Simulation of Encoding and Decoding with Randomly Constructed LDPC Check Matrix

In Mackey1's research methodology, we employed a randomly constructed parity-check matrix for LDPC codes and conducted comprehensive simulations of the encoding and decoding processes. The implementation involves generating sparse parity-check matrices using probabilistic methods where non-zero elements are strategically placed to maintain the desired row and column weights. For the decoding phase, we implemented the belief propagation algorithm (also known as message-passing algorithm) which iteratively updates probability messages between variable nodes and check nodes in the Tanner graph representation. This algorithm efficiently handles soft-decision decoding by calculating log-likelihood ratios and propagating extrinsic information through multiple iterations until convergence criteria are met. Through these systematic implementation steps, we gained deeper insights into LDPC code performance characteristics, error correction capabilities, and the practical application of belief propagation algorithms in modern coding theory. The simulation framework allows for performance evaluation under various channel conditions and code parameters.