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

点击翻译按钮获取中文摘要

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.