Date of Award

5-2015

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science

Advisor

Dr. Gretchen L. Matthews

Committee Member

Dr. Jim Brown

Committee Member

Dr. Shuhong Gao

Committee Member

Dr. Felice Manganiello

Abstract

In recent groundbreaking work, Arikan developed polar codes as an explicit construction of symmetric capacity achieving codes for binary discrete memoryless channels with low encoding and decoding complexities. In this construction, a specific kernel matrix G is considered and is used to encode a block of channels. As the number of channels grows, each channel becomes either a noiseless channel or a pure-noise channel, and the rate of this polarization is related to the kernel matrix used. Since Arikan's original construction, polar codes have been generalized to q-ary discrete memoryless channels, where q is a power of a prime, and other matrices have been considered as kernels. In our work, we expand on the ideas of Mori and Tanaka and Korada, Sasoglu, and Urbanke by employing algebraic geometric codes to produce kernels of polar codes, specifically codes from maximal and optimal function fields.

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