# Hybrid LDPC codes and iterative decoding methods - i3s

Hybrid LDPC codes and iterative decoding methods - i3s

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 random 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.

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