New classes of nonassociative divison algebras and MRD codes
Publication Date
August 4, 2021
Creator
Abstract
In the first part of the thesis, we generalize a construction by J Sheekey that employs skew polynomials to obtain new nonassociative division algebras and maximum rank distance (MRD) codes. This construction contains Albert’s twisted fields as special cases. As a byproduct, we obtain a class of nonassociative real division algebras of dimension four which has not been described in the literature so far in this form. We also obtain new MRD codes.
In the second part of the thesis, we study a general doubling process (similar to the one that can be used to construct the complex numbers from pairs of real numbers) to obtain new non-unital nonassociative algebras, starting with cyclic algebras. We investigate the automorphism groups of these algebras and when they are division algebras. In particular, we obtain a generalization of Dickson’s commutative semifields.
We are using methods from nonassociative algebra throughout.
Item Type
ethesis
Thesis Type
PhD
Supervisors
Subjects (LC)
Associated Schools / Departments
School of Mathematical Sciences (UK)
eprints ID
64396
UoN Repository URI
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Name
Thesis - Daniel Thompson with corrections.pdf
Type
Full-text
Description
Examined
Size
1.01 MB
Format
Adobe PDF
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