New Frontiers in Coding Theory: Reed-Muller and Polar Codes

编码理论是信息理论和实际系统之间的重要桥梁，信息理论研究数据传输的基本限制，实际系统需要低复杂度的计算效率算法。本研究计画主要探讨两种密切相关的码族--Reed-Muller码与极化码的基本理论与实际设计。在理论方面，我们计划攻击长期存在的猜想，Reed-Muller码的容量实现下的最大后验译码二进制无记忆对称信道。我们将从信道极化的角度来看待Reed-Muller码，并建立在Kudekar，Kumar，Modelli，Pfister，Sasoglu和Urbanke的令人兴奋的进展基础上，他们建立了二进制擦除信道的特殊情况下的代码的最优性。在实践方面，我们计划研究最近发明的极化码，这些极化码已被新发布的5G NR（新无线电）标准采用。首先，我们的目标是通过结合Peres的随机比特生成器的思想来提高极化码的有限长度性能。其次，我们计划探索新的应用程序中的极化原理超越点对点通信，如无线网络中的多用户通信，分布式压缩在新兴的数据存储系统，内容分发网络的缓存技术，感知矩阵设计组测试，等等。

Coding theory is an important bridge between information theory, which studies the fundamental limits of data transmission, and practical systems, which require low-complexity computationally efficient algorithms. This research program investigates fundamental theories as well as practical designs of two closely related families of codes---Reed-Muller codes and polar codes. On the theory side, we plan to attack the long-standing conjecture that Reed-Muller codes are capacity achieving under maximum a posteriori decoding over binary memoryless symmetric channels. We will perceive Reed-Muller codes from a channel polarization perspective and build on the exciting progress by Kudekar, Kumar, Modelli, Pfister, Sasoglu, and Urbanke, who established optimality of the code over the special case of binary erasure channels. On the practical side, we plan to investigate the recently invented polar codes, which have been adopted in the newly released 5G NR (New Radio) standard. First, we aim to improve the finite-length performance of polar codes by incorporating ideas from Peres's random-bit generator. Second, we plan to explore the polarization principle in new applications beyond point-to-point communication, such as multi-user communication in wireless networks, distributed compression in emerging data storage systems, caching techniques for content delivery networks, sensing matrix design for group testing, among others.