Status of Knowledge on Non-Binary LDPC Decoders Part II: … · 2010-12-06 · Status of Knowledge...

Status of Knowledge on Non-Binary LDPC DecodersPart II: Reduced Complexity Non-Binary Decoders

D. Declercq 1

1ETIS - UMR8051ENSEA/Cergy-University/CNRS

France

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

D. Declercq (ETIS - UMR8051) 1 / 56

Outline

1 Extended Min-Sum Decoding of NB-LDPC Codes

2 Further Refinements and ASIC Implementation

3 Analysis and Optimal Correction of the NB-LDPC decoders

4 Alternative Decoders: the Min-Max Decoder

5 Alternative Decoders: Symbol Flipping Decoder

6 Alternative Decoders: Non-Binary Stochastic Decoder

7 General Conclusion on Simplified Non-Binary Decoding


Bibliography of Low Complexity Decoders (1)

The bottleneck of the decoder complexity is the check node update

Extended Min-Sum Decoding[Declercq,2007] D. DECLERCQ AND M. FOSSORIER, “DECODING ALGORITHMS FOR NONBINARY LDPC CODES OVER

GF(Q)”, IEEE Transactions on communication, VOL. 55(4), PP. 633-643, APRIL 2007

[Voicila,2008] A. VOICILA, D. DECLERCQ, F. VERDIER, M. FOSSORIER AND P. URARD, “ARCHITECTURE OF A

NON-BINARY LDPC DECODER FOR THE NEXT GENERATION CODING SYSTEMS”, Proceedings of IEEE International

Conference on Consumer Electronics (ICCE), LAS VEGAS, USA, JAN. 2008

[Voicila,2010] A. VOICILA, D. DECLERCQ, F. VERDIER, M. FOSSORIER AND P. URARD, “LOW COMPLEXITY DECODING

ALGORITHM FOR NON-BINARY LDPC CODES IN HIGH ORDER FIELDS”, IEEE Transactions on communication, VOL. 58(5),

MAY 2010.

[Boutillon,2010] E. BOUTILLON AND L. CONDE-CANENCIA, “BUBBLE CHECK: A SIMPLIFIED ALGORITHM FOR

ELEMENTARY CHECK NODE PROCESSING IN EXTENDED MIN-SUM NON-BINARY LDPC DECODERS”, IEE Electronics

Letters, VOL. 46(9) PP. 633-634, APRIL 2010.


Bibliography of Low Complexity Decoders (2)

Min-Max Decoding[Savin,2008] V. SAVIN, “MIN-MAX DECODING FOR NON BINARY LDPC CODES”, Proceedings of ISIT’08, TORONTO,

CANADA, JULY 2008.

[Lin,2010] J. LIN, J. SHA, Z. WANG AND L. LI, “EFFICIENT DECODER DESIGN FOR NONBINARY QUASICYCLIC LDPC

CODES”, IEEE Transactions on Circuits and Systems, VOL.57(5), PP 1071–1082, MAY 2010.

Symbol Flipping Decoding[Kuo,2006] F.-C. KUO AND L. HANZO, “SYMBOL FLIPPING BASED DECODING OF GENERALIZED LOW DENSITY PARITY

CHECK CODES OVER GF(Q)”,Proceedings of IEEE WCNC’06, LAS-VEGAS, NV, APRIL, 2006.

[Liu,2010] B. LIU, J. GAO, G.DOU AND W. TAO, “WEIGHTED SYMBOL-FLIPPING DECODING FOR NONBINARY LDPC

CODES”, Int. Conf. on Network Security, Wireless Communications and Trusted Computing, WUHAN, HUBEI, CHINA,

APRIL 2010.

Stochastic Decoding[Sarkis,2009] G. SARKIS, S. MANNOR AND W.J. GROSS, “STOCHASTIC DECODING OF LDPC CODES OVER GF(Q)”,

Proceedings of IEEE ICC’09, DRESDEN, GERMANY, JUNE 2009.


