New classes of nonassociative division algebras and MRD codes

在论文的第一部分，我们研究了一个一般的加倍过程（类似于用实数对构造复数的过程）来获得新的非单位非结合代数，从循环代数开始。我们研究了这些代数的自同构群以及它们是除法代数时的自同构群。特别地，我们得到了Dickson交换半域的一个推广。在论文的第二部分，我们推广了J Sheekey的半域和最大秩距离码（mrd）的构造，该构造利用倾斜多项式得到新的非结合除法代数/ mrd。这个结构包含阿尔伯特扭曲场作为特例。作为一个副产品，我们得到了迄今为止文献中尚未描述的一类四维的非结合实数除法代数。我们也获得了新的MRD代码。我们自始至终都在使用非结合代数的方法。

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.