11 04 0992-02-000n Irregular Structured Ldpc Codes and Structured Puncturing

8/15/2019 11 04 0992-02-000n Irregular Structured Ldpc Codes and Structured Puncturing

1/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

Irregular Structured LDPC Codes and Structured

Puncturing

Victor Stolpman, Nico van Waes, Tejas Batt,

Carlie !ang, and "mitab Di#it

Tis presentation accompanies submission

I$$$ %&'())*&+-+%


2/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

.vervie/

0 LDPC Introduction

1 2egular versus Irregular ⇒ Irregular codes ave better per3ormance 1 Structured versus 4nstructured ⇒ Structured codes ave better latenc5

0 Irregular Structured LDPC Codes

1 Seed and Spreading 6atrices 1 Building bloc7s 3or structured codes

1 $#panded and $#ponential 6atrices 1 LDPC code construction

0 Simulations

1 BL$2 in "W8N ⇒ Per3ormance improves /it code/ord lengt

1 Conventional BP versus La5ered BP ⇒ La5ered BP o33ers good per3ormance /it 3ast convergence and e33icient silicon solutions

1 Signi3icant per3ormance improvement over te legac5 9$C solution 3or bot small and large pac7et si:es in %&'())n cannels

0 Structured Puncturing


3/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

Lo/*Densit5 Parit5*Cec7 ;LDPC< Codes

0 Wat is a LDPC code= 1 " LDPC code is simpl5 a bloc7 code de3ined b5 a parit5*cec7 matri#

tat as a lo/ densit5 o3 ones ;i(e( mostl5 :eros<

1 Decoding is done iterativel5 using Belie3 Propagation ;BP< 1 passing o3

e#trinsic in3ormation bet/een code/ord elements and parit5 cec7

e>uations

0 W5 do 5ou /ant to use LDPC codes=

1 Best per3orming 3or/ard error correction code available

1 Designs ave approaced capacit5 /itin &(&&+?dB

1 Structured designs o33er te great per3ormance /it 3aster convergence

and attractive silicon solutions

1 9or %&'())n, structured LDPC is a viable and attractive solution /it

signi3icant gains over te legac5 9$C s5stem


4/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

2egular vs( Irregular LDPC Codes

0 2egular LDPC Codes 1 9irst developed in earl5 )-@&As b5 2obert 8allager

1 $ac column o3 te parit5*cec7 matri# as te same number o3 ones

1 $ac ro/ o3 te parit5*cec7 matri# as te same number o3 ones

0 Irregular LDPC Codes

1 Superior per3ormance over regular LDPC constructions

1 .utper3orm Turbo*codes 1 especiall5 at ig code rates

1 Column*/eigt ma5 var5 across columns o3 te parit5*cec7 matri#

1 2o/*/eigt ma5 var5 across ro/s o3 te parit5*cec7 matri# 1 Can be designed 3or particular cannel statistics ;e(g( "W8N, B$C,

2a5leig, etc(<


5/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

4nstructured vs( Structured LDPC Codes

0 4nstructured LDPC Codes 1 2andom Constructions

1 2andoml5 constructed parit5*cec7 matri#

1 No structure to e#ploit in decoding ⇒ limited decoding coices

1 $ac code/ord lengt re>uires anoter construction ⇒ limited bloc7 si:esor ig storage re>uirements 3or multiple code lengts along /it comple#

interconnect

0 Structured LDPC Codes 1 "rcitecture "/are Constructions

1 2eduction o3 E?F or more in memor5 re>uirements

1 .33ers additional decoding coices tat ave 3ast convergence ;e(g( La5ered

Belie3 Propagation< ⇒ ig per3ormance /it lo/ latenc5

1 Supports man5 bloc7 si:es ⇒ reduction in :ero*padding ine33iciencies

1 $33icient decoder designs resulting in ceaper silicon solutions /it lo/er

po/er consumption and sorter interconnects


6/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

.vervie/

0 LDPC Introduction




1 $#panded LDPC and $#ponential 6atrices 1 Constructing a code

0 Simulations





er


7/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

Parit5*Cec7 Seed 6atri#

0 Small binar5 matri# ⇒ lo/ storage costs

0 "cts as a blueprint to te structure o3 te e#panded LDPC code0 Constructed 3rom an edge*distribution /it good as5mptotic