Outline









Complexity of BP-based decoders are too highIdea : Truncated messages

Objective: reduce the complexity of O(q2) while staying in the Log-Domain

Messages are composed of q Log-Density Ratios (LDR)

with GF (q) =˘

0, α0, α1, . . . , αq−2¯ =˘

0, α1, α2, . . . , αq−1¯

U(αk ) = logµc → p[αk ]

µc → p[0]∀k = 0, . . . , q − 1

We proposed a decoding algorithm, called extended min-sum (EMS), which stores only alimited number nm � q of reliabilities in the vector messages

n

q

values

values

m

dc−1 Input Edges Output Edge

nm

values


Compensation of the Truncated Messages (1)

First, the values in a message are sorted in decreasing order so that the kept valuescorrespond to the maximum reliabilities.

Let A be a size q LDR vector. We compensate the q − nm truncated values of A with asingle scalar value γA, which is the simplest model one can use to build the truncatedvector B (size nm)

Note that in principle γA depends on the values in the message A.

The truncated version B of A is composed of the nm largest values of A sorted indecreasing order:

A = [A[0] . . .A[nm − 1] . . .A[q − 1]]T

−−−−−−−→Truncation B = [B[0] . . .B[nm − 1] γA . . . γA]T


Compensation of the truncated messages (2)How to choose γA ?

Let PA/PB be the probability domain versions of A/B:

PA[k ] = PA[0]eA[k ] k ∈ {0, . . . , q − 1} PB [k ] = PA[0]eB[k ] k ∈ {0, . . . , nm − 1}

clearly:q−1Xk=0

PA[k ] = 1nm−1Xk=0

PB [k ] < 1

choose γA such thatnm−1Xk=0

PB [k ] + (q − nm)PγA = 1

γA = max ∗ (A[nm], . . . ,A[q − 1])− log(q − nm)

so, in a first order approximation, we can choose:

γA = A[nm − 1]− log(q − nm)− offset


New Definition of a MessageLet’s take the exemple of GF(8) and nm = 4.

with GF (q) =˘

0, α0, α1, . . . , αq−2¯ =˘

0, α1, α2, . . . , αq−1¯

2666666666666666666666666666664

0 = log µ(0)µ(0)

U[α1] = log µ(α1)µ(0)

U[α2] = log µ(α2)µ(0)

U[α3] = log µ(α3)µ(0)

U[α4] = log µ(α4)µ(0)

U[α5] = log µ(α5)µ(0)

U[α6] = log µ(α6)µ(0)

U[α7] = log µ(α7)µ(0)

3777777777777777777777777777775

⇒

U2666666666664

U[α3]

U[α7]

U[α1]

U[α4]

3777777777775

β2666666666664

α3

α7

α1

α4

3777777777775h 1c1

c1 c2 c3

h 3c3h 2c2

vp v

v pu

p cu vc p

n =4m

Permutation

Nodes

+


Description of the EMS algorithmVariable Node Update

dv−1 Input Edges

Output Edge

Option 1 Option 2

Truncate

ci

Udv [αk ] = Lch[αk ] +

dv−1Xi=1

Vi [αk ]

k ∈ {0, . . . , q − 1}

Vi [αk ] =

Vi [αk ] if αk ∈ βViγVi

if αk /∈ βVi

Complexity O(nm dv + nm log(q) dv ): still too large !

Complexity can be reduced without significant loss to O(nm log(nm) dv ) with intermediatetruncations and recursive implementation.


Description of the EMS algorithmVariable Node Update: special case of dv = 2

Output Edge

Single Input Edge

Option 1 Option 2

ci

U2[β2(k)] = Lch[β2(k)] + V1[β2(k)]

if β2(k) ∈ β1

U2[β2(k)] = Lch[β2(k)] + γV1

if β2(k) /∈ β1

k ∈ {0, . . . , nm − 1}

worst case complexity O(nm log(nm)), or average complexity O(nm + δ).


