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Graph-theoretic Aspects of Polar Codes
In 2008, Arikan introduced a powerful family of polar codes, which are the first codes to demonstrably achieve Shannon capacity. Polar codes are built on (and named for) the phenomenon of channel polarization: a mechanism for synthetically increasing and decreasing the reliabilities of a set of channels until some fraction are perfectly noiseless and the remaining channels are completely noisy. In Arikan’s initial publication, this is achieved with a simple recursive binary transform, and shortly thereafter, it was shown by Korada et al. that polarization is a very general phenomenon achieved by many different (not necessarily binary) recursive transforms. Transmission is straightforward once the reliable channels have been identified, but this task is difficult, both computationally and theoretically. Moreover, the guarantees of polar code performance come to fruition only as the code length goes to infinity. Polar codes do not compare favorably to, e.g., low-density parity-check codes at finite block lengths. Motivated by these concerns, this dissertation investigates graph-theoretic properties of polar codes in the context of belief propagation decoding. We first define the class of encoding trellises applicable to the polar coding framework and give a graph- theoretic criterion for polarization. A fundamental contribution is the introduction of the tensor-like join of trellises: a graph product which supplies a direct connection to the matrix construction of polar codes. We leverage the tensor-like join throughout as a mechanism for understanding both graph-theoretic structures and decoding outcomes. We build up a family of acyclic auxiliary graphs, called computation graphs, in order to study belief propagation analytically. In this context, we show that the tensor-like join is sufficient as a construction mechanism and moreover, that such constructions endow computation graphs with powerful structural properties. Following Wiberg, we extend the notion of a deviation as an indicator for decoding failure in the polar computation graph framework. We provide a recursive characterization of the deviation set for Arikan’s polar codes. Finally, looking to the future, we focus on further applications of the tools developed herein.
Bolkema, Jessalyn, "Graph-theoretic Aspects of Polar Codes" (2018). ETD collection for University of Nebraska - Lincoln. AAI10846006.