{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,6]],"date-time":"2024-06-06T08:29:37Z","timestamp":1717662577266},"reference-count":22,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["AMC"],"published-print":{"date-parts":[[2022]]},"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+u\\mathbb{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><\/jats:p>","DOI":"10.3934\/amc.2020111","type":"journal-article","created":{"date-parts":[[2020,9,18]],"date-time":"2020-09-18T09:41:25Z","timestamp":1600422085000},"page":"269","source":"Crossref","is-referenced-by-count":2,"title":["New self-dual codes of length 68 from a $ 2 \\times 2 $ block matrix construction and group rings"],"prefix":"10.3934","volume":"16","author":[{"given":"Maria","family":"Bortos","sequence":"first","affiliation":[]},{"given":"Joe","family":"Gildea","sequence":"additional","affiliation":[]},{"given":"Abidin","family":"Kaya","sequence":"additional","affiliation":[]},{"given":"Adrian","family":"Korban","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Tylyshchak","sequence":"additional","affiliation":[]}],"member":"2321","reference":[{"key":"key-10.3934\/amc.2020111-1","doi-asserted-by":"publisher","unstructured":"W. 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