Hybrid LDPC codes and iterative decoding methods - i3s

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Hybrid LDPC codes and iterative decoding methods - i3s

42 Chapitre 1 : Introduction to binary and non-binary LDPC codes

1

EXIT curves in three Galois fields

0.9

0.8

0.7

0.6

x out 0.5

0.4

0.3

0.2

0.1

x out

=x in

GF(2)

GF(8)

GF(256)

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x in

Figure 1.5 : EXIT curves of (2,4) GF(2), GF(8) and GF(256) regular codes. The SNR

is 0.7dB.

1.6.2 Structured ensembles

As pointed out in the introduction, a very efficient way to design code ensembles with

iterative decoding performance close to capacity and low error-floor, is to choose the permutations

in a structured way. Indeed, the aforementioned representation of LDPC codes

defines only the connection degrees of variable and check nodes, but any variable node can

be connected to any check node. A structured code ensemble has a representation which

defines to which type of check nodes each type of variable node can be connected. LDPC

codes with a detailed representation have been introduced in [52]. Some structured code

ensembles have been under the scope of many studies these last years: irregular repeataccumulate

(IRA) codes [25], protograph-based LDPC codes [26] and multi-edge type

LDPC [27]. The design of good D-GLDPC codes have been addressed for the BEC in

[29]. These techniques lead to codes with good code properties in terms, for instance, of

girth of the bipartite graph and possibility to perform the encoding procedure efficiently.

For a comprehensive survey of the design of those kinds of LDPC codes, we refer the

reader to [53].

1.7 Proof of theorems in Chapter 1

Lemma 3.Let P e (l) (x) denote the conditional error probability after the l-th BP decoding

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

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

Proof: The proof has the same structure as the proof of Lemma 1 in [11]. Thus, we

do not detail it, but instead refer the reader to [11] and rather only give the key elements.

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