Description of the EMS algorithmThe permutation step

c i

c ih ij

Uout [βout (k)] = U in[β in(k)]

βout [k ] = hij βin[k ]

k ∈ {0, . . . , nm − 1}

Since the message is now composed with the indices β, the permutation step is just acomputation of the new indices hij β


Parity Check Node UpdateNotion of Elementary Step

The bottleneck of the decoder complexity is the check node update

One Elementary Check Node Update corresponds to 2 input edges only.

c 1h1

c 2h2

c 3h3

V3[β3[k ]] = max(i,j)∈{0..nm−1}2

(U1[β1[i]] + U2[β2[j]])

such that β3[k ]⊕ β1[i]⊕ β2[j] = 0

k ∈ {0, . . . , nm}

Brute force complexity of this step is O(n2m + nm log(nm)).


Parity Check Node Update with dc = 5Forward/Backward implementation from elementary steps

p c1

U

p c1

U

size nm

size nm

size nm

size nm size nm

size nm size nm

size nm

size nm

size nm

c V5

p size nm

size nm

size nm

size nm

size nm

size nm

size nm

size nm

size nm p cU2

size nm U size nm I

size nm V

Backward

Forward

c V2

1 c V

c V4

p

p

p

BI 1 2

FI 1 2FI 1 1

3 c V p

U p c5

U p c5

p cU4

p cU4

U p c3

U p c3

1BI 1

2 3 4 5h v h v h v h v

p cU2

1h v1 2 3 4 5

(a) (b)


Reduced Complexity Algorithm for ElementaryCheck-Node

One elementary step assumes only two input messages U, I of size nm and one outputmessage V of size nm.

We assume here a serial architecture, i.e. which aims at filling the nm outputs of theElementary CheckNode in nm steps.

What is the minimum number of operations needed ?

[Voicila,2010] A. VOICILA, D. DECLERCQ, F. VERDIER, M. FOSSORIER AND P. URARD, “LOW COMPLEXITY DECODING

ALGORITHM FOR NON-BINARY LDPC CODES IN HIGH ORDER FIELDS”, IEEE Transactions on communication, VOL. 58(5), MAY

2010.


Reduced Complexity Algorithm for ElementaryCheck-NodeInitialization

V

U

I

GF adder


Reduced Complexity Algorithm for ElementaryCheck-NodeFirst output value

GF adder

U

I

V


Reduced Complexity Algorithm for ElementaryCheck-NodeSecond output value

GF adder

U

I

V


Reduced Complexity Algorithm for ElementaryCheck-Nodek − th output value

GF adder

U

I

V

The 6th output corresponds

to an entry already filled


Reduced Complexity Algorithm for ElementaryCheck-Nodek − th output value

GF adder

U

I

V


Minimum Complexity Algorithm for ElementaryCheck-NodeHow many candidates need to be consirered at each step ?

Let us denote the possible candidates at step k of the elementary check-node algorithm asbubbles,

We allow the moves of the bubbles to be both horizontal or vertical,

Let us remove the un-necessary candidates from the previous algorithm.

[Boutillon,2010] E. BOUTILLON AND L. CONDE-CANENCIA, “BUBBLE CHECK: A SIMPLIFIED ALGORITHM FOR ELEMENTARY

CHECK NODE PROCESSING IN EXTENDED MIN-SUM NON-BINARY LDPC DECODERS”, IEE Electronics Letters, VOL. 46(9) PP.

633-634, APRIL 2010.


Minimum Complexity Algorithm for ElementaryCheck-NodeWorst case scenario

Maximum number of bubbles: triangular shape,

Extraction of the k th value: comparison with a maximum of Ψ(k) candidates,

with a memory buffer of size Ψ(k), the elementary update needs nm + δ steps.

Ψ(k) =

&1 +

p1 + 8(k − 1)

2

’

[Boutillon,2010] E. BOUTILLON AND L. CONDE-CANENCIA, “BUBBLE CHECK: A SIMPLIFIED ALGORITHM FOR ELEMENTARY

CHECK NODE PROCESSING IN EXTENDED MIN-SUM NON-BINARY LDPC DECODERS”, IEE Electronics Letters, VOL. 46(9) PP.

633-634, APRIL 2010.


