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-G**LDPC** **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.