Status of Knowledge on Non-Binary LDPC Decoders Part III ...€¦ · Slightly Irregular LDPC codes...

Status of Knowledge on Non-Binary LDPC DecodersPart III: Why Non-Binary Codes/Decoders ?

D. Declercq 1

1ETIS - UMR8051ENSEA/Cergy-University/CNRS

France

IEEE SSC SCV Tutorial, Santa Clara, October 21st, 2010

D. Declercq (ETIS - UMR8051) 1 / 41

Outline

1 Advantages of Non-Binary LDPC Codes

2 Coding for high Spectral Efficiency

3 Coding for Very Low Rates

4 Decoding General Linear Block Codes

5 Concept of NB-decoder Diversity

6 Conclusion


Outline






6 Conclusion


Increased Interest from the Scientific Community since 2004

Theoretical Aspects:[Hu,2004] X.Y. HU AND E. ELEFTHERIOU, “BINARY REPRESENTATION OF CYCLE TANNER-GRAPH GF(2q ) CODES”,

IEEE Int. Conf. on Commun., PARIS, FRANCE, JUNE 2004

[Rathi,2006] V. RATHI AND R.L. URBANKE, “DENSITY EVOLUTION THRESHOLDS AND THE STABILITY CONDITIONS FOR

NON-BINARY LDPC CODES”, IEEE Transactions on communication, VOL. 152(6), DEC. 2005

[Bennatan,2006] A. BENNATAN AND D. BURSHTEIN, “DESIGN AND ANALYSIS OF NON-BINARY LDPC CODES FOR

ARBITRARY DISCRETE MEMORYLESS CHANNELS”, IEEE Transactions on Information Theory, VOL. 52, PP. 549–583,

FEB. 2006.

[Li,2003] G. LI, I. FAIR AND W. KRZYMIEN, “ANALYSIS OF NONBINARY LDPC CODES USING GAUSSIAN APPROXIMATION”,

Proceedings of ISIT, KANAGAWA, JAPAN, JULY 2003.

[Byers,2005] G. BYERS AND F. TAKAWIRA, “EXIT CHARTS FOR NON-BINARY LDPC CODES”, Proceedings of ICC,

SEOUL, KOREA, MAY 2005.

[Sassatelli,2010] L. SASSATELLI AND D. DECLERCQ, “NON-BINARY HYBRID LDPC CODES”, IEEE Transactions on

Information Theory, VOL. 56(10), PP 5314-5334, OCT. 2010

[Kasai,2008] K. KASAI, C. POULLIAT, D. DECLERCQ, T. SHIBUYA AND K. SAKANIWA, “WEIGHT DISTRIBUTION OF

NON-BINARY LDPC CODES”, in the proc. of ISITA’08, AUCKLAND, NEW ZEALAND, DEC. 2008



Theoretical Aspects:[Goupil,2007] A. GOUPIL, M. COLAS, G. GELLE AND D. DECLERCQ, “FFT-BASED DECODING OF GENERAL LDPC

CODES OVER ABELIAN GROUPS”, IEEE Transactions on communication, VOL. 55(4), APRIL 2007.

[Declercq,2011] D. DECLERCQ, “NON-BINARY DECODER DIVERSITY FOR DENSE OR LOCALLY-DENSE PARITY-CHECK

CODES”, to appear in IEEE Transactions on communication, 2011

Practical Finite Length Design:[Hu,2004] X.Y. HU AND E. ELEFTHERIOU, “BINARY REPRESENTATION OF CYCLE TANNER-GRAPH GF(2q ) CODES”,

IEEE Int. Conf. on Commun., PARIS, FRANCE, JUNE 2004

[Song,2006] S. SONG, L. ZENG AND K. ABDEL-GHAFFAR, “ALGEBRIC CONSTRUCTION OF NON-BINARY QUASI-CYCLIC

LDPC CODES”, Proceedings of ISIT, SEATTLE, USA, JULY 2006.

