2.3 Distributions optimization 69

any variable nodes.

The first solution is to modify the PEG algorithm to take into account the structure

specificity of such a hybrid **LDPC** ensemble where the global permutation is made of

various sub-interleavers. However, we did not have enough time to do this.

Another way is to build the graph thanks to the protograph method [53], in the same

way as multi-edge type **LDPC** **codes** are built. However, building the protograph of a

hybrid **LDPC** code fulfilling the detailed representation resulting from the optimization,

without additional restrictions on the detailed parametrization, can be quickly arduous.

We did not have enough time to investigate this method.

Moreover, since the best thresholds resulting from DE have been observed for high

order groups, this has been a hint to assume that we will not have an important loss in

the achievable thresholds when restricting the detailed representation in these conditions.

This restriction consists in considering all check nodes in the same group **and** with connection

degrees independent of the variable nodes they are connected. This allows to

switch from a non-linear optimization to a linear optimization, which is the topic of the

following section.

Finally, it is worthy to note that all the presented tools, i.e. decoders **and** EXIT charts,

may be used for optimization of hybrid protograph based **LDPC** **codes** by using equation

presented in [53] with functions J v (·, ·, ·) **and** J c (·, ·), or hybrid multi-edge **LDPC** **codes**

provided that the tools are adapted to the multi-edge type representation. However, some

problems would have to be solved for the definition of such a code ensemble, e.g. can the

linear maps be r**and**omly chosen on each edge of the code graph resulting from lifting, or

do they have to be the same as the one defining the protograph ?

2.3.3 Optimization with mono-dimensional EXIT charts

In this part, we consider the optimization of hybrid **LDPC** code ensembles with all check

nodes in the same group G(q l ) **and** with connection degrees independent of the variable

nodes to which they are connected. We present how general equations (2.12) turns into

mono-dimensional EXIT charts, **and** how this allows the use of linear programming for

optimization. Let x (t)

e denote the averaged mutual information of extended messages. It

is expressed in terms of the mutual information x vc

(i,k)(t) of messages going out of variable

nodes of degree i in G(q k ), by simplification of equation (2.10):

x (t)

e

= 1 −

1

log 2 (q l )

∑

From equation (2.10), we can see that, for any (i, k, l):

i,k

Π(i, k) log 2 (q k )(1 − x (i,k)(t)

vc ) .

lim

t→∞ x(i,k)(t) vc,l

= 1 ⇔ lim x vc (i,k)(t) = 1

t→∞

**and** then the successful **decoding** condition (2.15) reduces to

lim

t→∞ x(t) e = 1 .