1.3 General notation 27

When the parity-check matrix of the code, whose graph parameters are λ(x) **and** ρ(x), is

full rank, then those two quantities are related to the code rate by:

R = 1 −

∑ dcmax

j=2

ρ j /j

∑ dvmax

i=2

λ i /i

(1.2)

There is also a dual parametrization of the previous one, referred to as nodewise parametrization

[10]:

• Polynomial associated to data nodes:

˜λ(x) =

d∑

vmax

i=2

˜λ i x i−1

where ˜λ i is the proportion of degree i variable nodes.

• Polynomial associated to check nodes:

˜ρ(x) =

d∑

cmax

j=2

˜ρ j x j−1

where ˜ρ j is the proportion of degree j check nodes.

The transitions from one parametrization to another are given by:

˜λ i = ∑

λ i/i

k λ k/k

λ i =

i˜λ i

∑

, ˜ρ j = ∑

ρ j/j

k ρ k/k

k k˜λ , ρ j = j˜ρ j

∑

k k k˜ρ k

(1.3)

Thus, a ensemble of irregular **LDPC** **codes** is parametrized by (N, λ(x), ρ(x)). The regular

case is a particular case of this parametrization where λ(x) **and** ρ(x) are monomials.

Figure 1.2 is a graphical representation for this kind of code.

1.3 General notation

Throughout the thesis, vectors are denoted by boldface notations, e.g. x. R**and**om variables

are denoted by upper-case letters, e.g. X **and** their instantiations in lower-case,

e.g. x. The characterization **and** the optimization of non-binary **LDPC** **codes** are based

on DE equations, assuming that the **codes** are decoded using **iterative** BP [31]. An important

difference between non-binary **and** binary BP decoders is that the former uses

multidimensional vectors as messages, rather than scalar values. There are two possible

representations for the messages: plain-density probability vectors or Log-Density-Ratio