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- W3085293166 abstract "<p style='text-indent:20px;'>Many generator matrices for constructing extremal binary self-dual codes of different lengths have the form <inline-formula><tex-math id=M1>begin{document}$ G = (I_n | A), $end{document}</tex-math></inline-formula> where <inline-formula><tex-math id=M2>begin{document}$ I_n $end{document}</tex-math></inline-formula> is the <inline-formula><tex-math id=M3>begin{document}$ n times n $end{document}</tex-math></inline-formula> identity matrix and <inline-formula><tex-math id=M4>begin{document}$ A $end{document}</tex-math></inline-formula> is the <inline-formula><tex-math id=M5>begin{document}$ n times n $end{document}</tex-math></inline-formula> matrix fully determined by the first row. In this work, we define a generator matrix in which <inline-formula><tex-math id=M6>begin{document}$ A $end{document}</tex-math></inline-formula> is a block matrix, where the blocks come from group rings and also, <inline-formula><tex-math id=M7>begin{document}$ A $end{document}</tex-math></inline-formula> is not fully determined by the elements appearing in the first row. By applying our construction over <inline-formula><tex-math id=M8>begin{document}$ mathbb{F}_2+umathbb{F}_2 $end{document}</tex-math></inline-formula> and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to construct many new binary self-dual <inline-formula><tex-math id=M9>begin{document}$ [68, 34, 12] $end{document}</tex-math></inline-formula>-codes with the rare parameters <inline-formula><tex-math id=M10>begin{document}$ gamma = 7, 8 $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=M11>begin{document}$ 9 $end{document}</tex-math></inline-formula> in <inline-formula><tex-math id=M12>begin{document}$ W_{68, 2}. $end{document}</tex-math></inline-formula> In particular, we find 92 new binary self-dual <inline-formula><tex-math id=M13>begin{document}$ [68, 34, 12] $end{document}</tex-math></inline-formula>-codes.</p>" @default.
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- W3085293166 date "2022-01-01" @default.
- W3085293166 modified "2023-09-25" @default.
- W3085293166 title "New self-dual codes of length 68 from a $ 2 times 2 $ block matrix construction and group rings" @default.
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- W3085293166 doi "https://doi.org/10.3934/amc.2020111" @default.
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