Status of Knowledge on Non-Binary LDPC Decoders · 2010-12-06 · Status of Knowledge on Non-Binary...

Status of Knowledge on Non-Binary LDPC DecodersPart I: From Binary to Non-Binary Belief Propagation Decoding

D. Declercq 1

1ETIS - UMR8051ENSEA/Cergy-University/CNRS

France

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

D. Declercq (ETIS - UMR8051) 1 / 59

Outline

1 Introduction

2 Belief Propagation on a General Graph

3 Binary Belief Propagation Decoder

4 Non-Binary Belief Propagation Decoding


Outline

1 Introduction





Small History of Binary LDPC Codes: Landmarks

Gallager 1962 regular LDPC codes, proof of convergence (MLD), algo. A (bit

flipping), algo. B ,

Tanner 1981 composite codes on graphs, link with product codes and LDPCcodes,

MacKay 1995 Belief Propagation (BP) decoding, link with iterativeturbo-decoding, irregular LDPC codes,

Rich. et Urb. 2001 proof of convergence (BP), optimization of irregularity, codesapproaching capacity (BEC, BI-AWGN),

Since then Optimization for other types of channels (freq. selective, multilevel,multi-user, turbo-equalization, joint source-channel coding),finding good matrices for small sizes, lowering the error floor.

⇒ Golden age of LDPC codes, application in many standards.


Small History of Non-Binary LDPC Codes: Landmarks

Gallager 1963 LDPC codes in Galois Fields, iterative hard decoding algo. B fordv = 3,

MacKay 1998 Advantages for small blocks/high rates, ultra-sparse dv = 2 LDPCcodes in high-order Fields,

2003-2006 Development of practical decoders for Non-Binary LDPC codes,

2006-2010 Attempts to find applications where NB-LDPC codes outperformbinary LDPC codes,

2010-xxx Golden age of Non-Binary LDPC codes ?

[DAVEY 98] M. DAVEY AND D.J.C. MACKAY, “LOW DENSITY PARITY CHECK CODES OVER GF(Q)”,IEEE communication letter, VOL. 2, PP 165–167, JUNE 1998

[MACKAY 99] D.J.C. MACKAY AND M. DAVEY, “EVALUATION OF GALLAGER CODES FOR SHORT BLOCK LENGTH

AND HIGH RATE APPLICATIONS”, proc. of IMA workshop on codes, systems and graphical models, 1999


Definitions and Quantities

LDPC Code

CH =

c ∈ GF (q)×N | H.c

GF (q)= 0

ff LDPC Code

CG =˘

c = G.u,∀u ∈ GF (q)×K¯

with H(M × N), parity check matrix,

with G(N × K ), generator matrix.

Size

8<:of a codeword N R = K/Nof information K (if H is full rank)of the redundancy M

Density of H :

dH = nb. nonzero elements in HM.N LDPC : dH

N→+∞−→ 0


Tanner Graph Representation of a Binary LDPC Code

Tanner Graph is a Bi-partite graph with Adjacency Matrix H

LDPC: Low Density Parity Check Codes

10 2 3 5 6 74

10 2 3 5 6 74

Interleaver Π

Parity checks

Parity checks

codeword

codeword

: message

c c c c c c c c

cccccccc

01 1 10 0 0

1 1 1 10 0 0 0

0 0 0 1 111 0

10 0101

1

0 1

H =


Tanner graph Representation of a Non-Binary LDPCcode

Tanner graph of an Irregular Non-Binary LDPC code in GF (8)

03 7 20 0 0

4 3 5 10 0 0 0

1 0 0 6 735 0

00 0106

1

2

H =

0

.

.

.

.

.

.

.

.

10 2 3 5 6 74

10 2 3 5 6 74

Interleaver Π

Codeword

Codeword

: message

c c c c c ccc

c c c c c c c c

parity checks

parity checks


parameters for Non-Binary LDPC code irregularityIrregularity distribution, Irregularity profile

1 edges proportions :

λi : proportion of nonzero values {Hkl} in degree i columns,ρj : proportion of nonzero values {Hkl} in degree j rows,

2 node proportions :

λi : proportion of columns in H with degree i ,ρj : proportion of rows in H with degree j ,

λ(x) =dvmaxX

i=2

λi x i−1 ρ(x) =dcmaxX

j=2

ρj x j−1

3 non-zero values distribution :

hij(ω) : uniformly distributed in GF(q)\0.


