New classes of nonassociative division algebras and MRD codes

新类非结合除法代数和 MRD 代码

• 批准号: 1947057
• 金额: --
• 依托单位: University of Nottingham
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1947057
• 关键词: New; classes; nonassociative; division; algebras

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.

在论文的第一部分，我们从循环代数开始，研究了一个一般的倍化过程(类似于从实数对构造复数的过程)，从而得到了新的非单位非结合代数。我们研究了这些代数的自同构群以及当它们是除代数时。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构造的半域和最大秩距离码，它利用斜多项式得到新的非结合除法代数/最大秩距离码。这个结构包含了阿尔伯特扭曲的场作为特例。作为一个副产品，我们得到了一类到目前为止还没有在文献中描述的四维非结合实除代数。我们还得到了新的MRD码。我们从头到尾都使用非结合代数的方法。

项目成果

Division algebras that generalize Dickson semifields

推广迪克森半域的除法代数

• DOI: 10.48550/arxiv.1906.01329
• 发表时间: 2019
• 期刊: arXiv e-prints
• 作者: Thompson Daniel