2.5 Results 75

hybrid **LDPC** code number 1 is obtained by the method presented in section 2.3.3, when

setting the check node connection profile, all check nodes are in G(256), putting all the

redundancy variable nodes in G(256) **and** information variables in G(64). The connection

profiles for these two groups are then optimized with d vmax = 10. As already observed

in section 2.3.2, variable nodes in the highest order group are affected with as much high

connection degrees as possible, to balance the poor generalized component code. The

hybrid **LDPC** code number 2 is obtained by the method presented in section 2.3.3, when

setting the graph connections to be regular with constant variable degree d v = 2 **and** constant

check degree d c = 5. Although these thresholds are not better than the one of a

regular (d v = 2, d c = 4) GF(256) **LDPC** code, which is 0.5 dB [63], we can exhibit hybrid

**LDPC** distributions with better thresholds than the one of a regular (d v = 2, d c = 4)

GF(256) **LDPC** code, by allowing higher connection degrees. However, our purpose is

to point out the good finite length performance of hybrid **LDPC** **codes**, **and** that is why

we have focused on low connection degrees. For such low degrees, we are going to

see that hybrid **LDPC** **codes** have very good finite length performances, but they do not

approach the capacity as close as multi-edge type **LDPC** **codes** do. This is due to the

adopted detailed representation Π which cannot h**and**le degree one variable nodes. However,

it would be an interesting perspective to switch from the detailed representation to a

multi-edge type representation for **LDPC** **codes**. This will certainly enable to get capacityapproaching

distributions with low connection degrees. Indeed, it has been shown in [30]

that introducing degree-1 variable nodes in non-binary **LDPC** **codes** makes the **decoding**

threshold getting closer to the theoretical limit. Modifying the representation of hybrid

**LDPC** code ensemble is therefore very interesting for future work. We only present in

table 2.2 the thresholds of the distributions which are used for the following finite length

simulations.

Figure 2.5 represents some frame error rate (FER) curves for different **codes**, all with

K = 1024 information bits **and** code rate one-half. Figure 2.5 shows the performance

curves of hybrid **LDPC** **codes** number 1 **and** 2 compared with Quasi-cyclic Tanner **codes**

from [1], irregular **LDPC** **codes** from [10], a GF(256) **LDPC** code, a protograph based

**LDPC** code from [26] **and** a multi-edge type **LDPC** code from [27] with code length

N = 2560 bits (K = 1280 information bits). This code has been specially design for low

error-floor. The graphs of the binary, non-binary **and** hybrid **LDPC** **codes** have been built

with the r**and**om PEG algorithm described in [51].

We see that the hybrid **LDPC** code number 1 has performance very close to the protograph

based **LDPC** code, while the hybrid **LDPC** code number 2 has better waterfall

performance than the protograph based **LDPC** code but higher error floor. Also, the hybrid

**LDPC** code number 2 has a worse waterfall region than a regular (dv = 2, dc = 4)

GF(256) **LDPC** code, but a better error floor. These two observations are clues to investigate

a finite length optimization of the hybrid **LDPC** code, in order to refine the structure

of the graph to achieve better error floor performance.