74 Chapitre 2 : HybridLDPC Codes As in the previous section, we consider the binary images of cycles as component codes. Let Hc k be the binary image of the k-th stored cycle. Since we consider (2, d c ) codes, if some columns of Hc k are linearly dependent, so will be the columns of H (see  for more details). This means that a codeword of a cycle is also a codeword of the whole code. The proposed approach is hence to avoid low weight codewords by properly choosing the nonzero clusters implied in the cycles, so that no codeword of low-weight is contained in the cycles. This is achieved by ensuring that the binary matrices Hc k corresponding to the cycles have full column rank. The iterative procedure that we use in this optimization step is essentially the same as the one depicted in . In each step of the iterative procedure, we change the values of a limited number of non-zero clusters in order to maximize the number of cycle component codes Hc k which are full rank. Thus, the matrix of a cycle should be full rank to cancel the cycle. Contrarily to classical nonbinary LDPCcodes for which the matrix of a cycle is squared, the matrix of a cycle of a hybrid LDPC code is rectangular, with more rows than columns. This means that we will have more degrees of freedom to cancel the cycles in hybrid LDPCcodes. HybridLDPCcodes are therefore well-suited to this kind of finite length optimization procedure. 2.5 Results 2.5.1 Rate one-half codes Optimized distributions: thresholds and finite length performance Table 2.2 : Nodewise distributions of the hybrid LDPCcodes used for the finite length simulations. HybridLDPC code 1 HybridLDPC code 2 ˜Π(i = 2, q k = 32) 0.3950 ˜Π(i = 2, q k = 64) 0.4933 0.2050 ˜Π(i = 2, q k = 256) 0.4195 0.4000 ˜Π(i = 6, q k = 64) 0.0772 ˜Π(i = 6, q k = 256) 0.0100 ˜Π(j = 5, q l = 256) 0.5450 1 ˜Π(j = 6, q l = 256) 0.4550 ( ) ⋆ E b N o (dB) 0.675 0.550 Based on the optimization methods presented in section 2.3.3, we first present some code distributions and corresponding thresholds for code rate one-half, as given in table 2.2. For all the presented results, the channel is the BIAWGN channel with( BPSK ) ⋆ E modulation. Thresholds are computed by Monte-Carlo simulations. In table 2.2, b N o denotes the decoding convergence thresholds of the distributions in each column. The
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 , 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 LDPCcodes, and that is why we have focused on low connection degrees. For such low degrees, we are going to see that hybrid LDPCcodes have very good finite length performances, but they do not approach the capacity as close as multi-edge type LDPCcodes do. This is due to the adopted detailed representation Π which cannot handle degree one variable nodes. However, it would be an interesting perspective to switch from the detailed representation to a multi-edge type representation for LDPCcodes. This will certainly enable to get capacityapproaching distributions with low connection degrees. Indeed, it has been shown in  that introducing degree-1 variable nodes in non-binary LDPCcodes 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 LDPCcodes number 1 and 2 compared with Quasi-cyclic Tanner codes from , irregular LDPCcodes from , a GF(256) LDPC code, a protograph based LDPC code from  and a multi-edge type LDPC code from  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 LDPCcodes have been built with the random PEG algorithm described in . 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.