[Zeng,2008] L. ZENG, L. LAN, Y. YU TAI, B. ZHOU, SHU LIN, A. KHALED AND A.S. ABDEL-GHAFFAR, “CONSTRUCTION

OF NONBINARY CYCLIC, QUASI-CYCLIC AND REGULAR LDPC CODES: A FINITE GEOMETRY APPROACH”, IEEE

transactions on Communications, VOL. 56, PP. 1788-1793, MARCH 2008.

[Venkiah,2008] A. VENKIAH, D. DECLERCQ AND C. POULLIAT, “DESIGN OF CAGES WITH A RANDOMIZED PROGRESSIVE

EDGE GROWTH ALGORITHM”, IEEE Communication letters, VOL. 12, PP. 301-303, APRIL, 2008.

[Poulliat,2008] C. POULLIAT, M. FOSSORIER AND D. DECLERCQ, “DESIGN OF REGULAR (2,DC)-LDPC CODES OVER

GF(Q) USING THEIR BINARY IMAGES”, IEEE Transactions on communication, VOL. 56(10), PP. 1626 - 1635, OCT. 2008



Advantages for High Rates Application:[Ma,2006] L. MA, L. WANG AND J. ZHANG, “PERFORMANCE ADVANTAGE OF NON-BINARY LDPC CODES AT HIGH CODE

RATE UNDER AWGN CHANNEL”, Proceedings of ICCT’06, GUILIN, CHINA, NOV. 2006.

[Zeng,2008] L. ZENG, L. LAN, Y. YU TAI, B. ZHOU, SHU LIN, A. KHALED AND A.S. ABDEL-GHAFFAR, “CONSTRUCTION

OF NONBINARY CYCLIC, QUASI-CYCLIC AND REGULAR LDPC CODES: A FINITE GEOMETRY APPROACH”, IEEE

transactions on Communications, VOL. 56, PP. 1788-1793, MARCH 2008.

Advantages for Channels with Memory/Burst Error Correction:[Morinoni,2008] A. MORINONI, P. SAVAZZI AND S. VALLE, “EFFICIENT DESIGN OF NON-BINARY LDPC CODES FOR

MAGNETIC RECORDING CHANNELS, ROBUST TO ERROR BURSTS”, Proceedings of IEEE ISTC, LAUSANNE,

SWITZERLAND, SEPT. 2008.

[Chen,2005] J. CHEN, L. WANG AND Y. LI, “PERFORMANCE COMPARISON BETWEEN NON-BINARY LDPC CODES AND

REED-SOLOMON CODES OVER NOISE BURST CHANNELS”, Proceedings of IEEE ICCS’05, CHINA, MAY 2005.

Advantages for Low Rates Applications:[Sassatelli,2008] L. SASSATELLI, D. DECLERCQ AND C. POULLIAT, “LOW-RATE NONBINARY HYBRID LDPC CODES”, in

the proc. of IEEE Turbo-Coding Symposium, LAUSANNE, SWITZERLAND, SEPT. 2008

[Kasai,2010] K. KASAI, D. DECLERCQ, C. POULLIAT, K. SAKANIWA, “RATE-COMPATIBLE NON-BINARY LDPC CODES

CONCATENATED WITH MULTIPLICATIVE REPETITION CODES”, in the proc. of ISIT’10, AUSTIN, TEXAS, USA, JUNE 2010.



Advantages for High Throughput Vector Channels:[Declercq,2004] D. DECLERCQ, M. COLAS AND G. GELLE, “REGULAR GF (2q ) LDPC CODED MODULATIONS FOR

HIGHER ORDER QAM-AWGN CHANNELS”, Proceedings of IEEE ISITA, PARMA, ITALY, OCT. 2004.

[Peng,2006] R.-H. PENG AND R.-R. CHEN, “DESIGN OF NON-BINARY LDPC CODES OVER GF (q) FOR

MULTIPLE-ANTENNA TRANSMISSION”, Proceedings of MILCOM, WASHINGTON DC, USA, OCT. 2006.

[Jiand,2009] X. JIAND, Y. YAN, X.-G. XIA AND M.H. LEE, “APPLICATION OF NON-BINARY LDPC CODES BASED ON

EUCLIDEAN GEOMETRIES TO MIMO SYSTEMS”, Proceedings of Intern. conference on wireless comm. and Signal

processing, NANJING, CHINA, NOV. 2009.

