Hybrid LDPC codes and iterative decoding methods - i3s

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Hybrid LDPC codes and iterative decoding methods - i3s

58 Chapitre 2 : Hybrid LDPC Codes

we have not completed this proof of necessity, and hence do not present the mentioned

intermediate results. Although the necessity of stability condition has not been proved, it

is sufficient for comparing to stability condition of classical binary and non-binary LDPC

codes.

We first note that, for a usual non-binary GF(q) LDPC code, the hybrid stability

condition reduces to non-hybrid stability condition, given by [48], because

is equivalent in this case to

lim g(k,

n→∞ c(k) , Π,G ⊗n (x)) = 0

ρ ′ (1)λ ′ 1

(0)

q k − 1

q k −1


i=1

∫ √p(y|i)p(y|0)dy

< 1

When the transmission channel is BIAWGN, we have

∫ √p(y|i)p(y|0)dy

= exp(−

1

2σ 2n i)

Let ∆ nb be defined by

q

1

k −1


exp(− 1

q k − 1 2σ 2n i)

i=1

with n i , the number of ones in the binary map of α i ∈ G(q). Under this form, we can

prove that ∆ tends to zero as q goes to infinity on BIAWGN channel. This means that

any fixed point of density evolution is stable as q tends to infinity for non-binary LDPC

codes. This shows, in particular, that non-binary cycle-codes, that is with constant symbol

degree d v = 2, are stable if q tends to infinity, and can be used to design efficient coding

schemes if q is large enough [33, 57].

As an illustration, we compare the stability conditions for hybrid LDPC codes with all

variable nodes in G(q) and all check nodes in G(q max ) and for non-binary LDPC codes

defined on the highest order field GF(q max ). For hybrid codes of this kind, we have:

i=1

lim g(k,

n→∞ c(k) , Π,G ⊗n (x)) = 0

is equivalent to

(

)(

q−1

1 ∑

exp(− 1

q − 1 2σ 2n i) Π(i = 2) ∑ j

)

q − 1

Π(j)(j − 1) < 1

q max − 1

An advantage of hybrid LDPC codes over non-binary codes is that a hybrid LDPC

code, with same maximum order group, can be stable at lower SNR.

On figure 2.4, we consider rate one-half non-binary LDPC codes on GF(q), with

q = 2 . . .256, and rate R = 0.5 hybrid LDPC codes of type G(q) − G(q max ), with all

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