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.