40 Chapitre 1 : Introduction to binary **and** non-binary **LDPC** **codes**

1.5.5 The stability condition

Also obtained in [48], the stability condition, introduced in [10], is a necessary **and** sufficient

condition for the error probability to converge to zero, provided it has already

dropped below some value. This condition must be satisfied by the SNR corresponding

to the threshold of the code ensemble. Therefore, ensuring this condition, when implementing

an approximation of the exact density evolution, helps to have a more accurate

approximation of the exact threshold.

Given a ensemble of GF(q) **LDPC** **codes** defined by (λ, ρ), the following ensemble

parameter is defined:

Ω = ∑ ρ j (j − 1) (1.21)

j

For a given memoryless symmetric output channel with transition probabilities p(y|x),

the following channel parameter is also defined:

∆ = 1

q−1

∑

∫ √p(y|i)p(y|0)dy

(1.22)

q − 1

i=1

Theorem 2 [48] Consider a given GF(q) **LDPC** ensemble parametrized by (λ, ρ). Let

P t e = P e (R t ) denotes the average error probability at iteration t under density evolution.

• If Ω ≥ 1 ∆ , then there exists a positive constant ξ = ξ(λ, ρ, P 0) such that P t e > ξ for

all iterations t.

• If Ω < 1 , then there exists a positive constant ξ = ξ(λ, ρ, P ∆ 0) such that if Pe t < ξ

at some iteration t, then Pe t approaches zero as t approaches infinity.

1.5.6 Design example of GF(q) **LDPC** code ensemble on BIAWGN

channel

Optimization is performed for the BIAWGN channel. The goal of the optimization with

EXIT charts is to find a good ensemble of GF(q) **LDPC** **codes** with the lowest convergence

threshold, under a Gaussian approximation. This means that we look for the

parameters (λ(x), ρ(x)) of the ensemble of GF(q) **LDPC** **codes** with lowest convergence

threshold.

Let us denote the code rate R, **and** the target code rate R target . The optimization

procedure [10, 50] consists in finding (λ(x), ρ(x)) which fulfills the following constraints

at the lowest SNR:

Code rate constraint: R = R target (see equation (1.2))

∑

Proportion constraint: λ i = 1 **and** ∑ ρ j = 1

i

j

Successful **decoding** condition: x (t+1)

vc > x (t)

vc (see equation (1.20))

Stability constraint: Ω∆ < 1 (see equations (1.21) **and** (1.22))