit con-
sistencyconditions,itfollowsthat is
either or .
For , and
foreachedgeanditsendpointsinFig.10,wewritethepattern
consistencyconstraintsofLemma1(b),intermsof and
(16a)
(16b)
(16c)
(16d)
(16e)
(16f)
and (16g)
Fig.10. Depictionofthecandidate(cid:0)(cid:2)(cid:0)(cid:3)(cid:4)absorbingset.
(16h)
thenumberofbitnodeswiththreesatisfiedandoneunsatisfied Asimplificationofthelastsystemleads
checknodesiseven,andthus iseven.2First bythemin-
imumdistance, ofthecode,[31].If ,sincewe
have at most five edges going to unsatisfied checks, there are (17a)
twocases:(a)eitherthreeofthemgotooneunsatisfiedcheck
(17b)
andonetoanother,or(b)oneedgegoestoeachofthetwoun-
(17c)
satisfiedchecks.Incase(a),becausethegirthofthefactorgraph
is biggerthan [11],none ofthethree bitnodesthatsharean
We let and
unsatisfied check can share a satisfied check. Further, no two
.Notethatbythecheckconsistencycondition,all
bit nodes can share a satisfied check for the same reason. By
of and arenonzero.
counting, this eliminates case (a). In case (b), if we drop one
Wethenobtain(18)atthebottomofthepage.Thislastsystem
ofthebitnodesthathasanunsatisfiedcheckwewouldhavea
canberewrittenas
absorbingsetwhichwehavearguedinLemma5doesnot
existfor .
Thus,for weareleftwithconsideringthecase
sinceatmostfiveedgesgointounsatisfiedchecks.Thismeans (19)
thecandidateabsorbingsetcontainsonebitnodewithallchecks
satisfied and four bit nodes each with three satisfied and one
unsatisfiedcheck.Theonlywaythatsuchanabsorbingsetcould
Therefore, the determinant of the matrix multiplying the
existisifonehastheconfigurationshowninFig.10,wherethe
(nonzero) vector in (19) is itself zero, which
vertices represent bit nodes and edges represent their satisfied
simplifies to
checknodes.
Since and are all distinct elements of the set
(20)
, by the bit consistency condition, and by the sym-
metryofthecandidateconfigurationinFig.10,wemayassume
Since anddistinct,weconsiderall
assignmentsfor ,andforeachweevaluatetheleft-hand
2Therearefivebitnodesinacandidate(cid:0)(cid:2)(cid:0)(cid:0)(cid:4)absorbingset,somehaving
threeneighboringsatisfiedchecksandoneneighboringunsatisfiedcheck,and sideexpressionin(20).Notethatfordistinct ,
somehavingfourneighboringsatisfiedchecksandzeroneighboringunsatisfied thisexpressionisatmost inabsolutevalue,andthereforethe
(cid:0)
checks.Thenumberofformeristhen(cid:0)andthenumberoflatteris(cid:0)(cid:2) (cid:0)(cid:4).The constraintin(20)doesnothaveasolutionfor fordistinct
numberofedgesemanatingfromthebitsinthisabsorbingsetandendingat
(cid:0) .(Solutionsexistfor and ,which
satisfiedchecksiseven(sincethesechecksaresatisfied).Thus,(cid:5)(cid:0)(cid:6)(cid:3)(cid:0)(cid:2) (cid:0)(cid:4)
iseven.Thatis,(cid:0)(cid:7)(cid:8)(cid:0)(cid:0)(cid:4)isevenand(cid:0)itselfiseven. canbeverifiedbydirectnumericalsubstitution).
(from(17a))
(from(17a))
(from(17b))
(from
andbysubstitutingfrom(17c),(17b),and(17a),respectively). (18)
190 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.56,NO.1,JANUARY2010
For we oneofthebitnodesinthesecondgroup.Thesameistruewith
likewise establish the constraints as in (16) and (17). We theothertwobitnodesinthefirstgroup,namelytheytooshare
again let and their remaining three (satisfied) checks with the bit nodes in
andobtain thesecondgroup,andthesesatisfiedchecksconnecttotwobit
nodesintheabsorbingset.Therefore,thereexisttwobitnodes
inthefirstgroupandtwobitnodesinthesecondgroupsuchthat
(21) anytwoshareadistinctcheck.Thisconfigurationisnotpossible
byLemma5for .
Suppose now that there exists one unsatisfied check of de-
gree with respect to the bit nodes in the absorbing set. The
Sincetheentriesin areallnonzero,itfollows
remaining unsatisfied check then has degree with respect to
that the determinant of the matrix in (21) is zero. Simplifying
thebitnodesintheabsorbingset,andtheneighboringbitnodes
theexpressionforthedeterminantyieldsagaintheconditionin
intheabsorbingsetofthesetwounsatisfiedchecksaredifferent.
(20).Therefore,for , absorbingsetsdonotexist.