[Byers,2004] G.J. BYERS AND F.TAKAWIRA, “NON-BINARY AND CONCATENATED LDPC CODES FOR MULTIPLE-ANTENNA

SYSTEMS”, Proceedings of AFRICON, GABORONE, BOTSWANA, SEPT. 2004.

[Pfletschinger,2010] S. PFLETSCHINGER AND D. DECLERCQ, “GETTING CLOSER TO MIMO CAPACITY WITH

NON-BINARY CODES AND SPATIAL MULTIPLEXING”, in the proc. of Globecom’10, MIAMI, FLORIDA, USA, DECEMBER

2010.


Different Research Directions

Small Order Fields

Slightly Irregular LDPC codes in small order Fields GF (8) or GF (16).

for such low Field order, the issue of the decoder complexity is less crucial, and alldecoders mentioned in this presentation work nicely.

(FFT-logBP, EMS small nm, Min-Max, etc)

Large Order Fields

Stricly regular dv = 2 LDPC codes in high order fields GF (q), q ≥ 64.

Very Large Order Fields

Is there a need for decoders of NB-LDPC codes in GF (4096) or higher ?


Why ultra-sparse dv = 2 Tanner Graphs

Amazing girths can be obtained for small to moderate codeword lengths

dc = 3 dc = 4 dc = 6 dc = 8

Ns = 500 g=22 g=18 g=14 g=8

Ns = 3000 g=28 g=20 g=16 g=12

Number of independant decoding iterations = bg/4c. If g ↑, message passing decoder iscloser to MLD,

[Hu,2004] X.Y. HU AND E. ELEFTHERIOU, “BINARY REPRESENTATION OF CYCLE TANNER-GRAPH GF(2q ) CODES”, IEEE Int.

Conf. on Commun., PARIS, FRANCE, JUNE 2004

[Venkiah,2008] A. VENKIAH, D. DECLERCQ AND C. POULLIAT, “DESIGN OF CAGES WITH A RANDOMIZED PROGRESSIVE EDGE

GROWTH ALGORITHM”, IEEE Communication letters, VOL. 12, PP. 301-303, APRIL, 2008.


Link between dv = 2 NB-LDPC protograph and Parallel Turbo-Codes⇒ reconciliate Turbo-Codes and LDPC codes

Π

r 1

r 2

i

1 1

1 1

1 1

1 1

11

1 1

1 1

1 1

1 1

11

1

1

1

1

1

Π

0

0

Formally, a (dv = 2, dc = 3) NB-LDPC code with protograph description is equivalent to aparallel Turbo-Code, with

• Bloc codes instead of Conv. codes• Symbol-wise interleaver Π

Symbols

InformationNon Binary

Parity Check

Non Binary

Parity Check

i

r

r2

1

⇒ Same kind of limitations : Dmin ∼ log2(N) log2(q)


Non-Binary dv = 2 LDPC Codes for Short Block-Lengths

0 0.5 1 1.5 2 2.5 3 3.5 4 4.510

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N

0 (in dB)

Fra

me

Err

or

Ra

te

Performance Comparison, K=1024 information bits

JPL codesOptimized GF(256) codes

R=1/2

R=4/5

R=2/3

Is it Really Worth the Additionnal Complexity ?

[Poulliat,2008] C. POULLIAT, M. FOSSORIER AND D. DECLERCQ, “DESIGN OF REGULAR (2,DC)-LDPC CODES OVER GF(Q)

USING THEIR BINARY IMAGES”, IEEE Transactions on communication, VOL. 56(10), PP. 1626 - 1635, OCT. 2008


Outline






6 Conclusion


Non-binary Codes for Multiple AntennasTwo Possible Schemes

(a) q-ary information processing: plug the non-binary coded symbols directly on themodulated-MIMO channel in a Coded Modulation (CM),

(b) binary information processing: classical scheme using Bit-Interleaved Codedmodulation (BICM)

Assumptions: spatial multiplexing (no Space Time code), uncorrelated i.i.d. Rayleighfading, no CSIT, perfect CSIR.