Notion of Code Family

A LDPC code family is defined by (λ(x), ρ(x),N)

for characterization, proofs, theoretical studies

A fixed LDPC code is defined by (λ(x), ρ(x),N,Π, {hij})for practical application

Π

λ(x) =4

24x +

324

x2 +824

x3 +924

x8 ρ(x) = x5


Outline

1 Introduction





Concept of Iterative Decoder on GraphFrom Local Computation to Global Optimization

The general concept of LDPC decoders is based on message passing between nodes inthe Tanner graph of the code, so that iterative updates of the messages lead to a stablestate of the messages: convergence to a fixed point.

Messages represent probability density functions of the random variables. For a discreterandom variable in a set of q elements:

µ(0) = Prob (xn = 0|.) . . . µ(q − 1) = Prob (xn = q − 1|.)q−1Xi=0

µ(k) = 1

The decoding result is the a posteriori probability of one random variable xn:

Prob (xn|y0, y1, . . . , yN−1)

a particular scheduling of the computation of the messages defines a decoding iteration.


Terminology and some HistoryBelief Propagation: BP

1 Artificial Intelligence

Statistical learning : Pearl’s Belief Propagation (1981-86)Neural Networks : sum-product algorithm (ΣΠ) (1985-86)

2 Information Theory

Gallager iterative decoders for LDPC (1963),Viterbi (1967), BCJR (1974): can be analysed as BP on Factor graphs,

3 Statistical Physics

BP = Bethe approximation of the global free energy of complex systems (1935),Generalized BP = Kikuchi approximation of the free energy (1951)


ExampleTanner graph (for PC codes) is a special case of Factor Graph

Let A(ω), B(ω), C(ω), D(ω) be dependant random variables.

Let A′, B′, C′, D′ be their noisy observation.

A B

C D

A’

B’

C’ D’

p(B|A’,C’,D’)

With Belief propagation on a tree, we get the a posteriori density : optimal solution.


Notations for bi-partite graphs

xn

µx f

µf x

F(.)

node

variable function

node

(µx → f ,µf → x ) : messages = p.d.f.

The graph is not oriented: messages needed in both directions.

2 types of nodes = 2 types of local updates: data node update and function node update


Concept of Computational TreeExpansion of the graph from a symbol/check node.

Nodes seen after1 iteration

Nodes seen after2 iterations

LLRLLR LLR

F2F1


Concept of Computational TreePast of the graph = set of nodes.

SF1

SF2



LLRLLR LLR

F2F1

Past of F1: Past of F2:


Concept of Computational TreeIndependence assumption

SF1

SF2



LLRLLR LLR

F2F1

Past of F1: Past of F2:

Disjoint

Independent


Variable Node Update: Bayesian Merging⇔ bitnode/symbol updates for LDPC codes

F(.)1

µx f

x1

F(.)2

F(.)

F(.)4

3

1

µf x

3

µf x

4

µf x

2

µ1x → f [k ] ∝

4Yi=2

µif → x [k ] ∀k = 0 . . . q − 1

ASSUMPTION: input messages µif → x are independent

ASSUMPTION: noisy symbol sets leading to µif → x are disjoint

∆! Update equation is NOT normalized


Function Node Update: Bayesian Marginalization⇔ checknode updates for LDPC codes

x1

x4

x1

x3

x2

x2

x3

x4

1µ

f x

µx f

4F( , , , )

µx f

2

µx f

3

µ1f → x [k1] =

Xk2,k3,k4

F`x1 = k1, x2 = k2, x3 = k3, x4 = k4

´ 4Yi=2

µix → f [ki ]

∀k1 = 0 . . . q − 1

ASSUMPTION: input messages µif → x are independent

ASSUMPTION: noisy symbol sets leading to µif → x are disjoint


Function Node Update: Bayesian MarginalizationNon-Binary Parity-check case in GF(q)

