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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=715959
• 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.

我的整个研究领域是编码理论和密码技术，以实现可靠和安全的通信。编码理论的发展彻底改变了编码应用于实际系统的方式，也影响了高速调制解调器的设计。这些发展涉及Turbo码、Polar码和所有低密度奇偶校验码(LDPC)和格码。格型编码在通信中有几个应用：量化和信号星座。格可以分为两类，一类是具有最优译码的经典(低维)格，另一类是具有次优译码的现代(高维)格。在我的博士论文中，现代格码时代开始于LDPC格子的构建。 代数格有潜在的代码。其中一些码具有相当高密度的奇偶校验(HDPC)矩阵。与最大似然译码相比，应用迭代算法对最著名的代数码进行译码会导致较差的性能。为了解决这一问题，人们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令人惊叹的表现。研究表明，深度学习方法改进了HDPC码的最小和积算法。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它在传输消息中的影响。通过适当地设置权重，我们可以补偿小循环，这是导致高错误平台区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，用于解码具有相当HDPC矩阵的代数格。 2)构造了多电平准循环LDPC(QC-LDPC)格，并给出了这种格的多级译码算法。 在所有的LDPC码中，QC-LDPC码和空间耦合LDPC(SC-LDPC)码是两个基本的类别，由于它们的实用和简单的实现而比其他类型的LDPC码更受青睐。关于这些代码的构建，我的目标是： 1)提出了一种构造SC-LDPC码的新方法，建立了QC-LDPC码与SC-LDPC码之间的紧密联系。 2)改进了SC-LDPC码约束长度的解析下界，并提出了一些减小搜索空间的技术。 3)提供一种神经译码版本的滑动窗方法来译码SC-LDPC码。 关于LDPC码中的图形结构，如陷阱集和吸收集，我的目标是： 1)利用图论和组合学对不同围长的QC-LDPC码Tanner图的基本陷阱集进行了刻画。 2)构造Tanner图不含小尺寸陷阱集的QC-LDPC码。 3)SC-LDPC卷积码Tanner图中吸收集的刻画。