Capacities associated with the Two schemes

−10 −5 0 5 10 15 200

1

2

3

4

5

6

7

8

SNR [dB]

Cap

acity

[bits

per

cha

nnel

use

]2 × 2 MIMO, i.i.d. Rayleigh fading

Gaussian input8−QAM, CM8−QAM, BICM8−PSK, BICM16−QAM, CM16−QAM, BICM 4 5 6 7

2.6

2.8

3

3.28−

QAM, CM

8−PSK, BIC

M

8−QAM, BIC

M1.35 dB

Zoom: 8−QAM, 8−PSK

6 7 8 93.6

3.8

4

4.2

4.4

16−QAM

, CM

16−QAM, BIC

M1.56 dB

Zoom: 16−QAM


Why NB-Codes act as natural Space-Time codes ?

The key is to assign one and only one code symbol to each channel use nT log2(M) = log2(q)

As a result of this constraint, the L-values at the input of the decoder:- are uncorrelated from one code-symbol to another: we do not loose information due

to correlation at the demapper output.- are sufficient statistics with respect to the vector-channel (QAM+MIMO), so the code

aims at reaching the vector-channel capacity.

The following cases have been tested- GF(64) LDPC code, R = 1/2, nT = 2 antennas, 8-QAM/8-PSK,- GF(256) LDPC code, R = 1/2, nT = 2 antennas, 16-QAM.

Advantage at the receiver:

Mapping of one code symbol to one channel use leads to very simple APPdemapper (no need for reduced-complexity MIMO detector i.e. sphere decoder,MMSE, etc.)

[Pfletschinger,2010] S. PFLETSCHINGER AND D. DECLERCQ, “GETTING CLOSER TO MIMO CAPACITY WITH NON-BINARY

CODES AND SPATIAL MULTIPLEXING”, in the proc. of Globecom’10, MIAMI, FLORIDA, USA, DECEMBER 2010.


Finite Length Simulation Results (1)

3.5 4 4.5 5 5.5 6 6.5 7

10−4

10−3

10−2

10−1

100

CM

Shannon lim

it

BIC

M S

hannon limit

limit for Alamouti scheme

8−QAM, q = 64 8−PSK, binary

SNR = nT E

S / N

0 [dB]

BE

R

2 × 2 MIMO, 8−QAM/PSK


Finite Length Simulation Results (2)

6 6.5 7 7.5 8 8.5 9 9.5 10

10−5

10−4

10−3

10−2

10−1

100

CM

Shannon lim

it

BICM Shannon limit

limit for A

lamouti schem

eq = 256

binary

SNR = nT E

S / N

0 [dB]

BE

R

2 × 2 MIMO, 16−QAM


Outline






6 Conclusion


Existing Solutions for Low Rates codes

Good solutions at N = +∞

LDPC-Hadamard codes (special kind of Generalized Low Density codes),

Binary multi-edges types (MET-LDPC) or Protographs,

Both solutions have decoding threshold between 0.2dB and 0.4dB from the ShannonLimits.

Nothing works at small to moderate lengths

LDPC-Hadamard: NO results reported in litterature for lengths smaller than K = 104 bits,

Binary MET-LDPC codes / Protographs: NO results reported for rates below R = 1/6,

The only existing solution at small length is the Turbo-Hadamard or Zigzag-Hadamard, butsuffer for very high error floors ' 10−4.

[Kasai,2010] K. KASAI, D. DECLERCQ, C. POULLIAT, K. SAKANIWA, “RATE-COMPATIBLE NON-BINARY LDPC CODES

CONCATENATED WITH MULTIPLICATIVE REPETITION CODES”, in the proc. of ISIT’10, AUSTIN, TEXAS, USA, JUNE 2010.