µ1f → x [k1] =

Xk2,k3,k4

F`x1 = k1, x2 = k2, x3 = k3, x4 = k4

´ 4Yi=2

µix → f [ki ]

let αk ∈ GF(q) =˘

0, 1, α, α2, . . . , αq−1¯Parity-check case: the function node reduces to an indicator function:

F“

x1 = α1, x2 = α2, x3 = α3, x4 = α4

”= 1 if α1 + α2 + α3 + α4 = 0

F“

x1 = α1, x2 = α2, x3 = α3, x4 = α4

”= 0 if α1 + α2 + α3 + α4 6= 0

Parity-check case: results in one less sum dimension in the marginalization:

µ1f → x [α1] =

Xα2,α3

µ2x → f [α2] µ3

x → f [α3] µ4x → f [α1 + α2 + α3].


Scheduling and Definition of Iteration

This ordering of the messages is called flooding schedule

One decoding iteration = µ→ µ→ APP


Concept of Computational Tree (cont’d)



APP

LLRLLR LLR

Computational span of L iterations: in L iterations, a maximum ofdv ∗ (dv − 1)L−1 ∗ (dc − 1)L nodes are seen from the top of the tree.

As a consequence, a usual assumption is that the BP decoder needs at least L = log(N)iterations to converge (to see all LLRs).

As a consequence, the independence assumption for the BP decoder breaks after at mostL = log(N) iterations.


Concept of Computational Tree (cont’d)Breaking the independance assumption



LLRLLR LLR

wrong APP

wrong update

wrong update

a crucial parameter of the graph is its girth g, i.e. the size of the smallest closed path/cycle,

As a consequence, only bg/4c decoding iterations correspond to an exact inference !!!


Alternate Scheduling of messages (1)Layered BP or Shuffled Scheduling







Advantage: for bitnodes with degree dv ≥ 3 Messages are computed several times during ONEiteration⇒ Faster convergence.


Outline

1 Introduction





Binary Belief Propagation Algorithm in theLog-Domain

definition of messages in the log-domain

uk = logµk

f → x [0]

µkf → x [1]

vk = logµk

x → f [0]

µkx → f [1]

message update through the 2 types of nodes

VmU

k

U (LLR)0

C

vm = u0 +

dvXk=1,k 6=m

uktanh

uk

2=

dcYm=1;m 6=k

tanhvm

2Vm

Uk


From the Probability-Domain to the Log-Domain (1)Bitnode Update

µdvx → f [k ] =

dv−1Yi=1

µif → x [k ] ∀k = 0, 1

Let us consider a dv = 3 bitnode with v3 as output message:

v3 = logµ3

x → f [0]

µ3x → f [1]

= logµ0[0] µ1

f → x [0] µ1f → x [0]

µ0[1] µ1f → x [1] µ1

f → x [1]

= u0 + u1 + u2


From the Probability-Domain to the Log-Domain (2)Checknode Update

µdcf → x [αdc ] =

Xα1,...,αdc−1

dc−1Yi=1

µix → f [αi ] 1

Xk

αk = 0

!αk = 0, 1

Let us consider a dc = 3 bitnode with u3 as output message:

µ3f → x [0] = µ1

x → f [0]µ2x → f [0] + µ1

x → f [1]µ2x → f [1]

µ3f → x [1] = µ1

x → f [0]µ2x → f [1] + µ1

x → f [1]µ2x → f [0]


From the Probability-Domain to the Log-Domain (3)Checknode Update Intermediate step: decoding in the Fourier Domain

Now compute the factorization of the sum ...

(µ3f → x [0] + µ3

f → x [1]) = (µ1x → f [0] + µ1

x → f [1]) (µ2x → f [0] + µ2

x → f [1])

... and the factorization of the difference

(µ3f → x [0]− µ3

f → x [1]) = (µ1x → f [0]− µ1

x → f [1]) (µ2x → f [0]− µ2

x → f [1])

we can write in vector form:"1 11 −1

#"µ3

f → x [0]

µ3f → x [1]

#=

"1 11 −1

#"µ1

x → f [0]

µ1x → f [1]

#!⊗ "

1 11 −1

#"µ2

x → f [0]

µ2x → f [1]

#!


