Codes and Latices: Construction and Decoding Algorithms

我研究的总体领域是可靠和安全通信的编码理论和加密技术。编码理论的发展彻底改变了编码在实际系统中的应用方式，也影响了高速调制解调器的设计。这些发展与turbo码、极码和所有低密度奇偶校验码（LDPC）和格码有关。点阵编码在通信中有几个应用：量化和信号星座。晶格可以分为两类：具有最优解码器的经典（低维）晶格和具有次优解码器的现代（高维）晶格。现代格编码时代始于我博士论文中LDPC格的构建。

The overall area of my research is coding theory and cryptographic techniques for reliable and secure communications. Coding theory developments have revolutionized the way coding is applied to practical systems, affecting also the design of high speed modems. These developments are related to turbo, polar and all low density parity-check (LDPC) codes and lattices codes. Lattice coding has several applications in communications: quantization and signal constellation. Lattices can be divided into two categories including classical (low dimensional) ones with optimal decoders, and modern (high dimensions) lattices with suboptimal decoders. The modern lattice coding era starts with the construction of LDPC lattices in my PhD thesis. Algebraic lattices have underlying codes. Some of these codes have a rather high-density parity-check (HDPC) matrix. Applying iterative algorithms to decode the most well-known algebraic codes would result in poor performance when compared to maximum likelihood decoder. To resolve this problem, deep neural network decoders were proposed. In recent years, deep learning methods have shown amazing performances in a variety of subjects. It has been shown that deep learning methods improve the min-sum and sum-product algorithms for HDPC codes. The main benefit of neural network decoder is that the weight of every edge of its underlying graph relies on its influence in the transmitting messages. By setting weights properly, we can compensate for small cycles, which are the main cause of high error floor regions and deterioration of the decoding process. So, about construction and performance analysis of modern lattices my goals are: 1) To provide neural network lattice decoding algorithms to decode algebraic lattices with a rather HDPC matrix. 2) To construct multi-level Quasi-cyclic LDPC (QC-LDPC) lattices and to present a multi-stage decoder for these lattices. Among all LDPC codes, QC-LDPC codes and Spatially coupled LDPC (SC-LDPC) codes are two essential categories that are preferred to other types of LDPC codes because of their practical and simple implementations. Regarding the construction of these codes my goals are: 1) To propose a new approach to construct SC-LDPC codes and establish a close connection between QC-LDPC codes and SC-LDPC codes. 2) To improve analytical lower bounds on the constraint length of SC-LDPC codes and some techniques to reduce the search space. 3) To provide a neural decoder version of the Sliding Window method to decode SC-LDPC codes. Regarding to graphical structures such as trapping sets and absorbing sets in LDPC codes my goals are: 1) Using graph theory and combinatorics to characterize elementary trapping sets in Tanner graph of QC-LDPC codes with different girths. 2) Construction of QC-LDPC codes whose Tanner graphs are free of some trapping sets with small size. 3) Characterization of absorbing sets in the Tanner graph of SC-LDPC convolutional codes.