properties 3or te desired cannel ;e(g( "W8N, B$C, 9ading,6I6., etc(<

0 $#panded using permutation matrices ;e(g( circular*si3t matrices<

to construct te LDPC code used 3or 9$C0 "3ter e#pansion, te 3inal LDPC matri# /ill be o3 te same code

ensemble as te seed matri# /it te same as5mptotic per3ormance

=

))&)&&

)&&))&

&))&))

&&)&&)

S$$DH@S$$D = N

'S$$D = K G

)

S$$D

S$$D == N

K R

er


8/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

Permutation Spread 6atrices0 9inite set o3 matrices consisting o3 circular*si3t matrices, te identit5 matri#, and

te all :eros matri#

0 "ct as building bloc7s 3or te e#panded LDPC matri#0 $ac is inde#ed using teir e#ponent values ;i(e si3t*coe33icients<

=

&&)&&

&&&)&

&&&&)

)&&&&

&)&&&

'

SP2$"D

P

=

&)&&&

&&)&&

&&&)&

&&&&)

)&&&&

)

SP2$"D

P

=

)&&&&

&)&&&

&&)&&

&&&)&

&&&&)

&

SP2$"DP

=∞

&&&&&

&&&&&

&&&&&

&&&&&

&&&&&

SP2$"DP

=

&&&)&

&&&&)

)&&&&

&)&&&

&&)&&

G

SP2$"D

P

=

&&&&)

)&&&&

&)&&&

&&)&&

&&&)&

+

SP2$"DP

er


9/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

$#panded LDPC 6atri#

0 In matri# notation, /e /rite

0 $#panded LDPC matri# /ose sub*matrices belong to

0 Tus, te 3inal e#ponents ;i(e( si3t*coe33icients< are o3 te 3inite setH

=

−−− S$$D


10/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

4niversal $#ponential 6atri#

0 $#ponential matri# de3inition used 3or all structured LDPC codes

0 Because it is rule*based and not tied to a particular seed matri#, it

o33ers 3or/ard*compatibilit5 and ard/are reuse 3or di33erent device

classes

0 Supports all code/ord lengts and code rates /itout additional

storage 3or e#ponent values ;i(e( si3t*coe33icients<

∞∞∞

∞=

++−+−

−−−

−−

+−


11/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

9inal $#ponential 6atri#

0 Constructed via mas7ing te seed matri# /it te universal

e#ponent matri# ;NoteH operations can be reduced to just teones locations in te seed parit5*cec7 matri#<

0 We mas7 te seed matri# /it te universal e#ponentialH

( ) ( ) ( )

=

−−− S$$DS$$DS$$DS$$DS$$DS$$DS$$D

S$$D

S$$D

,',),

,'','),'

,)',)),)

N K N K N K N

N

N

F F F

F F F

F F F

F

{ } &,S$$D = jiH

{ } ),S$$D = ji

H

{ } ∞= ji,F

{ } { } ji ji ,$KP.N$NT,

EF =⇒⇒

er


12/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

Small Construction $#ample

=

))&)&&

)&&))&

&))&))

&&)&&)

S$$DH @S$$D =⇒ N

=

&)&

&&)

)&&)

SP2$"DP GSP2$"D

=⇒ N

∞∞

∞=

)%+&

)-@G&

)&%@+'&

@?+G')

$KP.N$NTE

∞∞∞

∞∞∞

∞∞∞∞∞∞

=

)%&

)G&

%@'&+)

F

=

&)&&&)&&&)&&&&&&&&

&&))&&&&&&)&&&&&&&

)&&&)&&&&&&)&&&&&&

&)&&&&&&&)&&)&&&&&

&&)&&&&&&&)&&)&&&&

)&&&&&&&&&&)&&)&&&

&&&&&))&&&&&&&))&&

&&&)&&&)&&&&)&&&)&

&&&&)&&&)&&&&)&&&)