From the Probability-Domain to the Log-Domain (4)Checknode Update Intermediate step: decoding in the Fourier Domain

with the definitions of the Fourier Transforms

F =

»1 11 −1

–F−1 = 1

2

»1 11 −1

–

we obtain the checknode update in the Fourier Domain:

"µ3

f → x [0]

µ3f → x [1]

#= F−1 ×

F ×

"µ1

x → f [0]

µ1x → f [1]

#⊗ F ×

"µ2

x → f [0]

µ2x → f [1]

#!


From the Probability-Domain to the Log-Domain (5)

link between the probability domain and the log-domain

µ3x → f [0] =

eu3

eu3 + 1µ3

x → f [1] =1

eu3 + 1

From previous equations, we have:

(µ3f → x [0]− µ3

f → x [1]) = (µ1x → f [0]− µ1

x → f [1]) (µ2x → f [0]− µ2

x → f [1])

⇒eu3 − 1eu3 + 1

=

„ev1 − 1ev1 + 1

« „ev2 − 1ev2 + 1

«⇒

eu3/2 − e−u3/2

eu3/2 + e−u3/2=

ev1/2 − e−v1/2

ev1/2 + e−v1/2

! ev2/2 − e−v2/2

ev2/2 + e−v2/2

!

⇒ tanhu3

2= tanh

v1

2× tanh

v2

2


Final Step: Remove all Products

Lets compute the BP checknode update in the Log-Domain:

log tanh|u3|2

= log tanh|v1|2

+ log tanh|v2|2

The sign of the message is computed in a parallel stream:

sign“

tanhu3

2

”= sign

“tanh

v1

2

”× sign

“tanh

v2

2

”

sign (u3) = sign (v1) × sign (v2)


Binary Belief Propagation Algorithm in theLog-Domain

definition of messages in the log-domain

uk = logµk

f → x [0]

µkf → x [1]

vk = logµk

x → f [0]

µkx → f [1]

message update through the 2 types of nodes

VmU

k

U (LLR)0

C

vm = u0 +

dvXk=1,k 6=m

uk

log tanh|uk |

2=

dcXm=1;m 6=k

log tanh|vm|

2

sign(uk ) =

dcYm=1;m 6=k

sign(vm)

Vm

Uk


From the Log-Domain BP to the Min-Sum decoder

From previous equations:

µ3f → x [0] = µ1

x → f [0]µ2x → f [0] + µ1

x → f [1]µ2x → f [1]

µ3f → x [1] = µ1

x → f [0]µ2x → f [1] + µ1

x → f [1]µ2x → f [0]

u3 = log

µ3

f → x [0]

µ3f → x [1]

!

= log„

ev1

ev1 + 1ev2

ev2 + 1+

1ev1 + 1

1ev2 + 1

«− log

„ev1

ev1 + 11

ev2 + 1+

1ev1 + 1

ev2

ev2 + 1

«= log

èv1+v2 + 1

´− log

èv1 + ev2

´= max ∗(v1 + v2, 0)−max ∗(v1, v2)

where max ∗(x , y) = logèx + ey´ denotes the Jacobian Logarithm.


From the Log-Domain BP to the Min-Sum decoder

After some transformations:

u3 = max ∗(v1 + v2, 0)−max ∗(v1, v2)

= max(v1 + v2, 0)−max(v1, v2) + log

1 + e|v1+v2|

1 + e|v1−v2|

!

= sign(v1) sign(v2) min (|v1| , |v2|) + log

1 + e|v1+v2|

1 + e|v1−v2|

!

The additionnal term log

1 + e|v1+v2|

1 + e|v1−v2|

!can be replaced by a constant value.

noting that this term is negative when v1 and v2 have the same sign, and the term ispositive when v1 and v2 have different signs, ...


