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，信息论研究数据传输的基本极限，而实际系统要求低复杂度、计算效率高的算法。本研究项目研究两类密切相关的码-Reed-Muller码和极性码的基本理论和实际设计。在理论方面，我们计划挑战长期存在的猜想，即Reed-Muller码是在二进制无记忆对称信道上的最大后验译码下获得的容量。我们将从信道极化的角度来感知Reed-Muller码，并建立在Kudekar、Kumar、modelli、Pfister、Sasoglu和Urbanke的令人兴奋的进展的基础上，他们建立了码在二进制擦除信道的特殊情况下的最优性。在实践方面，我们计划研究最近发明的极地码，这些码已经被新发布的5G NR(新无线电)标准所采用。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，我们计划在点对点通信之外的新应用中探索极化原理，例如无线网络中的多用户通信、新兴数据存储系统中的分布式压缩、内容交付网络的缓存技术、用于组测试的感知矩阵设计等。

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.