Digital communication pervades our daily lives while digital storage devices have become the principal means of preserving our information. During the "information age," in which we now live, the need for reliable transmission and storage of digital data is of paramount importance. What makes such reliable transmission and storage possible are error- correcting codes, first conceived by Claude Shannon over 50 years ago. The recent invention of polar codes is, without doubt, the most original and profound development in the theory of error-correcting codes in the past decade. Polar codes provably achieve the capacity of any memoryless symmetric channel, with low encoding and decoding complexity, thereby providing the first deterministic and constructive solution to the problem posed by Shannon in 1948. Nevertheless, the impact of polar codes in practice has been, so far, negligible. The objective of this project is to advance the theory of polar codes on one hand, and to bring polar codes much closer to practice on the other hand. If successful, the outcome of this research is likely to become an enabling technology for numerous communications applications, both commercial and for national security.
In order to make polar codes practical, major obstacles must be resolved. The first key problem in the field is how to efficiently construct polar codes. This project aims to develop a linear-time construction algorithm, with explicit guarantees on the quality of its output. The investigators also study algebraic and combinatorial structure of polar codes, with the goal of developing a good analytical handle on the rate of channel polarization. Currently available empirical results indicate that the rate of channel polarization is too slow for many applications. Thus the investigators intend to drastically improve the performance of polar codes, at short to moderate code lengths, by introducing certain key modifications in the successive-cancellation decoding algorithm. One especially promising idea in this regard is list decoding. A concerted effort is devoted to the analysis of list decoding algorithms for polar codes. Furthermore, a full implementation of such decoding algorithms in high-speed and low-power VLSI is pursued. This part of the research involves algorithmic transformations for the key steps of the decoder, effective high-throughput design techniques, careful VLSI complexity/area analysis, and new computation scheduling ideas. Finally, applications of polar codes and channel polarization beyond point-to-point communications are considered. Such applications include multiple-access channels, relay channels, Slepian-Wolf coding, and information-theoretic security.