We finally get the Corrected Min-Sum decoder

1 Bitnode update: same as for BP

vm = u0 +

dvXk=1,k 6=m

uk

2 Checknode update:

uk =

0@ dcYm=1;m 6=k

sign(vm)

1A . minm 6=k|vm|

3 Compensation/Correction:

uk = max(0, uk − γ) if uk > 0

uk = min(0, uk + γ) if uk < 0


Comments on the different Decoding Algorithms

Shuffled Scheduling can be parallelized if the LDPC is properly designed⇒ increasedthroughput,

Shuffled Scheduling converges approximately 2 to 3 times faster than flooding schedule⇒reduced latency,

Bit-flipping, Gal-A and Gal-B: easier to get theorems on theoretical performance,

Min-Sum with proper offset correction approaches BP for regular or slightly irregular LDPCcodes,

In some particular cases, the Min-Sum decoder can surpass the BP decoder in the errorfloor region.


Outline

1 Introduction





Belief Propagation in the Probability DomainCheck Node Equations: augmenting the Factor graph representation

Now the code is defined from Non Binary Parity Check equations

dcXj=1

hij .cj = 0 in GF (q)

with GF (q) =˘

0, α0, α1, . . . , αq−1¯

h 1c1

c1 c2 c3

h 3c3h 2c2

h c + h c + h c 1 1 2 2 3 3

x fµµ

f x

c1 c2 c3

p cµ

v pµ

µc p

µp v

Permutation

Nodes

+


Belief Propagation in the Probability DomainVariable Node Equations


dcXj=1


with GF (q) =˘

0, α0, α1, . . . , αq−1¯

Variable node Update is the Kronecker product of all incomming messages:

µdvv → p[k ] =

dv−1Yi=1

µip → v [k ] ∀k = 0, . . . , q − 1

Or in vector form:

µdvv → p

= µ0p → v

⊗ . . .⊗ µdv−1p → v


Explaining the Permutation Stepcyclic permutation / rotation

GF(q) is a cyclic Field, as such, multiplication by hij acts on the symbols as a cyclicpermutation of the Field elements:

µip → c [k ′] = µi

v → p[k ] αk′ = hij αk ∀k = 0, . . . , q − 1

α

α

α

α

α

α

0

1

2

3

4

5

6α

α

α

α

α

α

α

α

0

1

2

3

4

5

6

cji ih .

Internal operation

in GF(8)

00

ci

α =

α =

α =

α =

α =

0

1

2

3

4

5

6α =

α = α

α

α

α

α

α

α

2

2

2

2

2

2

2

GF(8)GF(8)


Belief Propagation in the Probability DomainCheck Node Equations


dcXj=1


with GF (q) =˘

0, α0, α1, . . . , αq−1¯

Check node Update is still a Bayesian marginalization. Case of dc = 3 and GF(4):

h 1c1 h 3c3h 2c2

p cµ

p cµ

µc p

1

2

3

+

c1 \ c2 0 1 2 30 0 1 2 31 1 0 3 22 2 3 0 13 3 2 1 0


Belief Propagation in the Probability DomainCheck Node Equations

h 1c1 h 3c3h 2c2

p cµ

p cµ

µc p

1

2

3

+

c1 \ c2 0 1 2 30 0 1 2 31 1 0 3 22 2 3 0 13 3 2 1 0

Check node Update is still a Bayesian marginalization. Case of dc = 3 and GF(4):

µ3c → p[0] = µ1

p → c [0]µ2p → c [0] + µ1

p → c [1]µ2p → c [1] + µ1

p → c [2]µ2p → c [2] + µ1

p → c [3]µ2p → c [3]

µ3c → p[1] = µ1

p → c [0]µ2p → c [1] + µ1

p → c [1]µ2p → c [0] + µ1

p → c [2]µ2p → c [3] + µ1

p → c [3]µ2p → c [2]

µ3c → p[2] = µ1

p → c [0]µ2p → c [2] + µ1

p → c [2]µ2p → c [0] + µ1

p → c [1]µ2p → c [3] + µ1

p → c [3]µ2p → c [1]

µ3c → p[3] = µ1

p → c [0]µ2p → c [3] + µ1

p → c [0]µ2p → c [3] + µ1

p → c [1]µ2p → c [2] + µ1

p → c [2]µ2p → c [1]

The number of terms in the above equations grows as q2.


