| Fountain codes were introduced to provide higher reliability, lower complexities, and more scalability for networks such as the Internet. In this thesis, we study Luby-Transform (LT) codes which are the realization of Fountain codes. In the LT codes, a sparse random factor graph is dynamically generated on both the encoder and decoder sides of the communications channel. The graph is generated from an ensemble degree distribution. The LT codes are also known as rateless codes, in the sense that they can generate potentially limitless codeword symbols from original data and self-adjust to channels with different erasure probabilities. LT Codes also have a very low encoding and decoding complexities when comparing with some traditional block codes, e.g., Reed Solomon (RS) codes and Low-Density-Parity-Check (LDPC) codes. Therefore, LT Codes are suitable for many different kinds of applications such as broadcast transmission.;LT codes achieve the capacity of the Binary Erasure Channel (BEC) asymptotically and universally. For finite lengths, the search is continued to find codes closer to the capacity limits at even lower encoding and decoding complexities. Most previous work on single-layer Fountain coding targets the design via the right degree distribution. The left degree distribution of an LT code is left as Poisson to protect the universality. For finite lengths, this is no longer an issue; thus, we focus on the design of better codes for the BEC and noisy channels as well at practical lengths. We propose two encoding schemes for BEC and noisy channels by shaping the left degree distribution. Our left degree shaping provides codes outperforming regular LT code and all other competing schemes in the literature. For instance, at a bit error rate of 10-7 and k = 256, our scheme provides a realized rate of 0.6 which is 23.5% higher than Sorensen et al.'s scheme over BEC. In addition, over noisy channels our proposed scheme achieves an improvement of 14% in the released rates at k = 100 and 30 Belief Propagation (BP) iterations. |
