Hybrid LDPC codes and iterative decoding methods - i3s

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Hybrid LDPC codes and iterative decoding methods - i3s

2.3 Distributions optimization 71

Due to the fact that we a priori set the group orders of variable nodes necessarily

equal or lower than check nodes group orders they are connected, the rate of the hybrid

bipartite graph, whose nodes are in different order groups, is higher than the code rate

(i.e., the actual rate of the transmission). Setting the proportion of variable nodes in

G(q k ) for all k also sets the rate of the graph, which becomes the target graph rate in

the optimization procedure. From the target code rate R target , we can compute the target

graph rate, denoted by R graph by:

R graph =


R target

α k log 2 (q k )

k=info1,info2


R target

α k log 2 (q k ) + 1 − R target

α red log 2 (q red )

k=info1,info2

(2.21)

The result of the optimization is finally the set of the three profiles Π(i, k), ∀(i, k) ∈

[1, d vmax ] × [red,info1,info2], for which the following constraints are fulfilled at lowest

SNR:


Proportion constraint: ∀i = 2... d vmax , Π(i, red) + Π(i, info1) + Π(i, info2) = 1

Code rate constraint:

∀k = red,info1,info2,

i


i

Π(i,k)

i

=

α k


1 − R graph

j

Π(j)

j

Sorting constraint: Π(i,j,k,l) = 0, ∀(i,j,k,l) such that q k > q l (2.22)

Successful decoding condition:

x (t+1)

e

= F(x (t)

e ,Π(i,k),σ2 ) > x (t)

e

Set connexion profile, open group-order profile

Another way to optimize hybrid LDPC ensembles is to set the connection profile and optimize

the group orders of variable nodes. As in the previous section, we set the check

node parameters (group order G(q red ) and connection profile), independently of the variable

nodes parameters. This time, the decomposition of Π(i, j, k, l) is:

Π(i, j, k, l) = Π(i, k|l)Π(j)Π(l)

Similarly to equation (2.19), we aim at optimizing several group order profiles, as many as

the number of different variable node connection degrees. In a finite length performance

purpose, we start from an ultra-sparse Tanner graph with a regular connection profile (e.g.

(d v = 2, d c = 3)). Hence the previous decomposition falls into:

Π(i, j, k, l) = δ(i, d v )δ(j, d c )Π(k)δ(l, red)

Since the group order profile of the redundancy is set, the result of the optimization will

be the group order profiles of information variable nodes. We denote by I the indexes of

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