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