&&&&&&&)&&&&&&&&)&

&&&&&&&&)&&&&&&&&)

&&&&&&)&&&&&&&&)&&

H

))=⇒ p p N ≤+ 'S$$D p N ≤SP2$"D

Parit5 S5stematic

er


13/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

.vervie/

0 LDPC Introduction





0 Simulations





er


14/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

BPS*"W8N Simulations0 Simulated code/ord lengtsH

1 J?E@,E'&, E@%, %@+, -@&, )&&%, ))?', )'-@, )G++, )++&, )?G@, )?%+, )E'%, )%E', )-'&,

'&)@, '))', ')@&, 'G&+

1 Larger code/ord lengts are alread5 supported b5 te speci3ied seed matrices

0 Permutation spreading sub*matri# dimensionsH 1 J)',)?,)@,)%,'&,'),'+,'E,'%,G&,G',GG,G@,G-,+&,+',++,+?,+%

0 2ate )' seed matrices o3 dimension ;'+#+%


15/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

0 0.5 1 1.5 2 2.5 3 3.5 4

10-4

10-3

10-2

10-1

100

BLER R1/2

Eb/No [dB]

B L E R

N= 576, K= 288,( 36 bytes,Ns=12)

N= 720, K= 360,( 45 bytes,Ns=15)

N= 768, K= 384,( 48 bytes,Ns=16)

N= 864, K= 432,( 54 bytes,Ns=18)

N= 960, K= 480,( 60 bytes,Ns=20)N=1008, K= 504,( 63 bytes,Ns=21)

N=1152, K= 576,( 72 bytes,Ns=24)

N=1296, K= 648,( 81 bytes,Ns=27)

N=1344, K= 672,( 84 bytes,Ns=28)

N=1440, K= 720,( 90 bytes,Ns=30)

N=1536, K= 768,( 96 bytes,Ns=32)

N=1584, K= 792,( 99 bytes,Ns=33)

N=1728, K= 864,( 108 bytes,Ns=36)

N=1872, K= 936,( 117 bytes,Ns=39)

N=1920, K= 960,( 120 bytes,Ns=40)

N=2016, K=1008,( 126 bytes,Ns=42)N=2112, K=1056,( 132 bytes,Ns=44)

N=2160, K=1080,( 135 bytes,Ns=45)

N=2304, K=1152,( 144 bytes,Ns=48)

2ate )' BL$2 1 "W8N BPS

er


16/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

1 1.5 2 2.5 3 3.5 4

10-4

10-3

10-2

10-1

100

BLER R2/3

Eb/No [dB]

B L E R

N= 576, K= 384,( 48 bytes,Ns=12)

N= 720, K= 480,( 60 bytes,Ns=15)

N= 768, K= 512,( 64 bytes,Ns=16)

N= 864, K= 576,( 72 bytes,Ns=18)

N= 960, K= 640,( 80 bytes,Ns=20)N=1008, K= 672,( 84 bytes,Ns=21)

N=1152, K= 768,( 96 bytes,Ns=24)

N=1296, K= 864,( 108 bytes,Ns=27)

N=1344, K= 896,( 112 bytes,Ns=28)

N=1440, K= 960,( 120 bytes,Ns=30)

N=1536, K=1024,( 128 bytes,Ns=32)

N=1584, K=1056,( 132 bytes,Ns=33)

N=1728, K=1152,( 144 bytes,Ns=36)

N=1872, K=1248,( 156 bytes,Ns=39)

N=1920, K=1280,( 160 bytes,Ns=40)


N=2160, K=1440,( 180 bytes,Ns=45)

N=2304, K=1536,( 192 bytes,Ns=48)

2ate 'G BL$2 1 "W8N BPS

er


17/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

1.5 2 2.5 3 3.5 4 4.5 5

10-4

10-3

10

-2

10-1

100

BLER R3/4

Eb/No [dB]

B L E R

N= 576, K= 432,( 54 bytes,Ns=12)

N= 720, K= 540,( 67.5 bytes,Ns=15)

