20 Introduction trade-off between waterfall performance and error floor. The first chapter is dedicated to introduce the useful notions about binary and nonbinary LDPCcodes, as well as the existing tools for their analysis. In the second chapter, we introduce and study a new class of LDPCcodes that we call multi-binary hybrid LDPCcodes. The class of hybrid LDPCcodes is a generalization of existing classes of LDPCcodes, both binary and non-binary. For hybrid LDPCcodes, we allow the connectivity profile to be irregular and the orders of the symbols in the codeword to be heterogeneous. The asymptotic analysis of this class of codes is performed with a given detailed representation to derive stability condition and EXIT charts analysis. The study is performed on the BIAWGN channel, whereas studies of generalized LDPCcodes usually consider the BEC [30, 29] where the one parameter approximation of message densities is straightforward, unlike for the BIAWGN channel. Thus, for the EXIT chart analysis, we have tried to provide an as complete as possible analysis of the accuracy of the projection of message densities on only one scalar parameter. Distributions are optimized and some thresholds computed. We show how the finite length optimization method of  can be adapted and applied to get very low error floor. We finally present experimental results for code rate one half, as well as for code rate one sixth. The third chapter reviews the investigation done on the initial topic of this thesis: how some machine learning methods might be applied to the bipartite graph of a code for finite length optimization purpose? The final goal was to use hybrid LDPCcodes as a tool for building codes with good finite length properties by means of a learning algorithm to be determined. First, we are interested in code design. We look for a way to build the Tanner graph of a code by means of a supervised learning process applied to the graph of a mother code in order to decide which edges should be pruned away in order to lower the sub-optimality of the BP decoder. Then, we move towards decoder design for a given LDPC code. We investigate how to modify the BP decoder by adapting it to the graph of a given code, in order to lower its sensibility to graph cycles. For this purpose, the BP decoder has been considered as a classifier with room for improvement. The fourth chapter also aims at finding good decoders well performing on finite length LDPCcodes, but with good asymptotic behavior too. In this chapter, we switch from continuous BP decoding to quantized decoding. The idea is still to find a decoding rule adapted to topologies hard to decode, like trapping sets . To do so, a class of two-bit message passing decoders is proposed for the binary symmetric channel. The thresholds for various decoders in this class are derived using density evolution. For a specific decoder, the sufficient conditions for a column-weight-four LDPC code to correct all patterns up to three errors are derived. A code satisfying the conditions is constructed and numerical assessment of the code performance is provided via simulation results.
Contributions 21 Contributions In the present thesis, we proposed the following contributions: • A new class of non-binary LDPCcodes, named hybrid LDPCcodes, is studied. ◦ The asymptotic analysis is presented: the property of Linear-Application invariance is exhibited for the code ensemble, leading to a stability condition and an EXIT charts analysis for AWGN channels. Two kinds of EXIT charts of hybrid LDPCcodes are studied: multi-dimensional and mono-dimensional EXIT charts. ◦ Study of the condition allows to conclude that there exist many cases where any fixed point of density evolution for hybrid LDPCcodes can be stable at lower SNR than for non-binary codes. ◦ For the EXIT chart analysis, a detailed analysis of the accuracy of the approximation of message densities by one scalar parameter is provided. ◦ Distribution optimization are performed to get finite-length codes with very low connection degrees and better waterfall region than protograph or multiedge type LDPCcodes. ◦ A cycle cancellation technique is applied to hybrid LDPCcodes, which are well fitted to such a technique, thanks to their specific structure. ◦ The resulting codes appear to have, additionally to a better waterfall region, 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 coding techniques. • An investigation on how machine learning methods could be used for finite length optimization of LDPC coding schemes has been led: ◦ It has been shown that no learning algorithm can be used to build a code from pruning the Tanner graph of a mother code, when the aim is simultaneously to have a high minimum distance and to exploit the value of the messages during the iterativedecoding. ◦ Decoder design, with machine learning methods, has been investigated. The decoding has been defined as a classification problem to which a better decoder than BP may be found, in order to handle message statistical dependencies. The neural network corresponding to the BP decoding has been expressed. To determine optimal synaptic weights to perform better than BP on a finite length code, we proposed a cost function based on the difference between an estimated mutual information and the EXIT chart. The reason why this approach fails has been detailed.