Hybrid LDPC codes and iterative decoding methods - i3s

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Hybrid LDPC codes and iterative decoding methods - i3s

46 Chapitre 2 : Hybrid LDPC Codes

to an ensemble of p 1 -sized vectors whose elements lie in Z . This is the reason why we

2Z

adopt the fully denomination of these codes as being multi-binary hybrid LDPC codes. In

the remainder, we use a shortcut and refer to them as hybrid LDPC codes.

Let q 1 and q 2 , such that q 1 < q 2 , denote the group orders of a column and of a row of

H, respectively. They will be similarly called variable and check order. Let G(q 1 ) denote

the group of variable j and G(q 2 ) the group of parity-check i. The non-zero elements of

the parity-check matrix are applications which have to map a value in the column group

(variable node group), to a value in the row group (check node group, see figure 2.1). This

is achieved thanks to functions named h ij such that

h ij : G(q 1 ) → G(q 2 )

c j → h ij (c j )

Hence, an hybrid parity-check equation is given by


h ij (c j ) = 0 in G(q 2 ) (2.2)

j

We notice that, on equation (2.1) as well as on equation (2.2), the additive group

structure defines the local constraints of the code. Moreover, as mentioned in [11], and

deeply studied in, e.g., [45], the additive group structure possesses a Fourier transform,

whose importance for the decoding is pointed out in section 2.1.7.

Since the mapping functions h ij can be of any type, the class of hybrid LDPC codes

is very general and includes classical non-binary and binary codes.

c 1 ∈ G(q 1 ) c 2 ∈ G(q 2 )

c 3 ∈ G(q 3 )

2 4 8

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

q 1 ≤ q 2 ≤ q 3

h i1(c 1) h i2(c 2) h i3(c 3)

8

8

8

parity-check in G(q 3 )

h i1 (c 1 ) + h i2 (c 2 ) + h i3 (c 3 ) = 0, h ij (c j ) ∈ G(q 3 )

defines a component code in the group G = G(q 1 ) × G(q 2 ) × G(q 3 )

Figure 2.1 : Factor graph of parity-check of an hybrid LDPC code.

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