Proposed scheme based on NB repetition nodesStart from an Ultra-Sparse R = 1/3 NB-LDPC Code

Optimized Interleaver for Very High Girth

h hhh hh h h hhh h1 11 2 22 3 3 313 2


Proposed scheme based on NB repetition nodesRepeat all symbols with extra non-zero values


h hhh hh h h hhh h

h h h h h4 h4

1 11 2 22 3 3 313 2

4 4 4 4


Proposed scheme based on NB repetition nodesContinue repetition ...


h hhh hh h h hhh h

h4 h4 h4 h4 h4

h5 h6 h5 h5 h5 h5 h5h6 h6 h6 h6 h6

h4

1 11 2 22 3 3 313 2


Performance ResultsGF(4096) LDPC code vs. MET/Protograph Binary code

−1 −0.5 0 0.5 1 1.510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

(Eb/N

0) in dB

Fra

me

Err

or

Ra

te

Multi−Edge type LDPC, K=1024 bits, R=1/6

GF(4096), K=1008 bits, R=1/6


Performance ResultsGF(4096) LDPC code vs. GF(256) LDPC codes, K = 1008 info. bits

−1.5 −1 −0.5 0 0.5 110

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

(Eb/N

0) in dB

Fra

me

Err

or

Ra

te

GF(256) Codes

GF(4096) Codes

SP59 Lower Bound: R=1/18

R=1/3

R=1/6R=1/18


Outline






6 Conclusion


Binary Representation of Non-Binary CodesBinary Maps and Companion Matrices

hji ∈ GF (q) ci ∈ GF (q) hji .ci ∈ GF (q)

Let us replace hji with a binary matrix representation A(hji ):

0

1

00

1 1 11 1

11

110 00

1

11

0 1

1

0

0

Companion matrix of hji

Binary map of ci

Binary map of hji*ci


Binary Representation of Non-Binary Codesnon-Binary codes are ”locally dense” codes

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

18

20

0 50 100 150 200 250 3000

20

40

60

80

100

120

140

160

0 50 100 150 200 250 3000

20

40

60

80

100

120

140

160

Non−Binary Parity Check Matrix in GF(256)

Binary Representation of the Parity Check Matrix

Random Permutation of Rows and Columns of H


Background

Let start with Hb , a binary (Mb × Nb) parity check matrix of a dense block code,

One can obtain a non-binary Tanner graph of Hb by bit-clustering of order p,- by combining p adjacent columns to form a non-binary symbol of order q = 2p ,- by combining p adjacent rows to form a NB-parity check node in the group Fp

2

the obtained Tanner graph has M = Mb/p generalized parity checks and N = Nb/psymbol nodes,

if Mb and Nb are not integer multiples of p:→ complete with all-zeros columns (pruned bits),→ complete with redundant rows.

a decoding instance is obtained by several iterations of generalized BP decoder on thenon-binary Tanner graph [Goupil,2007],

[Goupil,2007] A. GOUPIL, M. COLAS, G. GELLE AND D. DECLERCQ, “FFT-BASED DECODING OF GENERAL LDPC CODES OVER

ABELIAN GROUPS”, IEEE Transactions on communication, VOL. 55(4), APRIL 2007.


Generalized parity-Check Equations

Generalized Parity Check Equation

dcXj=1

hij (xj ) = 0 in F(2p)

where non-zero values are now function hij (.) : F(2p)→ F(2p)

Or in Vector Form

dcXj=1

Hij .x j = 0p in (F2)p

Hij is a binary matrix (p × p) called Binary Cluster


Functions Associated to Binary Clusters

0000000100100011010001010110

011110001001101010111100110111101111

0000000100100011010001010110

011110001001101010111100110111101111

1 0 1 01 1 1 01 0 0 11 1 0 1

Hij=(b)

b jb i

Nonzero Cluster Associated function Nonzero Cluster Associated function

Full Rank Case

Hij=

1 1 0 01 0 0 1

0 1 1 11 0 1 0 (a)

b jb i

Rank Deficient Case


Decoding ResultsCase of the extended quadratic-residue linear code (N = 48, K = 24, Dmin = 12)

0 1 2 3 4 5 6 7 8 9 1010

−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

(Eb/N

0) in dB

Fra

me E

rror

Rate

Binary Min−SumNon−Binary F

16

Non−Binary F256

Non−Binary F4096

Lower Bound


Outline






6 Conclusion


A small example (1)Diversity of Tanner Graph Representations

Good and Bad Tanner Graphs after clustering from the same Hb :

0

1

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

1

1

1

0

1

1

1

1

1

0

0

0

0

0

1 1

0

0

0

1

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

1

1

1

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

0

1

1

1

1

1

0

0

1

1

1

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

1

0

0

0

1

1

0

1

0

0

0

1

etc ...

