New classes of nonassociative division algebras and MRD codes

In the first 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.

In the second part of the thesis, we generalize a construction of semifields and maximum rank distance codes (MRDs) by J Sheekey that employs skew polynomials to obtain new nonassociative division algebras/MRDs. 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. We also obtain new MRD codes.

We are using methods from nonassociative algebra throughout.