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，可以在噪声信道上实现最大限度的健壮通信。由PI设计的LDPC码已被NASA选择用于各种应用，并被10G Base-T以太网采用。然而，在LDPC码普及应用之前，还有许多挑战。具体地说，需要一种数学框架来确定所构造的码的属性，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一时刻突然停止。这种称为差错平层的现象可能会阻止LDPC码应用于需要极低误码率的应用，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，提出了研究高效可编、可译码LDPC码的相关性质的方法，以确保在使用迭代算法进行译码时具有良好的性能。研究还提出了一种使用回溯技术的高效译码算法，以降低LDPC码由于陷阱集而产生的误码平层。这种译码技术为LDPC码在要求极低误码率的通信和存储系统中的应用扫清了障碍。此外，研究还提出了一种新颖的、计算高效的基于可靠性的迭代译码算法，用于纠正随机和突发错误的组合。这种译码技术只需要整数和有限域运算，并在性能和译码复杂度之间提供了有效的折衷。

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.