Belief Propagation in the Probability DomainHow to simplify the checknode update ?

h 1c1 h 3c3h 2c2

p cµ

p cµ

µc p

1

2

3

Fourier ?

+


Tensorial Notation of Messages

in case of binary extension fields GF(2p), the symbols c ∈GF(q) could be represented bya binary map, or a polynomial:

c = [c1, . . . , cp] with {c1, . . . , cp} ∈ {0, 1}

c(x) =

pXi=1

ci x i−1 with {c1, . . . , cp} ∈ {0, 1}

Let put the probability weights µ(c = αk ) in a size-2, p-dimensional tensor indexed bybinary values {c1, . . . , cp}.

C = =Prob(u(x)=x)

Prob(c(x)=1)

Prob(c(x)=0)

Prob(c(x)=1+x)

C[0,0]

C[1,0]

C[0,1]

C[1,1]

C[i,j]=


Tensorial Notation of Messagesa GF(8) example

Prob(c(x)=1+x+x )

Prob(c(x)=1)

Prob(c(x)=x)

Prob(c(x)=1+x)

Prob(c(x)=0)

Prob(c(x)=x )2

Prob(c(x)=x+x )2

2

Prob(c(x)=1+x )2

C =

C[1,0,0]C[0,1,0]C[0,0,1]C[1,1,0]C[0,1,1]C[1,1,1]C[1,0,1]

=C[i,j,k]=

C[0,0,0]



Prob(c(x)=1)

Prob(c(x)=x)

Prob(c(x)=1+x)

Prob(c(x)=0)

Prob(c(x)=1+x+x )

Prob(c(x)=x )2

Prob(c(x)=x+x )2

2

Prob(c(x)=1+x )2

C =

C[1,0,0]C[0,1,0]C[0,0,1]C[1,1,0]C[0,1,1]C[1,1,1]C[1,0,1]

=C[i,j,k]=

C[0,0,0]



Prob(c(x)=1)

Prob(c(x)=x)

Prob(c(x)=1+x)

Prob(c(x)=0)

Prob(c(x)=1+x+x )

Prob(c(x)=x )2

Prob(c(x)=x+x )2

2

Prob(c(x)=1+x )2

C =

C[1,0,0]C[0,1,0]C[0,0,1]C[1,1,0]C[0,1,1]C[1,1,1]C[1,0,1]

=C[i,j,k]=

C[0,0,0]



Prob(c(x)=1)

Prob(c(x)=x)

Prob(c(x)=1+x)

Prob(c(x)=0)

Prob(c(x)=1+x+x )

Prob(c(x)=x )2

Prob(c(x)=x+x )2

2

Prob(c(x)=1+x )2

C =

C[1,0,0]C[0,1,0]C[0,0,1]C[1,1,0]C[0,1,1]C[1,1,1]C[1,0,1]

=C[i,j,k]=

C[0,0,0]


Fast Fourier Transform applied to Tensors

Expression of the Fourier Transform

◦C= F(C) = C ⊗1 F ⊗2 F . . .⊗p F F =

»1 11 −1

–where ⊗k denotes the tensor product in the k -th dimension of the tensor C(i1, . . . , ip).

in the k -th dimension, ∀ (i1, . . . , ik−1, ik+1, . . . , ip) ∈ {0, 1}p−1

◦C (i1, . . . , ik−1, 0, ik+1, . . . , ip) = C(i1, . . . , ik−1, 0, ik+1, . . . , ip) + C(i1, . . . , ik−1, 1, ik+1, . . . , ip)◦C (i1, . . . , ik−1, 1, ik+1, . . . , ip) = C(i1, . . . , ik−1, 0, ik+1, . . . , ip)− C(i1, . . . , ik−1, 1, ik+1, . . . , ip)

for the Fourier Transform in one dimension, we perform 2p = q operations,

the total number of operations for F(.) is then p 2p = q log(q) operations:⇒ Fast Fourier Transform


Illustration of the FFT in multiple dimensions

1 1

1 −1

1 1

1 −1

1 1

1 −1

11

1

−1

GF(4) : 2 dimensions

GF(8) : 3 dimensions

1 1

1 −1


Belief Propagation Decoding Steps in the FourierDomain

F F F F F F F F

Π

Information Symbols

Fourier Tranform

Permutation Nodes

Interleaver

Product Node

U

UV

V vppv

pccp

product

permutation

Fourier

product


Belief Propagation in the Log-Domain (1)recursive use of the max∗ operator.

