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

where the components of r(r (0) ,l (1) , ...,l (d i−1) ) are defined as in equation (1.7).

P(R (0) = r (0) ) **and** P(L t = l (n) ) are obtained recursively from the previous iterations

of belief propagation. The probability function of R t is now obtained by

P(R t = x) =

v∑

i=1

λ i P(R (d i)

t = x) (1.16)

Theoretically, the above algorithm is sufficient to compute the desired densities. If one

consider that the all-zero codeword has been sent, which is relevant in the context presented

in section 1.5.2, it is easy to compute the probability of doing an error when applying

decision rule (1.11) to a probability vector, e.g. R t .

As aforementioned, Richardson **and** Urbanke in [11] proved a concentration theorem

that states that as the block length N approaches infinity, the bit error rate at iteration t

converges to a similarly defined probability of error. It has been shown in [48] that the

concentration theorem can be applied to frame error rate of GF(q) **LDPC** code ensembles.

In this way, the performance of correction of a GF(q) **LDPC** ensemble, as defined

in Section 1.2, can be exactly predicted. However, in practice, a major problem is the fact

that the quantities of memory required to store the probability density of a q-dimensional

message grows exponentially with q. That is why it is important to look for a computationally

easier way to follow the message densities in order to be able to predict the code

ensemble performance.

As mentioned in [48], if Pe t = P e(R t ) is a sequence of error probabilities produced

by density evolution, then Pe t is a non-increasing function of t. The demonstration is

similar to the proof of theorem 7 in [10]. This non-increasing property ensures that the

sequence corresponding to density evolution by iterating between equation (1.15) **and**

equation (1.16) converges to a fixed point. Implementing the density evolution allows to

check whether not this fixed point corresponds to the zero error probability, which means

that the **decoding** in the infinite codeword length case has been successful. That is why

GF(q) **LDPC** **codes**, like binary **LDPC** **codes**, are said to have a threshold behavior.

In the sequel, it is explained why such an implementation is not possible for GF(q)

**LDPC** **codes**, unlike their binary counterparts. The proposed method from the literature

aims at approximating the densities, thereby simplifying the recursion **and** making possible

the study of its convergence. This method is presented in the next section.

1.5.4 Approximation of message densities by only one scalar parameter

Analogously to the binary case, a Gaussian approximation of the message densities is

used to be able to practically track these densities **and** predict error probabilities of GF(q)

**LDPC** code ensembles. To reduce the densities to only one scalar parameter, things are

a little more elaborated than in the binary case since messages are no more scalars but

q-sized probability vectors, which entails that the densities are multi-variate densities.