68 Chapitre 2 : HybridLDPC Codes (i, q k ) (2, 64) (2, 128) (2, 256) (3, 64) (3, 128) (3, 256) (4, 64) (4, 128) (4, 256) ˜Π(j, l) (j, q l ) (5, 64) 0.0073 × × 0 × × 0 × × 0.0086 (5, 128) 0 0.0089 × 0 0 × 0.0080 0.0175 × 0.0405 (5, 256) 0.0003 0.0290 0.0001 0.0226 0 0 0 0.0001 0 0.0614 (6, 64) 0.0087 × × 0.0470 × × 0.0554 × × 0.1091 (6, 128) 0.0367 0.0003 × 0.0521 0.0063 × 0.0218 0.0931 × 0.2065 (6, 256) 0.4248 0.0197 0.0043 0.0851 0.0021 0.0101 0.0042 0.0151 0.0193 0.5739 ˜Π(i, k) 0.5916 0.0717 0.0055 0.1707 0.0069 0.0083 0.0554 0.0779 0.0120 Table 2.1 : Distribution Π(i,j,k,l) of a hybrid LDPC code ensemble with code rate onehalf and threshold 0.1864 dB under Gaussian approximation. The marginals ˜Π(i,k) and ˜Π(j,l) correspond to the proportions of variable nodes of type (i,k) and check nodes of type (j,l), respectively. When a proportion is forced to zero by the sorting constraint, × is put in the box. charts, the computation time for the cost function, i.e., for the threshold, is much higher than in the binary case too. Result of the optimization It results from the optimization with DE that distributions with best thresholds are not obtained for a majority of binary variable and check nodes. It is worthy to recall that only small connection degrees are allowed for check nodes (5 or 6). Also, as mentioned in section 2.1.5, the detailed representation adopted in this work is less general than the multiedge type representation . Indeed, it is possible to consider proportions of different (i, k) type punctured symbols, but it is not possible to assume degree one variable nodes because we cannot describe check nodes with exactly one edge to such a variable. This is the reason why we logically do not get back the code distributions of multi-edge type LDPCcodes , i.e., binary LDPCcodes with low connection degrees and thresholds close to capacity. Instead, we obtain distributions Π with very low connection degrees (2 to 4) and very good thresholds under the above discussed Gaussian approximation, when only high order groups (G(16) to G(256)) are allowed. This is in agreement with the results of . An example of such a resulting distribution is given in table 2.1. Firstly, we see from this table that the optimization procedure puts a maximum of powerful component codes (or "generalized codes", see section 2.1.3), i.e. variable nodes in the smallest order group (G(64)) connected to check nodes in the highest order group (G(256)). Secondly, the variable nodes in a high order group tend to correspond to poor component codes, and hence, higher connection degrees are affected to this type of variable nodes in order to have a code length high enough to balance the high K, which is in turns log 2 (q k ). This interpretation can also be made in terms of code doping [1, 61]. Graph construction We now discuss the graph construction of such a hybrid LDPC code: how to build a graph satisfying the detailed representation, i.e., where all check nodes cannot be connected to
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 , in the same way as multi-edge type LDPCcodes 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 LDPCcodes by using equation presented in  with functions J v (·, ·, ·) and J c (·, ·), or hybrid multi-edge LDPCcodes 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 randomly 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 .