Low-density parity-check (LDPC) codes have been the focus of much research over the past decade thanks to their near Shannon limit performance and to their efficient message-passing (MP) decoding algorithms. However, the error floor phenomenon observed in MP decoding, which manifests itself as an abrupt change in the slope of the error-rate curve, has hindered the adoption of LDPC codes and MP decoders in some applications requiring very low error rates. As an alternative to MP decoding, linear programming (LP) decoding is an approximation to maximum-likelihood decoding by relaxing the optimal decoding problem into a linear optimization problem. It has been noticed that, when the symbols error probability in channel output is low, LP decoding has superior error correction performance. However, due to the inefficiency of general-purpose LP solvers which are not optimized for solving LP problems, LP decoding is computationally more complex than MP decoding, especially for codes of large block size. In this dissertation, we first design an efficient exhaustive search algorithm to find all small error-prone substructures, some of which are commonly blamed for certain decoding failures of MP decoding. Then, we investigate the cause of error floors in LDPC codes from the perspective of the MP decoder implementation, with special attention to limitations that decrease the numerical accuracy of messages passed during decoding, and propose a quantization method for fixed-point implementation of MP decoding which significantly improves the error-floor performance by overcoming the limitations of standard quantization rules. For LP decoding, we improve the error-correcting capability of LP decoding by using an effective algorithm to generate additional redundant parity-check constraints which eliminate certain undesired solutions to LP decoding problem. We further propose an efficient message-passing algorithm to solve the LP decoding problem. This algorithm is based on the alternating direction method of multipliers (ADMM), a classic technique in convex optimization theory, and our key contribution is a novel, efficient projection algorithm that can improve the decoding speed of the ADMM-based LP decoder. The last part of this dissertation is a separate piece of work on optimizing video transmission over distributed cognitive radio network. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]