132Chapitre 4 : Two-Bit Message Passing Decoders for LDPC Codes Over the Binary Symmetric Channel correction capability for higher rate codes than one-bit Gallager type decoding. We have computed thresholds for various two-bit decoders, and shown that the decoder for which the previous conditions has been derived has better thresholds than one-bit decoders, like Gallager A and B. Finally, we have compared the frame error rate performance of the two-bit decoder and Gallager B algorithm for decoding a column-weight-four code with high rate. The two-bit decoder performs better than Gallager B both in the waterfall and in the error-floor region. This illustrates that it is interesting to use two bits rather than one bit for decoding.
Conclusions and Perspectives Conclusions In this thesis, we have first proposed a new class of non-binary LDPCcodes, named hybrid LDPCcodes. The asymptotic analysis of this new class has been carried out. Specific properties of considered hybrid LDPC code ensembles, like the Linear-Map invariance, have been expressed to be able to derive both stability condition and EXIT charts. The stability condition of such hybrid LDPC ensembles shows interesting advantages over non-binary codes. The EXIT charts analysis is performed on the BIAWGN channel. In order to optimize the distributions of hybrid LDPC ensembles, we have investigated how to project the message densities on only one scalar parameter using a Gaussian approximation. The accuracy of such an approximation has been studied, and has led to two kinds of EXIT charts for hybrid LDPCcodes: multi-dimensional and mono-dimensional EXIT charts. The distribution optimization allows to get finite length codes with very low connection degrees and better waterfall region than protograph or multi-edge type LDPCcodes. Moreover, hybrid LDPCcodes are well fitted for the cycle cancellation presented in , thanks to their specific structure. Additionally to a better waterfall region, the resulting codes have a very low error-floor for code rate one-half and codeword length lower than three thousands bits, thereby competing with multi-edge type LDPC. Thus, hybrid LDPCcodes allow to achieve an interesting trade-off between good error-floor performance and good waterfall region with non-binary codes techniques. We have also shown that hybrid LDPCcodes can be very good candidates for efficient low rate coding schemes. For code rate one sixth, they compare very well to existing Turbo Hadamard or Zigzag Hadamard codes. More particularly, hybrid LDPCcodes exhibit very good minimum distances and error floor properties. In the second part of the thesis, we have tried to determine which kind of machine learning methods would be useful to design better LDPCcodesand better decoders in the short code length case. We have first investigated how to build the Tanner graph of a code by pruning away edges from the Tanner graph of a mother code, using a machine learning algorithm, in order to optimize the minimum distance. We showed that no relevant cost function can be found for this problem. Hence, no pruning method could be applied. We have pointed out that this pruning problem was not a classification problem, and that is why this approach failed. 133