CIF: Small: Theory and Structure of Quasi-Cyclic LDPC Codes and Algorithms to Lower the Error Floor and Decode Non-Binary LDPC Codes

低密度奇偶校验码（LDPC）被认为是目前最有前途的编码技术，以实现在噪声信道上的鲁棒通信的最终限制。由pi设计的LDPC代码已被NASA选择用于各种应用，并被用于10G Base-T以太网。然而，在LDPC码在应用中变得通用之前，还存在许多挑战。特别是，需要一个数学框架来确定构造代码的属性，以便于编码器和解码器的实现。此外，已经观察到，随着信道的改进，代码性能的显着改善会在某个点突然停止。这种被称为错误层的现象可能会使LDPC码无法用于需要非常低错误率的应用，例如高速卫星通信和高密度数据存储系统。另一个挑战是在被噪声损坏后检索正确数据所需的计算复杂性。本研究解决了这三个挑战。本研究开发了研究高效可编解码LDPC码的相关特性的方法，以确保在使用迭代算法解码时具有良好的性能。研究还开发了一种使用回溯技术的高效解码算法，以降低LDPC码由于捕获集而导致的错误层。这种解码技术消除了LDPC码在通信和存储系统中应用的障碍，这些系统需要非常低的错误率。此外，该研究还提出了一种新的、计算效率高的基于可靠性的非二进制LDPC码译码迭代算法，用于校正随机和突发错误的组合。这种解码技术只需要整数和有限域操作，并且在性能和解码复杂性之间提供了有效的权衡。

Low-density parity-check (LDPC) codes are currently recognized as the most promising coding technique to achieve the ultimate limits of robust communications over noisy channels. LDPC codes devised by the PIs have been selected by NASA for various applications and adopted for the 10G Base-T Ethernet. However, there are many challenges before LDPC codes become universal in applications. In particular, there is a need for a mathematical framework to determine the properties of the constructed codes for easy encoder and decoder implementations. Furthermore, it has been observed that the dramatic improvement in code performance as the channel improves comes to a sudden halt at some point. This phenomenon, known as error-floor, may preclude LDPC codes from applications requiring very low error rates, such as high-speed satellite communication and high-density data storage systems. Another challenge is the computational complexity needed to retrieve the correct data after being corrupted with noise. This research addresses these three challenges.The research develops methods to study the relevant properties of efficiently encodable and decodable LDPC codes that ensure good performance when decoding using iterative algorithms. The research also develops an efficient decoding algorithm using a backtracking technique to lower the error-floors of LDPC codes due to trapping sets. This decoding technique removes the obstacles for applications of LDPC codes in communication and storage systems where very low error rates are required. Furthermore, the research also proposes a novel and computationally efficient reliability-based iterative algorithm for decoding non-binary LDPC codes for correcting combinations of random and bursts of errors. This decoding technique requires only integer and finite field operations and offers an effective trade-off between performance and decoding complexity.