56 Chapitre 2 : **Hybrid** **LDPC** Codes

of error probability in terms of the channel parameter for physically degraded channels.

Thus hybrid **LDPC** **codes**, like binary or non-binary **LDPC** **codes**, exhibit a threshold

phenomenon.

Like for GF(q) **LDPC** **codes**, implementing the density evolution for hybrid **LDPC**

**codes** is too computationally intensive. Thus, in the sequel, we present a useful property

of hybrid **LDPC** code ensembles, which allows to derive both a stability condition **and**

an EXIT chart analysis for the purpose of approximating the exact density evolution for

hybrid **LDPC** code ensembles.

2.2.3 Invariance induced by linear maps (LM-invariance)

Now we introduce a property that is specific to the hybrid **LDPC** code ensembles. Bennatan

et al. in [48] used permutation-invariance to derive a stability condition for nonbinary

**LDPC** **codes**, **and** to approximate the densities of graph messages using onedimensional

functionals, for extrinsic information transfer (EXIT) charts analysis. The

difference between non-binary **and** hybrid **LDPC** **codes** lies in the non-zeros elements

of the parity-check matrix. Indeed, they do not correspond anymore to cyclic permutations,

but to extensions or truncations which are linear maps (according to definitions 6

**and** 7). Our goal in this section is to prove that linear map-invariance (shortened by LMinvariance)

of messages is induced by choosing uniformly the extensions. In particular,

LM-invariance allows to characterize message densities with only one scalar parameter.

Until the end of the current section, we work with probability domain r**and**om vectors,

but all the definitions **and** proofs also apply to LDR r**and**om vectors.

Definition 8 A r**and**om vector Y of size q l is LM-invariant if **and** only if for all k **and**

(A −1 , B −1 ) ∈ T k,l × T k,l , the r**and**om vectors Y ×A−1 **and** Y ×B−1 are identically distributed.

Lemma 7 If a r**and**om vector Y of size q l is LM-invariant, then all its components are

identically distributed.

Proof of lemma 7 is given in section 2.7.3.

Definition 9 Let X be a r**and**om vector of size q k , we define the r**and**om-extension of size

q l of X, denoted ˜X, as the r**and**om vector X ×A , where A is uniformly chosen in E k,l **and**

independent of X.

Lemma 8 A r**and**om vector Y of size q l is LM-invariant if **and** only if there exist q k **and**

a r**and**om vector X of size q k such that Y = ˜X.

Proof of lemma 8 is given in section 2.7.3.

Thanks to lemma 6, the messages going into check nodes are LM-invariant in the

ensemble of hybrid **LDPC** **codes** with uniformly chosen extensions. Moreover, r**and**omtruncations,

at check node output, ensures LM-invariance of messages going into variable

node (except the one from the channel).