Complexity and Memory Evaluation of EMSWorst Case scenario

The bubble algorithm can be used as well for the variable node update.

The Worst Case complexity of EMS algorithm is dominated by O(nm log(nm)) for bothparity and variable nodes computation

No. max No. real add. No. GF(q) add.

Check Node 3(dc − 2)nm log(nm) 3(dc − 2)(2 nm) 3(dc − 2)(2 nm)

Variable Node (dv + 2(dv − 2))nm log(nm) (dv + 2(dv − 2))(2 nm) 0

Permutation Node 0 0 nm(dv + dc )

The Average complexity of EMS algorithm is O(nm + δ)

The edge memory scales as N ∗ dv ∗ nm ∗ (Nbits + log2 q) bits


Performance of the EMS vs. Non-Binary BP

The EMS GF(64) nm = 32 algorithm has 0.05dB performance loss in the waterfall regionand performs even better than the BP decoder in the error floor region.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 310

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N0 (in db)

FE

R

EMS GF(64) nm=16

BP GF(64)

BP GF(256)

EMS GF(256) nm=32

EMS GF(64) nm=32

R=1/2,dv = 2, dc = 4Nb ≈ 848 bits


Performance vs. Binary LDPC codes

EMS vs binary MS algorithm,

LDPC code (R=0.5, Nb=504 bits),

BI-AWGN channel

1 1.5 2 2.5 3 3.5 410

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N0 (in dB)

FER

EMS GF(64) nm=6MS binary correctedEMS GF(64) nm=12EMS GF(64) nm=18

EMS vs binary MS algorithm,

LDPC code (R=0.5, Nb=1008 bits),

256-QAM-AWGN channel

8 8.5 9 9.5 10 10.5 11 11.5 12 12.510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N0 (in dB)FE

R

EMS GF(256) nm=6

EMS GF(256) nm=12

EMS GF(256) nm=18

EMS GF(256) nm=36

MS binary corrected


Conclusions

∆! Like the Binary Min-Sum, the EMS needs to be compensated by an offset ∆!

The EMS is the best known low complexity decoding algorithm for non binary LDPC codes.

The main originality of the proposed algorithm is to truncate the vector messages to a fixednumber of values nm � q.

The complexity of the proposed algorithm is dominated by O(nm log2(nm))

The EMS decoding algorithm can approach the performance of the BP decoder and evenin some cases beat the BP decoder.

The proposed low complexity, low memory EMS decoding algorithm becomes a goodcandidate for the hardware implementation of non binary LDPC decoders.


Outline









The DAVINCI project

The DAVINCI Project

European projet funded by the 7th-Framework-Program of the European Commission(http://www.ict-davinci-codes.eu/),

Consortia Partners: Samsung-UK, ENSEA, ST-Microelectronics, IMEC, univ.Bretagne-sud, CTTC, WISER, ITTI,

David Declercq scientific coordinator of the project.

Outcomes of the Project

QC-LDPC Code Design adapted for parallel implementation of the decoder and linearencoding,

Refinements of the EMS decoder (bubble-check algorithm, new definition of messages),

Hardware model with increased degree of parallelism.


Slightly Modified Definition of the Messages


U(αk ) = logµc → p[α∗]

µc → p[αk ]∀k = 0, . . . , q − 1

such that U(α∗) ≥ U(αk ), ∀k .the definition of the LDR depends on the message !not a problem for the EMS since the indices are stored in the message,With this new definition, we need only Nbit = 5 bits of quantization for each LDR value.


