| Channel coding is the art of communicating efficiently over a noisy channel. Low-density parity check codes are a type of linear channel in which there is at least one low-density parity check matrix H. Background and traditional encoding and decoding procedures are defined in chapter 1.; The primary concern of this thesis is the class of protograph ensembles of low density parity check codes, which are defined in chapter 2. Protograph ensembles are characterized by a template graph called a protograph. The Tanner-graph representation of a code in the ensemble is a random lift of the protograph.; A persistent problem which appears to be common in all codes optimized for density evolution threshold is that of error floors. In chapter 3, we consider weight enumerators and stopping set enumerators of protograph LDPC codes. Codeword and stopping set enumerators can be efficiently computed if a certain (non-concave) function can be efficiently maximized. We give a criterion which the protograph should meet in order to have minimum distance or minimum stopping set size growing linearly with the size of the code.; A difficulty in using protograph ensembles is finding a suitable protograph. Since graphs are discrete objects, the techniques primarily used to optimize unstructured ensembles which are based on functional optimization cannot be applied. Instead, we apply the technique of simulated annealing (chapter 4) and show a remarkable degree of success. For example, on the AWGN channel, given a constraint on the node degrees, protograph ensembles can be found that achieve a threshold only half as far (measured in dB) from the Shannon limit as unstructured irregular ensembles.; Protograph-based LDPC codes have significant advantages in available strategies for hardware implementation. One such strategy is described in chapter 6, and is based on quantization described in chapter 5. Under this strategy, we instantiate a single unit of hardware for each node in the protograph which is responsible for sequentially performing all decoder computations corresponding to that node. |
