Hybrid LDPC codes and iterative decoding methods - i3s

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Hybrid LDPC codes and iterative decoding methods - i3s

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 random vectors,

but all the definitions and proofs also apply to LDR random vectors.

Definition 8 A random 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 random vectors Y ×A−1 and Y ×B−1 are identically distributed.

Lemma 7 If a random 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 random vector of size q k , we define the random-extension of size

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

independent of X.

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

a random 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, randomtruncations,

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

node (except the one from the channel).

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