ST-Microelectronics ASIC implementationPower Estimation

Power estimation done at gate netlist level

Based on real simulation patterns, in different modes and SNR conditions

Passive/active phases of the LDPC decoding taken into account

Mode Mode 1N=96

Mode 2N=192

Mode 3N=288

Mode 4N=384

FER3.3E-4 1.3E-4 < 1E-5 < 1E-5

Mean n.o. iterations3.72 4.03 4.23 4.38

Mean frame power (W)0.0986 0.101 0.103 0.104

Consumption (µJ)0.284 0.583 0.89 1.2

~ 100 mW in continuous reception


ST-Microelectronics ASIC implementationChip Area

Post-synthesis area: 1.28 mm2 on a 45nm technology

100 Mbits/s worst case throughput (user throughput) @ 300 MHz clock

C. Liu [7] T. Brack [8] G. Gentile [9]DaVinci decoder

ASIC version

Code length (bits) 576-2304 576-2304 576-2304 288-2880

F clock (MHz) 150 333 400 300

Iterations 20 10-15 15 18

throughput 105 133-928 128-746 100-200

Technology 90 nm 130 nm 65 nm 45 nm

area 6.25 3.83 0.59 1.28

Area equiv. 45nm 1.56 0.48 0.3 1.28

x4 area


Outline









The Role Model Framework

For small values of nm, very precice compensation of the EMS decoder is an issue

A single offset correction might not be powerfull enough:

1 compensation for the truncation of messages,2 compensation for the max operator instead of max∗,3 compensation for the truncation of intermediate messages during the

Forward/Backward recursion,4 compensation for the presence of cycles in the Tanner graph.

Motivation: make use of a theoretical framework for the optimal design ofreduced-complexity receiver components operating on finite precision messages

General Idea: Let’s assume that one can have access to the Bayesian optimumreceiver/decoder outputs, which will serve as Role Model.

[Sayir,2010] J. SAYIR, “DESIGN OF NON-BINARY DECODERS USING THE ROLE MODEL FRAMEWORK”, proceedings of the

ISTC’10, BREST, FRANCE, SEPTEMBER 2010.


Classical Design of Receiver Components

ChannelSource Bayesian EstimatorX

Y1

Y2

Y3

PX|Y1Y2Y3

X !1

X !2

Bayesian Estimation

Advantages:Guarantees optimalityCan be repeated at will in a classical serial or iterative message-passing setup

Drawbacks:Calculates probabilities, i.e., real numbersOperations can be overly complex (e.g. MIMO MAP detector, Non-Binary LDPCdecoder)


Practical Implementation of Receiver Algorithms

Impact of Hardware Implementation with Finite Precision, Reduced Complexity

Bayesian operations are replaced by simpler alternatives

⇒ Approximate Bayesian estimation

Loss of optimality (even under constraints)

Often, we turn to an heuristic decoder design by simulation.


Approach by Constrained Optimization

ChannelSourceLow Complexity

Component (non-Bayesian)

X

Y1

Y2

Y3

X !1

X !2

BayesianPost-Processing

PX|ZZ

Bayesian Post-Processing

Optimal under constraints, given the low complexity component

Can be derived analytically for simple components (e.g. Min-Sum decoder for binary LDPCcodes)

Allow to consider elaborate parametrized correction functions (vector, non-linear, adaptive,etc) and to optimize the parameters offline.


Role Model Estimation

Source Estimatorin TrainingChannel

Low ComplexityComponent

XY Z = f(Y )

BayesianRole ModelEstimator

QX|Z

PX|Y

X !1

X !2

arg minQX|ZEPYZ

ˆDKL(PX |Y ||QX |Z )

˜= PX |Z

This is convex optimization problem: optimum values of the parameters exist.

The issue is to choose an efficient model for EMS correction/compensation.

With the simple model γA = A[nm − 1]− log(q − nm)− offset , the Role Model frameworkgives the same offset value as the ones we got by Monte Carlo simulation.


Outline









Alternative to the EMS: the Min-Max decoder


U(αk ) = logµc → p[α∗]

µc → p[αk ]∀k = 0, . . . , q − 1

such that U(α∗) ≥ U(αk ), ∀k .

The EMS step:V3[β3[k ]] = min

(i,j)∈{0..nm−1}2(U1[β1[i]] + U2[β2[j]])


k ∈ {0, . . . , nm}is replaced by:

V3[β3[k ]] = min(i,j)∈{0..q−1}2

(max (U1[β1[i]],U2[β2[j]]))


k ∈ {0, . . . , q − 1}


Alternative to the EMS: the Min-Max decoderAdvantages

possible to perform a block implementation of the check-node (no Forward/Backward),Throughput is improved compared to the full-complexity Non-Binary Min-Sum decoder,

Drawbacks ?

