Mathematics http://hdl.handle.net/10034/6981 2022-09-29T16:59:12Z DNA codes from skew dihedral group ring http://hdl.handle.net/10034/627180 DNA codes from skew dihedral group ring Dougherty, Steven; Korban, Adrian; Şahinkaya, Serap; Ustun, Deniz &lt;p style='text-indent:20px;'&gt;In this work, we present a matrix construction for reversible codes derived from skew dihedral group rings. By employing this matrix construction, the ring &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$\mathcal{F}_{j, k}$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and its associated Gray maps, we show how one can construct reversible codes of length &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$n2^{j+k}$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; over the finite field &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$\mathbb{F}_4.$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; As an application, we construct a number of DNA codes that satisfy the Hamming distance, the reverse, the reverse-complement, and the GC-content constraints with better parameters than some good DNA codes in the literature.&lt;/p&gt; From Crossref journal articles via Jisc Publications Router; Publication status: Published 2022-01-01T00:00:00Z Miyamoto groups of code algebras http://hdl.handle.net/10034/627075 Miyamoto groups of code algebras Castillo-Ramirez, Alonso; McInroy, Justin A code algebra A_C is a nonassociative commutative algebra defined via a binary linear code C. In a previous paper, we classified when code algebras are Z_2-graded axial (decomposition) algebras generated by small idempotents. In this paper, for each algebra in our classification, we obtain the Miyamoto group associated to the grading. We also show that the code algebra structure can be recovered from the axial decomposition algebra structure. Split spin factor algebras http://hdl.handle.net/10034/627049 Split spin factor algebras McInroy, Justin; Shpectorov, Sergey Motivated by Yabe's classification of symmetric $2$-generated axial algebras of Monster type \cite{yabe}, we introduce a large class of algebras of Monster type $(\alpha, \frac{1}{2})$, generalising Yabe's $\mathrm{III}(\alpha,\frac{1}{2}, \delta)$ family. Our algebras bear a striking similarity with Jordan spin factor algebras with the difference being that we asymmetrically split the identity as a sum of two idempotents. We investigate the properties of these algebras, including the existence of a Frobenius form and ideals. In the $2$-generated case, where our algebra is isomorphic to one of Yabe's examples, we use our new viewpoint to identify the axet, that is, the closure of the two generating axes. 2021-12-22T00:00:00Z Enumerating 3-generated axial algebras of Monster type http://hdl.handle.net/10034/627048 Enumerating 3-generated axial algebras of Monster type McInroy, Justin; Shpectorov, Sergey; Khasraw, Sanhan An axial algebra is a commutative non-associative algebra generated by axes, that is, primitive, semisimple idempotents whose eigenvectors multiply according to a certain fusion law. The Griess algebra, whose automorphism group is the Monster, is an example of an axial algebra. We say an axial algebra is of Monster type if it has the same fusion law as the Griess algebra. The 2-generated axial algebras of Monster type, called Norton-Sakuma algebras, have been fully classified and are one of nine isomorphism types. In this paper, we enumerate a subclass of 3-generated axial algebras of Monster type in terms of their groups and shapes. It turns out that the vast majority of the possible shapes for such algebras collapse; that is they do not lead to non-trivial examples. This is in sharp contrast to previous thinking. Accordingly, we develop a method of minimal forbidden configurations, to allow us to efficiently recognise and eliminate collapsing shapes. 2021-06-17T00:00:00Z