CIF: Small: Ordered Metrics and Their Applications

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The central theme of this project is research into properties and applications of polar codes, a new method of coding information for transmission over noisy channels that for the first time realizes the full potential of Shannon's theorems that relate to data rate and transmission reliability. Polar codes have been shown to advance a range of classical and new information-theoretic problems that rely on efficient encoding of the data. This project addresses properties of polar codes in nonbinary communication channels, the design of optimal polarizing transformations, and applications to unequal error protection, hierarchical source coding, broadcast channels, signal design, and other problems of importance for network communication.The analysis of polar codes and their applications in this project relies on the concept of ordered distances that have been shown to control the reliability of transmitted symbols on nonbinary channels. Ordered distances contribute new ideas to the study of linear codes and related concepts such as multipartite and hierarchical secret sharing schemes. The project draws on these ideas to study new algebraic polarization transformations as well as the advances in the theory of linear code-based constructions for a number of models of practical communication systems.