Throughput is only slightly better than the EMS decoder with nm small enough,Looses more performance in the waterfall than the EMS (0.2dB reported in [Lin,2010] forGF(32) and Nb = 600 bits),Worse robusteness to quantization: [Lin,2010] reports good performance with 7 bits ofquantization.

Advantage ?

Could bring an extra complexity reduction in case of hybrid version Min-Max/EMS ?

[Savin,2008] V. SAVIN, “MIN-MAX DECODING FOR NON BINARY LDPC CODES”, Proceedings of ISIT’08, TORONTO, CANADA,

JULY 2008.

[Lin,2010] J. LIN, J. SHA, Z. WANG AND L. LI, “EFFICIENT DECODER DESIGN FOR NONBINARY QUASICYCLIC LDPC CODES”,

IEEE Transactions on Circuits and Systems, VOL.57(5), PP 1071–1082, MAY 2010.


Outline









Alternative to the EMS: the Symbol Flipping decoder

Messages are now symbols of GF (q) =˘

0, α1, α2, . . . , αq−1¯

: U = αi V = αj

Let us consider the simplest Check-Node update

dc−1 incomming edges

VU

c 1h1

c 2h2

h dc−1

cdc−1

Vdc−1 = U1 ⊕ . . .⊕ Udc−2

Issue is now how to define the variable node update ?

dv−1 Input Edges

Output Edge

Option 1 Option 2

U

L

V

ci

Udv−1 = Vi if all {Vi}i=0..dv−2 are equal

Udv−1 = χ if not


Alternative to the EMS: the Symbol Flipping decoderDifferent Solutions for the Variable Node update

1 [Gallager,1963]: Random choice of χ ∈ GF(q)\L2 Weighted Symbol Flipping: choose χ according to the channel soft-values,3 Put memory on the edges and choose χ from a previous computed value.

Advantages

Increadibly low complexity hard decoding,Requires minimal storage for the messages values.Complexity added for Weighted Symbol Flipping is low,Could be an alterative to Hard Decoding of Reed-Solomon codes.

Drawbacks

Too simple: performance loss in the waterfall is large: [Liu,2010] report 1.5dB to 2.5dB loss,for very small block-lengths (a hundred symbols).so far, it’s only an alternative to Hard Decoding of Reed-Solomon codes.need more research to decide if weighted symbol flipping is promising.

[Liu,2010] B. LIU, J. GAO, G.DOU AND W. TAO, “WEIGHTED SYMBOL-FLIPPING DECODING FOR NONBINARY LDPC CODES”,

Int. Conf. on Network Security, Wireless Communications and Trusted Computing, WUHAN, HUBEI, CHINA, APRIL 2010.


Outline









Stochastic DecoderList/Vector version of Hard-Decoders

Messages are now lists of symbols of GF (q) =˘

0, α1, α2, . . . , αq−1¯

:

U =˘

U(t)¯t=1...T V =

˘V (t)¯

t=1...T

The occurences of the symbols in the list represent the probabilities of the symbols.

exemple for GF(4) = {0, α1, α2, α3}

µ(0) =9

25µ(1) =

325

µ(2) =1025

µ(3) =3

25

. . . 0 α1 α2 α2 α2 0 0 α3 α1 α1 0 0 0 α2 α3 0 α2 α2 0 α2 0 α3 α2 α2 α2 α2 . . .

Check-Nodes update is restricted to Parity-Check computation (same as symbol-flipping).