N= 768, K= 576,( 72 bytes,Ns=16)

N= 864, K= 648,( 81 bytes,Ns=18)

N= 960, K= 720,( 90 bytes,Ns=20)N=1008, K= 756,( 94.5 bytes,Ns=21)

N=1152, K= 864,( 108 bytes,Ns=24)

N=1296, K= 972,(121.5 bytes,Ns=27)

N=1344, K=1008,( 126 bytes,Ns=28)

N=1440, K=1080,( 135 bytes,Ns=30)

N=1536, K=1152,( 144 bytes,Ns=32)

N=1584, K=1188,(148.5 bytes,Ns=33)

N=1728, K=1296,( 162 bytes,Ns=36)

N=1872, K=1404,(175.5 bytes,Ns=39)

N=1920, K=1440,( 180 bytes,Ns=40)


N=2160, K=1620,(202.5 bytes,Ns=45)

N=2304, K=1728,( 216 bytes,Ns=48)

2ate G+ BL$2 1 "W8N BPS

er,


18/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

La5ered Belie3 Propagation

=

&)&&&)&&&)&&&&&&&&

&&))&&&&&&)&&&&&&&

)&&&)&&&&&&)&&&&&&

&)&&&&&&&)&&)&&&&&

&&)&&&&&&&)&&)&&&&

)&&&&&&&&&&)&&)&&&

&&&&&))&&&&&&&))&&

&&&)&&&)&&&&)&&&)&

&&&&)&&&)&&&&)&&&)

&&&&&&&)&&&&&&&&)&

&&&&&&&&)&&&&&&&&)

&&&&&&)&&&&&&&&)&&

H

0 Parit5*cec7 matri# is partitioned into la5ers and messages are passed bet/een

0 Speeds convergence time signi3icantl5 ⇒ Mig per3ormance /it lo/ latenc5

0 Signi3icant reduction in memor5 re>uirements ;E?F reduction<

0 6ost structured LDPC codes can implement la5ered*BP in cost e33ective solutions

er,


19/31

er,2004

Victor

doc.: IEEE 802.11-04/992

Submission

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.810-4

10-3

10-2

10-1

100

Eb/N0 (dB)

B L E R

Comparing Conventional and Layered Belief Propagation, AWGN, BPSK, N=1152

Nokia, Conventional BP

Nokia, Layered BPTI, Conventional BP

TI, Layered BP

Conventional BP : 50 Iterations

Layered BP : 15 Iterations

La5ered vs( Conventional BP ;2ate )'<

La5ered BP;)? iterations<

Conventional BP

;?& iterations<

er,


20/31

e ,2004

Victor

doc.: IEEE 802.11-04/992

Submission

Structured LDPC, N )-'&, Di33erent Code*2ates

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.210-3

10-2

10-1

100

Eb/N0 (dB)

B L E

R

Layered Belief Propagation, 12-iterations, AWGN, BPSK, N=1920

R-1/2

R-2/3R-3/4

er,2004


21/31

,2004

Victor

doc.: IEEE 802.11-04/992

Submission

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.810-4

10-3

10-2

10-1

100

Eb/N0 (dB)

B L E R

Comparing Conventional and Layered Belief Propagation, AWGN, BPSK, N=1920, R-1/2

Conventional BP, 12-iter

Layered BP, 12-iter

Layered BP, 8-iterParallel Layered BP, 12-iter

Structured LDPC, N )-'&, 2ate )'

er,2004


22/31

,2004

Victor

doc.: IEEE 802.11-04/992

Submission

%&'())n Cannel Simulations

0 Cannel B 1 '#' 6I6. /it ' spatial streams in '&6M:

1 G& iterations o3 conventional belie3 propagation ;i(e( SP"<

1 Large pac7et si:es using concatenated code/ords o3 lengt 'G&+

0 Cannel D

1 )#) SIS. in '&6M:

1 '& iterations o3 conventional belie3 propagation ;i(e( SP"<

1 Small pac7et si:es using a single code/ord o3 lengt 'G&+

0 Cannel $

1 '#' 6I6. /it ' spatial streams in '&6M: 1 G& iterations o3 conventional belie3 propagation ;i(e( SP"<

