2.7 Proofs of theorems in Chapter 2 81

analysis is performed on the BIAWGN channel, whereas studies of generalized **LDPC**

**codes** usually consider the BEC [30, 29]. In order to optimize the distributions of hybrid

**LDPC** ensembles, we have investigated how to project the message densities on only

one scalar parameter using a Gaussian approximation. The accuracy of such an approximation

has been studied, **and** used to lead to two kinds of EXIT charts of hybrid **LDPC**

**codes**: multi-dimensional **and** mono-dimensional EXIT charts. Distribution optimization

allows to get finite length **codes** with very low connection degrees **and** better waterfall

region than protograph or multi-edge type **LDPC** **codes**. Moreover, hybrid **LDPC** **codes**

are well fitted for the cycle cancellation presented in [34], thanks to the specific structure

of the linear maps. The resulting **codes** appear to have, additionally to a better waterfall

region, a very low error-floor for code rate one-half **and** codeword length lower than three

thous**and**s bits, thereby competing with multi-edge type **LDPC**. Thus, hybrid **LDPC** **codes**

allow to achieve an interesting trade-off between good error-floor performance **and** good

waterfall region with non-binary **codes** techniques.

We have also shown that hybrid **LDPC** **codes** can be very good c**and**idates for efficient

low rate coding schemes. For code rate one sixth, they compare very well to existing

Turbo Hadamard or Zigzag Hadamard **codes**. In particular, hybrid **LDPC** **codes** exhibit

very good minimum distances **and** error floor properties.

As future work, it would be of first interest to allow degree one variable nodes in the

representation of hybrid **LDPC** **codes**, by, e.g., adopting a multi-edge type representation

[27]. As shown in [30], this would allow to have better **decoding** thresholds, in particular

for low rate **codes**.

This would give rise to the study **and** optimization, with the same tools, of non-binary protograph

based or multi-edge type **LDPC** **codes**. However, the extension may be theoretically

not completely straightforward as the non-zero values have to be carefully h**and**led

to define the code ensemble.

On the other h**and**, it would be interesting to study hybrid **LDPC** **codes** on other channels.

Let us mention that we made some experiments on an AWGN channel with 16-

QAM modulation. We restricted the connection profile to be regular, in order to not bias

the results by the absence of special allocation on differently protected symbols. Only

two group orders where allowed to avoid correlation between channel LLRs: G(16) **and**

G(256). The optimization of fractions of variable nodes in these two different orders have

been done. The results where slightly degraded compared to a (2, 4) GF(256) **LDPC**

**codes**. A study of these **codes** on the BEC would be also interesting, according to what

has been done for D-G**LDPC** **codes** on the BEC [56].

2.7 Proofs of theorems in Chapter 2

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

iteration of a GF(q) **LDPC** code, assuming that codeword x was sent. If the channel is

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