Mathematics, Department of
Department of Mathematics: Dissertations, Theses, and Student Research
First Advisor
Christine A. Kelley
Date of this Version
5-2025
Document Type
Dissertation
Citation
A dissertation presented to the faculty of the Graduate College at the University of Nebraska in partial fulfilment of requirements for the degree of Doctor of Philosophy
Major: Mathematics
Under the supervision of Professor Christine A. Kelley
Lincoln, Nebraska, May 2025
Abstract
When communicating over a noisy channel, the probability of message interference sets a maximum possible transmission rate known as the channel capacity. Any family of codes which have rates converging to the channel capacity and arbitrarily low probability of decoding failure is called capacity achieving. Such codes have been known to exist since the birth of information theory, but are difficult to find explicitly. It has recently been shown that the permutation groups of a family of codes can be used to show that the family is capacity achieving on the q-ary erasure channel.
This thesis seeks to apply the permutation results to specific families of codes to find and classify new families of capacity achieving codes. In particular, we focus on methods of combining existing component codes to obtain new capacity achieving codes. Chapters 1 and 2 deal with the necessary background information for the research. Chapters 3 and 4 deal with product codes and half-product codes, first in two dimensions and then in higher dimensions. Chapter 5 focuses on an attempt to re-purpose the original low density parity code construction to be a deterministic capacity achieving family. Chapter 6 deals with cyclic codes and develops a simplified method of obtaining capacity achieving codes in the cyclic case.
Advisor: Christine A. Kelley
Comments
Copyright 2025, Daniel Welchons. Used by permission