Hybrid LDPC codes and iterative decoding methods - i3s

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Hybrid LDPC codes and iterative decoding methods - i3s

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

thousands 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 candidates 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 handled

to define the code ensemble.

On the other hand, 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-GLDPC 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.

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