Open AccessNew self-dual codes from $ 2 \times 2 $ block circulant matrices, group rings and neighbours of neighbours
Author(s) -
Joe Gildea,
Abidin Kaya,
Adam Michael Roberts,
Rhian Taylor,
Alexander Tylyshchak
Publication year - 2023
Publication title -
advances in mathematics of communications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.601
H-Index - 26
eISSN - 1930-5346
pISSN - 1930-5338
DOI - 10.3934/amc.2021039
Subject(s) - circulant matrix , mathematics , combinatorics , block (permutation group theory) , group (periodic table) , matrix (chemical analysis) , construct (python library) , dual (grammatical number) , discrete mathematics , arithmetic , algebra over a field , pure mathematics , computer science , physics , art , literature , quantum mechanics , programming language , materials science , composite material
In this paper, we construct new self-dual codes from a construction that involves a unique combination; \begin{document}$ 2 \times 2 $\end{document} block circulant matrices, group rings and a reverse circulant matrix. There are certain conditions, specified in this paper, where this new construction yields self-dual codes. The theory is supported by the construction of self-dual codes over the rings \begin{document}$ \mathbb{F}_2 $\end{document} , \begin{document}$ \mathbb{F}_2+u \mathbb{F}_2 $\end{document} and \begin{document}$ \mathbb{F}_4+u \mathbb{F}_4 $\end{document} . Using extensions and neighbours of codes, we construct \begin{document}$ 32 $\end{document} new self-dual codes of length \begin{document}$ 68 $\end{document} . We construct 48 new best known singly-even self-dual codes of length 96.