40 Chapitre 1 : Introduction to binary and non-binary LDPCcodes 1.5.5 The stability condition Also obtained in , the stability condition, introduced in , 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) LDPCcodes 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  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) LDPCcodes 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) LDPCcodes 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))
1.6 Other design techniques 41 d c = 4 d c = 5 d c = 6 d c = 7 d c = 8 q = 4 2.56 0.95 0.66 0.52 0.48 q = 64 0.76 0.53 0.51 0.58 0.90 q = 256 0.65 0.54 0.59 0.79 1.27 Table 1.1 : Thresholds of GF(q) LDPC code ensembles with constant check degree d c and code rate one half, optimized with EXIT charts on the BIAWGN channel. The maximum variable degree allowed in the optimization procedure is d vmax = 30. Thresholds are given in term of the SNR E b N 0 in dB, and are obtained using the Gaussian approximation. We briefly illustrate what can be the results of such an optimization, and how it allows to find again known results from the literature. Table 1.1 gathers some thresholds obtained by optimization of the irregularities for various field order and check degrees. These thresholds are hence computed by EXIT charts, with a Gaussian approximation. The code rate is one-half. Since degree-1 variable nodes are not allowed in the optimization process, the code ensemble with d c = 4 is regular with d v = 2. In this case, we observe that the threshold is better for higher order field. This observation ca, be identified to the following claim of Hu and Eleftheriou in . They considered GF(q) random ensembles defined by the probability p that an element of the parity-check matrix be non-zero. When p is very low, the binary random ensemble defined by p is far away from the Shannon equiprobable random ensemble. In this case, they illustrated that the Hamming weight distribution of the GF(q) random ensemble tends to the binomial distribution as q increases. As an additional example, EXIT curves of regular (2,4) codes in GF(2), GF(8) and GF(256) are plotted on figure 1.5, confirming results of the first column of Table 1.1: the curve of GF(256) is the only one for which the tunnel is open. 1.6 Other design techniques 1.6.1 Finite length design of LDPCcodes We do not detail the design techniques relative to finite length design of LDPCcodes, but just mention some works on that. First, the PEG construction has been proposed in  to build the graph of codes, given the irregularities. This technique has recently been improved . For non-binary LDPCcodes, additionally to the PEG construction, Poulliat et al.  expressed a criterion and developed a technique for cancelling cycles of GF(q) LDPCcodes by an appropriate choice of the non-zero values.