An information theoretic study on linear dispersion codes and low-density parity-check codes

In this dissertation, we focus on an information theoretic study of linear dispersion (LD) codes and low-density parity-check (LDPC) codes. For bit-linear LD codes, we find necessary and sufficient conditions on the dispersion matrices for the a posteriori distribution of the information bit vector to be a product distribution. We also study the design of dispersion matrices to maximize the mutual information between the information bits and the output of a multiple-antenna channel. We derive several bounds on mutual information, and based on the bounds, we propose design guidelines for dispersion matrices. We design two sets of dispersion matrices based on a random search technique. In order to obtain rigorous bounds on the mutual information trajectory of the belief propagation decoding of LDPC codes, we study the extremal problems of moments and information combining. Among all binary-input symmetric-output channels with a fixed mutual information value, the binary symmetric channel (BSC) and the binary erasure channel (BEC) are the extremal channel distributions for an optimization problem related to the second conditional moment of the channel soft-bit. The properties of moments are used to solve the original information combining problem. In order to obtain a better prediction of the convergence behavior of the belief propagation decoding of LDPC codes, we extend the information combining problem at the check nodes by adding a constraint on the second conditional moment of the channel soft-bit. This problem is also solved from a moments approach. The solution to the extension problem can be used to derive potentially better performance bounds on mutual information, provided that another optimization problem be solved.