Therearetwobitnodesintheabsorbingsetthathaveallchecks
We can now proceed with the analysis of absorbing satisfied. Partitionthe bit nodes inthe absorbing set intothree
sets.Sincethenumberofbitnodeswiththreesatisfiedandone groups:thefirstgroupcontainsthethreebitnodesthatsharea
unsatisfied check node is even, is even. First, is not degree- unsatisfied check, the second group contains the one
possiblesince [31].Thefollowinglemmaaddresses bitnodethathasoneunsatisfiedcheck,andthethirdgroupcon-
thecaseof . tainsthetwobitnodesthathaveallfourcheckssatisfied.Each
ofthethreebitnodesinthefirstgrouphasoneunsatisfiedand
Lemma7: For ,thefactorgraphfamily doesnot
three satisfied checks and thus it shares a satisfied check with
containany absorbingsets.
eachofthebitnodesinthesecondandthirdgroupsinceitcannot
Proof: Wefirstclaimthatthereisnochecknodeofdegree
share a satisfied check with another bit node in the first group
atleast withrespecttothebitnodesintheabsorbingset.Let
by the girth condition [11]. The bit node in the second group
usfirstsupposethatthereexistsonesuchchecknodeandthatit
alsohasoneunsatisfiedandthreesatisfiedchecks,andthelatter
hasanevendegreewithrespecttothebitnodesintheabsorbing
are shared then with the bit nodes in the first group. The two
set.Sinceweareconsideringanabsorbingsetwithsixbitnodes,
bit nodes in the third group have all four checks satisfied, the
suchachecknodewouldhavedegreeeither or withrespect
threeofwhichtheyeachsharewitheachofthebitnodesinthe
tothebitnodesintheabsorbingset.Ifthissatisfiedcheckisof
first group. Since all three satisfied checks of the bit node in
degree , there would exist two bit nodes in the absorbing set
thesecondgroupareusedupwiththechecksitshareswiththe
whichwouldshareanadditionalsatisfiedcheck.Thissituation
bitnodesinthefirstgroup,thetwobitnodesinthethirdgroup
wouldimplytheexistenceofacycleoflength ,whichisim-
share a satisfied check with each other. Therefore, there exist
possiblebythegirthcondition[11].
two bit nodes in the first group and two bit nodes in the third
Supposenowthatthissatisfiedcheckhasdegree .Eachbit
groupsuchthatanytwoshareadistinctcheck.Thisconfigura-
nodethatparticipatesinthischeckhasatleasttwomoreneigh-
tionisnotpossiblebyLemma5for .
boring satisfied checks, which it then necessarily must share
We conclude that no check incident to the bit nodes in the
withtheremainingtwobitnodesintheabsorbingsetthatthem-
absorbing set has degree larger than , namely that all neigh-
selvesdonotparticipateinthisdegree- checkbythegirthcon-
boringsatisfied(respectively,unsatisfied)checkshavedegree
dition[11].Ifthereexistsabitnodethatparticipatesinthisde-
(respectively, ).Byrequiringthateachvertexcorrespondingto
gree- check and has all checks satisfied, it then shares its re-
abitnodeintheabsorbingsethaseitherthreeorfouroutgoing
mainingneighboringcheckwithoneofthebitnodeswithwhich
edges,andthattherearenoparalleledges,itfollowsthatthere
italreadysharesacheck.Thissituationviolatesthegirthcon-
aretwopossibleconfigurations,asshowninFig.11,thatrelate
straint[11].Ifallbitnodesintheabsorbingsetthatparticipate
bitnodesinthe absorbingset(vertices) andtheirshared satis-
in this degree- check have three satisfied and one unsatisfied
fiedchecks(edges).
check,threeofthemwouldhavetoparticipateinthesameun-
ObservethattheconfigurationinFig.11(b)containsa
satisfied check to make the total number of unsatisfied checks
absorbingsetwhichconsistsof and
betwo.Thisagainviolatesthegirthcondition[11].
. By Lemma 5, such configuration is not possible for
Therefore,allsatisfiedcheckswithrespecttothebitnodesin
. The analysis of the configuration in Fig. 11(a) is con-
the absorbing set have degree . Suppose there exists a check
siderablymoreinvolvedanditstechnicaldetailsaredeferredto
nodethatisunsatisfiedwithrespecttothebitsintheabsorbing
AppendixI-A,inwhichwederiveacongruencyconstraintthat
setandthathasdegreebiggerthan .Ifsuchachecknodehas
cannotholdforprime underallpossibleconfiguration
degree ,therewouldnecessarilyexisttwobitnodesintheab-
labelings.Withthatresult,theproofofLemma7iscomplete.
sorbingsetthatsharethisdegree- checkandanothersatisfied
check,whichisimpossiblebythegirthcondition[11]. Havingeliminatedsmallercandidateabsorbingsets,wenow
Supposethatthereexisttwodegree- checksincidenttothe provethefollowingresult.
bit nodes in the absorbing set. First, these degree- checks do
Lemma 8: For all , the factor graph family has
nothaveanyneighboringbitnodesincommonsincewerequire
(fully)absorbingsets.Thesesetsarecompletelycharac-
that each bit node has at most one unsatisfied check. We can
terizedbythesolutionsgiveninTablesIV,V,VI,andVII.
then group the bit nodes in the absorbing set into two disjoint
Proof: We will first show that all satisfied checks neigh-
groups,eachofsize ,suchthatthebitsinthesamegroupshare
boringbitnodesinonesuchabsorbingsetmusthavedegree .
the same degree- check. Consider a bit node in, say, the first
Notethatthere cannotbeadegree- checkwithrespecttothe
group.Itsharesitsremainingthree(satisfied)checkswitheach