More

Sparse !

0

0

0

1

1

1

Good and Bad ?


A small example (2)Diversity of Decoding behaviors

test case: 10000 noise vectors at Channel Error Probability α = 0.1

5019 cases where both decoders converge to the right codeword,

384 cases where both decoders fail to converge after 50 iterations,

589 cases where both decoders converge to a wrong codeword,

1213 cases where one decoder fails to converge, and the other one converges to the rightcodeword,

1041 cases where one decoder fails to converge, and the other one converges to a wrongcodeword,

1754 cases where one decoder converges to a wrong codeword, and the other oneconverges to the right codeword.

Good cases Bad cases


Capitalizing on the Decoding DiversityDiversity Sets

Definition of Diversity Set: set of d distinct non-binary Tanner graphs of the same code:nG(1)

p1, . . . ,G(i)

pi, . . . ,G(d)

pd

o

1 H(i)b = P(i) (Hb) = A(i).Hb.Π

(i) is called pre-processing,2 G(i)

piis the Tanner graph obtained by clustering H(i)

b with an order pi .

[Declercq,2011] D. DECLERCQ, “NON-BINARY DECODER DIVERSITY FOR DENSE OR LOCALLY-DENSE PARITY-CHECK CODES”,

to appear in IEEE Transactions on communication, 2011


Capitalizing on the Decoding DiversityMerging Strategies

• Serial merging: Use the decoders sequentially, i.e. switch to another decoder only whenthe current decoder failed to converge to a codeword,⇒O(Serial) = (1 + ε)×O(p.2p.N.Nit )

• Parallel merging: Run all d decoders in parallel and choose appropriately an estimatedcodeword (majority voting, maximum likelihood, etc)⇒O(Parallel) = d ×O(p.2p.N.Nit )

• Other merging strategies: O(Serial) ≤ Comp ≤ O(Parallel),

• Lower bound: Is there at least 1 graph among the d candidates such that the decoderconverges to the right codeword ? (we do not address the PB of finding the good oneamong the d)


BCH codes on the BSC ChannelDiversity set and parameters

NO information about Error Location is available

Constant clustering order in the diversity sets p,

Considered Preprocessing:

1 Build Hb from low weight codewords of the BCH dual code⇒ Hb is as sparse as possible,

2 consider random row permutations and random column permutations:

H(i)b = Π

(i)1 .Hb.Π

(i)2


BCH codes on the BSC ChannelBCH(N = 127, K = 71, Dmin = 19)

3 3.5 4 4.5 5 5.5 6 6.5 710

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N

0 in dB

Fra

me

Err

or

Ra

te

BCH (N=127,K=71,T=9) over the BSC channel

GDD (p=8,d=1)

GDD (p=8,d=200) − lower bound

GDD (p=8,d=2000) − lower bound

Bounded Distance Decoding


Turbo-Codes from the DVB-RCS standardSimulation results.

Diversity decoder performance for a (R = 0.5,N = 848) duobinary TC

0.5 1 1.5 2 2.5 3 3.510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N

0 in dB

Fra

me

Err

or

Ra

teTurbo−decoder5 group decoders: serial merging5 group decoders: lower bound


Outline






6 Conclusion


Conclusions

People are generally happy with the modern error correcting performance of LDPC andTurbo-Codes.They are right: binary modern codes have excellent performance.

For binary-input memoryless channels, NB-LDPC codes have a slight advantage in thewaterfall and a more important advantage in the error floor.Does the increased decoding complexity justify those small gains ?

For more complex systems, like MIMO and M-QAM channels, or channels with memorywhere errors appear in burst, there is no doubt that NB-LDPC codes have an advantage:the mutual information loss due to marginalization in the binary case does not apply for thenon-binary case.

anyhow, the advances in NB-LDPC decoding have shown that the decoding complexity ofGF(q) codes is not q times larger than binary decoders, which is promising for future anemerging non-binary applications.


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Status of Knowledge on Non-Binary LDPC Decoders Part III ...€¦ · Slightly Irregular LDPC codes...

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