Analyse et construction de codes LDPC non-binaires pour des canaux à évanouissement. (Analysis and Design of Non-Binary LDPC Codes over Fading Channels)

Over the last 15 years, spectacular advances in the analysis and design of graph-based codes and iterative decoding techniques paved the way for the development of error correction systems operating very close to the theoretical Shannon limit. A prominent role has been played by the class of Low Density Parity Check (LDPC) codes, introduced in the early 60's by Gallager's and described latter in terms of sparse bipartite graphs. In the early 2000's, LDPC codes were shown to be capacity approaching codes for a wide range of channel models, which motivated the increased interest of the scienti c community and supported the rapid transfer of this technology to the industrial sector. Over the past few years there has been an increased interest in non-binary LDPC codes due to their enhanced correction capacity. Although Gallager already proposed in his seminal work the use of non-binary alphabets (by using modular arithmetic), non-binary LDPC codes de ned over nite elds have only been investigated starting with the late 90's. They have been proven to provide better performance than their binary counterparts when the block-length is small to moderate, or when the symbols sent through channel are not binary, which is the case for high-order modulations or for multiple-antennas channels. However, the performance gain comes at a non-negligible cost in the decoding complexity, which may prohibit the use of non-binary LDPC codes in practical systems, especially when the price to pay in decoding complexity is too high for the performance gain that one can get. This thesis addresses the analysis and design of non-binary LDPC codes for fading channels. The main goal is to demonstrate that besides the gain in the decoding performance, the use of non-binary LDPC codes can bring additional bene ts that may o set the extra cost in decoding complexity. Flexibility and diversity are the two bene ts that we demonstrate in this thesis. The exibility is the capacity of a coding system to accommodate multiple coding rates through the use of a unique encoder/decoder pair. The diversity of a coding system relates to its capacity to fully exploit the communication channel's heterogeneity. The rst contribution of the thesis is the development of a Density Evolution approximation method, based on the Monte-Carlo simulation of an in nite code. We show that the proposed method provides accurate and precise estimates of non-binary ensemble thresholds, and makes possible the optimization of non-binary codes for a wide range of applications and channel models. The second contribution of the thesis consists of the analysis and design of exible coding schemes through the use of puncturing. We show that the non-binary LDPC codes are more robust to puncturing than their binary counterparts, thanks to the fact that non-binary symbol-nodes can be only partially punctured. For regular codes, we show that the design of puncturing patterns must respect di erent rules depending on whether the symbol-nodes are of degree 2 or higher. For irregular codes we propose an optimization procedure and we present optimized puncturing distributions for non-binary LDPC codes,