54 Chapitre 2 : **Hybrid** **LDPC** Codes

2.2 Asymptotic analysis of hybrid **LDPC** code ensembles

In this section, we describe the density evolution analysis for hybrid **LDPC** **codes**. Density

evolution is a method for analyzing **iterative** **decoding** of code ensembles. In this section,

we first prove that, on a binary input symmetric channel (BISC), we can assume that the

all-zero codeword is transmitted because the hybrid decoder preserves the symmetry of

messages, which entails that the probability of error is independent of the transmitted

codeword.

We express the density evolution for hybrid **LDPC** **codes**, **and** mention the existence of

fixed points, which can be used to determine whether or not the **decoding** of a given hybrid

**LDPC** code ensemble is successful for a given SNR, in the infinite codeword length case.

Thus, convergence thresholds of hybrid **LDPC** **codes** are similarly defined as for binary

**LDPC** **codes** [10]. However, as for GF(q) **LDPC** **codes**, the implementation of density

evolution of hybrid **LDPC** **codes** is too computationally intensive, **and** an approximation

is needed.

Thus, we derive a stability condition, as well as the EXIT functions of hybrid **LDPC**

decoder under Gaussian approximation, with the goal of finding good parameters for having

good convergence threshold. We restrict ourselves to binary input symmetric channels,

but all the demonstrations can be extended to non-symmetric channels by using, e.g.,

a coset approach [48].

2.2.1 Symmetry of the messages

The definitions **and** properties induced by channel symmetry have been developed in section

1.5.2. All the lemmas carry unchanged over the hybrid **LDPC** ensemble.

Lemma 5 Let P e (t) (x) denote the conditional error probability after the t th BP **decoding**

iteration of a hybrid **LDPC** code, assuming that codeword x was sent. If the channel is

symmetric, then P e (t) (x) is independent of x.

The proof of this lemma is provided in Section 2.7. For Lemma 4, we add the two following

lemmas to the proof.

Lemma 6 If W is a symmetric LDR r**and**om vector, then its extension W ×A , by any

linear extension A with full rank, remains symmetric. The truncation of W by the inverse

of A, denoted by W ×A−1 , is also symmetric.

Proof of lemma 6 is given in section 2.7. The specificity of hybrid **LDPC** **codes** lies in

function nodes on edges. Thus, when hybrid **LDPC** **codes** are decoded with BP, both data

pass **and** check pass steps are the same as classical non-binary **codes** **decoding** steps. Since

these steps preserve symmetry [10], lemma 6 ensures that the hybrid decoder preserves

the symmetry property if the input messages from the channel are symmetric.