Mathematics, Department of

 

First Advisor

Christine A. Kelley

Date of this Version

12-2023

Document Type

Article

Citation

A dissertation presented to the faculty of the Graduate College at the University of Nebraska in partial fulfillment of requirements for the degree of Doctor of Philosophy

Major: Mathematics

Under the Supervision of Professor Christine A. Kelley

Lincoln, Nebraska, December 2023

Comments

Copyright 2023, Meraiah Martinez.

Abstract

In order to communicate information over a noisy channel, error-correcting codes can be used to ensure that small errors don’t prevent the transmission of a message. One family of codes that has been found to have good properties is low-density parity check (LDPC) codes. These are represented by sparse bipartite graphs and have low complexity graph-based decoding algorithms. Various graphical properties, such as the girth and stopping sets, influence when these algorithms might fail. Additionally, codes based on algebraically structured parity check matrices are desirable in applications due to their compact representations, practical implementation advantages, and tractable decoder performance analysis.

This dissertation focuses on codes based on parity check matrices that are dyadic, n-adic, or quasi-dyadic (QD), meaning the parity check matrix representation is block structured with dyadic matrices as blocks. Depending on the number of nonzero positions in the leading row of each block, these codes may be either low density or moderate density. Since each block is reproducible, the resulting QD codes have similar advantages to quasi-cyclic (QC) codes. We examine basic code properties of dyadic, n-adic, and QD parity check codes, including bounds on the dimension and minimum distance, cycle structure of the corresponding Tanner graph, and their possible use in quantum code constructions. We also consider the relationship between cycle codes of graphs and cycle codes of their lifts.

Advisor: Christine A. Kelley

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