Quantization impacts on the performance are very strong in the Probability Domain

u(k) = logµc → p[k ]

µc → p[0]∀k = 0, . . . , q − 1 v(k) = log

µp → c [k ]

µp → c [0]∀k = 0, . . . , q − 1

Case of dc = 3 and GF(4):

µ3c → p[0] = µ1

p → c [0]µ2p → c [0] + µ1

p → c [1]µ2p → c [1] + µ1

p → c [2]µ2p → c [2] + µ1

p → c [3]µ2p → c [3]

µ3c → p[1] = µ1

p → c [0]µ2p → c [1] + µ1

p → c [1]µ2p → c [0] + µ1

p → c [2]µ2p → c [3] + µ1

p → c [3]µ2p → c [2]

µ3c → p[2] = µ1

p → c [0]µ2p → c [2] + µ1

p → c [2]µ2p → c [0] + µ1

p → c [1]µ2p → c [3] + µ1

p → c [3]µ2p → c [1]

µ3c → p[3] = µ1

p → c [0]µ2p → c [3] + µ1

p → c [0]µ2p → c [3] + µ1

p → c [1]µ2p → c [2] + µ1

p → c [2]µ2p → c [1]


Belief Propagation in the Log-Domain (2)Limitations

After some manipulations:

u3(1) = max ∗(v1(1) , v2(1) , v1(2) + v2(3) , v1(3) + v2(2))− K

u3(2) = max ∗(v1(2) , v2(2) , v1(1) + v2(3) , v1(3) + v2(1))− K

u3(3) = max ∗(v1(3) , v2(3) , v1(1) + v2(1) , v1(2) + v2(1))− K

K = max ∗(0 , v1(1) + v2(1) , v1(2) + v2(2) , v1(3) + v2(3))

The number of max ∗ operators grows in O(q2),

Its a recursive implementation: approximations (e.g. use of max instead of max ∗, smallLUT) become rapidly catastrophic,

Log-Domain implementation and the FFT complexity reduction O(q2)→ O(q log(q)) arenot compliant.


Conclusion on Non-Binary Belief PropagationDecoding

The bottleneck of the decoder complexity is the check node update

1 Belief Propagation in the Time/Probability-Domain,[Davey,1998] M. DAVEY AND D.J.C. MACKAY, “LOW DENSITY PARITY CHECK CODES OVER GF(Q)”, IEEE communication

letter, VOL. 2, PP 165–167, JUNE 1998

2 Belief Propagation in the Time/Log-Domain (limits to GF(16)),[Wymeersch,2004] H. WYMEERSCH, H. STEENDAM AND M. MOENECLAEY, “LOG DOMAIN DECODING OF LDPC CODES

OVER GF (q)”, Proceedings of IEEE ICC, PARIS, FRANCE, JUNE 2004.

3 Belief Propagation in the Frequency/Probability Domain,[Davey,1998] M. DAVEY AND D.J.C. MACKAY, “LOW DENSITY PARITY CHECK CODES OVER GF(Q)”, IEEE communication

letter, VOL. 2, PP 165–167, JUNE 1998

[Barnault,2003] L. BARNAULT AND D. DECLERCQ, “FAST DECODING ALGORITHM FOR LDPC CODES OVER GF (2q )”,

Proceedings of IEEE Information theory workshop, PARIS, FRANCE, MARCH, 2003.

4 Belief Propagation in the Frequency/Log-Domain (partially - limits to GF(16)),[Song,2003] H. SONG AND J.R. CRUZ, “REDUCED-COMPLEXITY DECODING OF Q-ARY LDPC CODES FOR MAGNETIC

RECORDING”, IEEE Transactions on Magnetics, VOL. 39(2), MARCH 2003


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