Codes and Latices: Construction and Decoding Algorithms

代码和逻辑：构造和解码算法

• 批准号: RGPIN-2019-06180
• 负责人: RafsanjaniSadeghi, MohammadReza
• 金额: $ 1.38万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=738760
• 关键词: Codes; Latices; Construction; Decoding; Algorithms

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.

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。代数格具有底层代码。其中一些代码具有相当高密度的奇偶校验（HDPC）矩阵。与最大似然解码器相比，将迭代算法应用于最知名的代数码解码会导致性能差。为了解决这一问题，提出了深度神经网络解码器。近年来，深度学习方法在各种学科中表现出惊人的表现。研究表明，深度学习方法改进了HDPC代码的最小和和积算法。神经网络解码器的主要优点是其底层图的每条边的权重取决于其对传输消息的影响。通过适当地设置权值，我们可以补偿小周期，这是导致高误差层区域和译码过程恶化的主要原因。因此，关于现代格的构造和性能分析，我的目标是：1)提供神经网络格解码算法来解码具有相当HDPC矩阵的代数格。2)构造了多级拟循环LDPC （QC-LDPC）格，并给出了这些格的多级解码器。在所有LDPC码中，QC-LDPC码和SC-LDPC （spatial coupled LDPC）码由于其实用和简单的实现，是比其他类型的LDPC码更受青睐的两个基本类别。对于这些代码的构建，我的目标是：1)提出一种新的SC-LDPC代码的构建方法，并建立QC-LDPC代码与SC-LDPC代码之间的紧密联系。2)改进SC-LDPC码约束长度的解析下界和减小搜索空间的一些技术。3)提供滑动窗口方法的神经解码器版本来解码SC-LDPC码。对于LDPC码中的捕获集和吸收集等图形结构，本文的研究目标是：1)利用图论和组合学对不同周长QC-LDPC码Tanner图中的基本捕获集进行表征。2)构造Tanner图不存在小尺寸捕获集的QC-LDPC码。3) SC-LDPC卷积码Tanner图中吸收集的表征。