Hybrid LDPC codes and iterative decoding methods - i3s

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Hybrid LDPC codes and iterative decoding methods - i3s

2.1 The class of hybrid LDPC codes 53

• Check node update: Consider a check node c and a variable node v. Let {v 1 , . . ., v dc−1}

be the set of all variable nodes connected to c, except v. Let G be the Cartesian product

group of the groups of the variable nodes in {v 1 , . . .,v dc−1}. For all a ∈ G(q v )

l (t)

cv (a) = µ cv


(b 1 ,...,b dc−1 )∈G:

L dc−1

i=1 Av i cb i=A vca

d∏

c−1

n=1

r (t)

v nc(b n ) (2.5)

where µ cv is a normalization factor, and the ⊕ operator highlights that the addition

is performed over G(q c ), the group of the row corresponding to c, as defined in

Section 2.1.4.

• Stopping criterion: Consider a variable node v. Let {c 1 , . . .,c dv } be the set of all

check nodes connected to v. Equation (2.6) corresponds to the decision rule on

symbols values, at iteration t:

ˆx (t)

v

= arg maxr (0)

a

v (a) d v


n=1

l c (t)

nv (a) . (2.6)

Variable and check node updates are performed iteratively until the decoder has

converged to a codeword, or until the maximum number of iterations is reached.

It is possible to have an efficient Belief propagation decoder for hybrid LDPC codes.

As mentioned in [11][45], the additive group structure possesses a Fourier transform, so

that efficient computation of the convolution can be done in the Fourier domain. One

decoding iteration of BP algorithm for hybrid LDPC codes, in the probability domain

with a flooding schedule, is composed of:

• Step 1 Variable node update in G(q j ) : pointwise product of incoming messages

• Step 2 Message extension G(q j ) → G(q i ) (see definition 6)

• Step 3 Parity-Check update in G(q i ) in the Fourier domain

◦ FFT of size q i

◦ Pointwise product of FFT vectors

◦ IFFT of size q i

• Step 4 Message truncation from G(q i ) → G(q j ) (see definition 7)

Although we do not focus on low-complexity decoders, it is important to note that hybrid

LDPC codes are compliant with reduced complexity non-binary decoders which have

been presented recently in the literature [46, 47]. In particular, [46] introduces simplified

decoding of GF(q) LDPC codes and shows that they can compete with binary LDPC

codes even in terms of decoding complexity.

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