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（LDPC）码是目前公认的最有前途的编码技术，以实现在噪声信道下的鲁棒通信的极限。由PI设计的LDPC码已被NASA选择用于各种应用，并用于10 G Base-T以太网。然而，LDPC码要普及应用还面临着许多挑战。特别地，需要一个数学框架来确定构造的代码的属性，以便于编码器和解码器的实现。此外，已经观察到，随着信道的改善，代码性能的显著改善在某个点突然停止。这种被称为错误平层（error-floor）的现象可能使LDPC码无法应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破坏后检索正确数据所需的计算复杂性。本研究针对这三个挑战，开发出研究可有效编码和可有效解码的LDPC码的相关特性的方法，以确保在使用迭代算法进行解码时具有良好的性能。本研究亦发展一种有效的译码演算法，利用回溯技术来降低低密度奇偶校验码因陷阱集所造成的错误平层。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，其中需要非常低的错误率。此外，该研究还提出了一种新的和计算效率高的可靠性为基础的迭代算法解码的非二进制LDPC码纠正组合的随机和突发错误。这种解码技术只需要整数和有限域操作，并提供了一个有效的性能和解码复杂度之间的权衡。

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.