1 Large pac7et si:es using concatenated code/ords o3 lengt 'G&+

er,2004


23/31

2004

Victor

doc.: IEEE 802.11-04/992

Submission

Cannel B '#' Simulation 2esults

.ver GdBgain in

'#'

2 4 6 8 10 12 14 16 18 20 22

10-2

10-1

100

SNR [dB]

P E R

Packet Error Rate for LDPC Codes in ChB 2x2 using N=2304

CC .11a ChB R1/2 2x2 4-QAM 24Mbps (1008 bytes)

LDPC(30-SPA) ChB R1/2 2x2 4-QAM 24Mbps (1008 bytes)

CC .11a ChB R3/4 2x2 16-QAM 72Mbps (1080 bytes)

LDPC(30-SPA) ChB R3/4 2x2 16-QAM 72Mbps (1080 bytes)CC .11a ChB R2/3 2x2 64-QAM 96Mbps (960 bytes)

LDPC(30-SPA) ChB R2/3 2x2 64-QAM 96Mbps (960 bytes)

er,2004


24/31

2004

Victor

doc.: IEEE 802.11-04/992

Submission

Cannel D )#) Simulation 2esults

5 10 15 20 25

10-3

10-2

10-1

100

SNR [dB]

P E R

Packet Error Rate for LDPC Codes in ChD 1x1 using N=2304

CC .11a ChD R1/2 1x1 4-QAM 12Mbps (144 bytes)

LDPC(20-SPA) ChD R1/2 1x1 4-QAM 12Mbps (144 bytes)

CC .11a ChD R3/4 1x1 16-QAM 36Mbps (216 bytes)

LDPC(20-SPA) ChD R3/4 1x1 16-QAM 36Mbps (216 bytes)CC .11a ChD R2/3 1x1 64-QAM 48Mbps (192 bytes)

LDPC(20-SPA) ChD R2/3 1x1 64-QAM 48Mbps (192 bytes)

O'dB 8ain

in )#)

er,2004


25/31

2004

Victor

doc.: IEEE 802.11-04/992

Submission

Cannel $ '#' Simulation 2esults

3 4 5 6 7 8 9 10

10-2

10-1

100

SNR [dB]

P E R

Packet Error Rate for LDPC Codes in ChE 2x2 using N=2304

CC .11a ChE R1/2 2x2 4-QAM 24Mbps (1008 bytes)

LDPC(30-SPA) ChE R1/2 2x2 4-QAM 24Mbps (1008 bytes)

.ver GdBgain in

'#'

er,2004


26/31

2004

Victor

doc.: IEEE 802.11-04/992

Submission

9eatures

0 9or/ard compatibilit5 and ard/are reuse

1 $#isting seed sets alread5 support longer code/ord lengts

1 "dditional seed are easil5 added 3or di33erent cannel models, additionalcode rates, and to accommodate tradeo33s in silicon

0 "rcitecture "/are constructions tat allo/ 3or La5ered*BP 1 9ast convergence ⇒ ig per3ormance and lo/ latenc5

1 $33icient silicon solutions

0 Wide range o3 bloc7 si:es reduces :ero*padding ine33iciencies

0 4pper triangular seed matrices ⇒ linear time encoding

0 In te pipeline 1 Seed matrices 3or additional code rates ?@ and E%

1 "dditional seed si:es 3or di33erent number o3 data sub*carriers ;e(g+&6M: cannel bonding<

er,2004


27/31

2004

Victor

doc.: IEEE 802.11-04/992

Submission

.vervie/

0 LDPC Introduction





0 Simulations



1 Signi3icant per3ormance improvement over te legac5 9$C solution 3or

bot small and large pac7et si:es in %&'())n cannels


er,2004


28/31

2004

Victor

doc.: IEEE 802.11-04/992

Submission

Structured Puncturing o3 LDPC Codes

04sed to o33er all possible code rates in bet/een and above te basic code rate set J)','G,G+,E%

