New classes of nonassociative divison algebras and MRD codes

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Thompson, Daniel (2021) New classes of nonassociative divison algebras and MRD codes. PhD thesis, University of Nottingham.

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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: Thesis (University of Nottingham only) (PhD)
Supervisors: Pumpluen, Susanne
Keywords: nonassociative algebra, division algebra, skew polynomial rings
Subjects: Q Science > QA Mathematics > QA150 Algebra
Faculties/Schools: UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID: 64396
Depositing User: Thompson, Daniel
Date Deposited: 04 Aug 2021 04:40
Last Modified: 04 Aug 2021 04:40
URI: https://eprints.nottingham.ac.uk/id/eprint/64396

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