Author ORCID Identifier
Date Available
5-9-2024
Year of Publication
2024
Degree Name
Doctor of Philosophy (PhD)
Document Type
Doctoral Dissertation
College
Graduate School
Department/School/Program
Mathematics
First Advisor
Dr. Heide Gluesing-Luerssen
Abstract
A skew-polynomial ring is a polynomial ring over a field, with one indeterminate x, where one must apply an automorphism to commute coefficients with x. It was first introduced by Ore in 1933 and since the 1980s has been used to study skew-cyclic codes. In this thesis, we present some properties of skew-polynomial rings and some new constructions of skew-cyclic codes. The dimension of a skew-cyclic code depends on the degree of its generating skew polynomial. However, due to the skew-multiplication rule, the degree of a skew polynomial can be smaller than its number of roots and hence tricky to predict. In Chapter 2, we introduce tools offered by Leroy in 2012 which connect the degree of a skew polynomial to linear independence of field elements which are related to the roots. In Chapter 3, we study a particular type of skew polynomial called a W-Polynomial. These are skew polynomials of smallest degree which vanish on some set of field elements. More specifically, we classify when skew polynomials of the form x n − a are W-polynomials. In Chapter 4 we will make use of this work to study more general skew-cyclic codes than in the current literature and establish the skew-Roos bound for their distance. Finally, in Chapter 5, we study subfield subcodes of skew-cyclic codes, and compare skew-BCH codes of the first and second kind.
Digital Object Identifier (DOI)
https://doi.org/10.13023/etd.2024.103
Recommended Citation
Hechtel, Kathryn, "Properties of Skew-Polynomial Rings and Skew-Cyclic Codes" (2024). Theses and Dissertations--Mathematics. 108.
https://uknowledge.uky.edu/math_etds/108