0 Puncturing does not re>uire canging te parit5*cec7

connective net at eiter te encoder or decoder

0 Supports eas5 lin7 adaptation( In 6I6. applications,

puncturing allo/s 3or di33erent spatial streams to ave di33erent

code rates /itout using multiple coding bloc7s

0 "pproac can be reused in M5brid*"2Q s5stems

0 Structured approac reduces storage re>uirements and e#pands

easil5 to multiple bloc7 lengts

er,2004 d IEEE 802 11 04/992


29/31

2004

Victor

doc.: IEEE 802.11-04/992

Submission

0 1 2 3 4 5 6 710-4

10-3

10-2

10-1

100

sim punc output N624 M312 53Nokia12 50iters.mat

B L E R

Eb/No dB

Rate 0.500, N= 624, K= 312Rate 0.525, N= 594, K= 312

Rate 0.553, N= 564, K= 312

Rate 0.584, N= 534, K= 312

Rate 0.619, N= 504, K= 312

Rate 0.658, N= 474, K= 312

Rate 0.703, N= 444, K= 312

Rate 0.754, N= 414, K= 312

Rate 0.813, N= 384, K= 312

Rate 0.884, N= 353, K= 312

2ate )' Puncture $#am ple;6oter Code, N@'+<

er,2004 d IEEE 802 11 04/992


30/31

2004

Victor

doc.: IEEE 802.11-04/992

Submission

Summar5

0 Irregular Structured LDPC codes ave great per3ormance

0 .33ers 3or/ard*compatibilit5 and ard/are reuse

0 "lread5 supports code/ord lengts greater tan 'G&+

0 "rcitecture "/are constructions ⇒ La5ered*BP decoding

0 $33icient silicon solutions /it ig trougput and lo/ latenc5

0 Wide range o3 bloc7 si:es reduces :ero*padding ine33iciencies

0 4pper triangular seed matrices ⇒ linear time encoding

0 Structured puncturing allo/s 3or additional code rates 3or use

/it spatial stream adaptation in 6I6. s5stems

er,2004 doc : IEEE 802 11 04/992


31/31

2004

Victor

doc.: IEEE 802.11-04/992

Submission

2e3erences

)( T( R( 2icardson, 6( "( So7rollai, and 2( L( 4rban7e, Design o3 Capacit5"pproacing Irregular Lo/*Densit5 Parit5*Cec7 Codes, IEEE Transactions on Information Theory, vol( +E, pp( @)-*@GE, 9eb( '&&)(

'( Sae*oung Cung, On the Construction of Some Capacity-Approaching CodingSchemes, PD Dissertation, 6IT, '&&&(

G( R( Mou, P( Siegel, and L 6ilstein, Per3ormance "nal5sis and Code .ptimi:ation o3Lo/ Densit5 Parit5*Cec7 Codes on 2a5leig 9ading Cannels, IEEE ! Se"ect! Areas Commun!# Issue on The Tur$o %rincip"e& From Theory to %ractice I , vol( )-,

no( ?, pp( -'+*-G+, 6a5 '&&)(+( 6( 6( 6asour and N( 2( Sanbag, Turbo decoder arcitectures 3or lo/*densit5

parit5 cec7 codes, IEEE '"o$a" Comm! Conf! (')O*ECO+,, Nov( '&&', pp()G%G*)G%%(

?( 6( 6( 6ansour and N( 2( Sanbag, Lo/ po/er VLSI arcitectures 3or LDPCcodes, in '&&' International Lo/ Po/er $lectronics and Design, '&&', pp( '%+*'%-(

@( D( $( Mocevar, LDPC code construction /it 3le#ible ard/are implementation, %roc!& IEEE Int" Conf! On Comm! (ICC,, "ncorage, ", 6a5 '&&G(

E( 6( 6( 6ansour and N( 2( Sanbag, Mig*Trougput LDPC Decoders, I$$$Trans( .n VLSI S5stems, vol( )), No( @, pp( -E@*--@, December '&&G(

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11 04 0992-02-000n Irregular Structured Ldpc Codes and Structured Puncturing

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