Stochastic Decoder: Variable Nodes (1)Agree or Freeze

Memory

One Step

ci

U(t)dv−1 = V (t)

i if all {Vi} are equal

U(t)dv−1 = U(t−1)

dv−1 if not


Stochastic Decoder: Variable Nodes (1)Agree or Freeze

Freeze

Memory

One Step

ci

U(t)dv−1 = V (t)


U(t)dv−1 = U(t−1)

dv−1 if not


Stochastic Decoder: Variable Nodes (2)Agree or Memory

Memory

Edge

ci

U(t)dv−1 = V (t)


U(t)dv−1 = U(t−τ(ω))

dv−1 if not


Stochastic Decoder: Variable Nodes (2)Agree or Memory

Freeze

Memory

Edge

accessRandom Memory

ci

U(t)dv−1 = V (t)


U(t)dv−1 = U(t−τ(ω))

dv−1 if not


Error-correcting Performance

(256,128) LDPC code over GF(16)

10-6

10-5

10-4

10-3

10-2

10-1

100

1 1.5 2 2.5 3 3.5 4

Fra

me E

rror

Rate

Eb/N0 (dB)

rmax = 10

rmax = 1000

SPA maxitrs=103

Stoc. maxitrs=106

Stoc. maxitrs=104

Simple decoder, but latency is catastrophic.


Relaxed Half-Stochastic Decoding

For high dv and q, the variable node is the bottleneck in stochastic decoders

Relaxed Stochastic Decoder: replace edge memory by a q-vector message ofprobabilities µv → p estimated in a relaxed way. Then sample the Check-Node input fromµv → p .

Relaxed Half-Stochastic Decoder: replace variable node inputs by (dv − 1) q-vectormessages of probabilities

˘µip → v

¯estimated in a relaxed way. Perform BP-based

variable node update to get µv → p , then sample the Check-Node input from µv → p .

[Sarkis,2010] G. SARKIS, S. HEMATI, S. MANNOR AND W.J. GROSS, “RELAXED HALF-STOCHASTIC DECODING OF LDPC

CODES OVER GF(q)”, Proceedings of the Allerton conf., MONTICELLO, ILLINOIS, USA, SEPTEMBER 2010.


Performance of Relaxed Half-Stochastic Decoder

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

1 1.5 2 2.5 3 3.5 4 4.5 5

Eb/N0 (dB)

GF(32) (4,28)n = 868, k = 747rmax = 10

GF(16) (2,4)n = 256, k = 128rmax = 10

GF(64) (2,4)n = 192, k = 96rmax = 1

GF(256) (2,4)n = 192, k = 96rmax = 1

SPARHS


Average Iteration is still Too Large !!

(256,128) LDPC code over GF(16).

SNR (dB) 2.0 3.0 4.0

Stochastic 2834 475 253

RHS, β = 164 373 167 103

RHS, SNR-adapted β 373 50 17

No hardware implementation proposed yet, too soon to conclude about interest (or not) ofRHS decoder.


Outline









ConclusionsPros/Cons of existing Solutions

EMS with nm sufficiently large: good performance but too complex (3 to 6 times more thanbinary),

Memory/Routing requirements are reduced ... but not greatly,

Min-Max: same class of algorithm than EMS: roughly same advantages and drawbacks.

Stochastic Decoding: Simple, but good performance achieved only at the price of a veryhigh latency,

Symbol Flipping: Very Simple, but too weak for the moment,

The most interesting issue in my opinion comes from a proper compensation of thesub-optimal decoders.How, where and when correct the messages is the key question.


ConclusionsThe Holy-Grail of Non-Binary Decoding

Algorithms do not scale well with q for fixed complexity (even EMS),

GF(8) GF(16) GF(32) GF(64) GF(128) GF(256) GF(512)0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Galois Field Order

Thre

shold

Valu

e δ

=(E

b/N

0) d

B

factor correctionoffset correction

BP

EMS(n

m,n

c)=(7,2)

EMS(n

m,n

c)=(13,3)

In principle we would like that the algorithms for GF(q) LDPC decoders have complexitygrowing in p = log2(q) and not with q.

How to achieve that with still good performance is the Holy-Grail of Non-Binary Decoding.


Status of Knowledge on Non-Binary LDPC Decoders Part II: … · 2010-12-06 · Status of Knowledge...

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Transcript of Status of Knowledge on Non-Binary LDPC Decoders Part II: … · 2010-12-06 · Status of Knowledge...