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New self-dual codes of length 68 from a 2 × 2 block matrix construction and group rings
Bortos, Maria Gildea, Joe Kaya, Abidin Korban, Adrian Tylyshchak, Alexander
Bortos, Maria
Gildea, Joe
Kaya, Abidin
Korban, Adrian
Tylyshchak, Alexander
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2020-09-15
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Abstract
Many generator matrices for constructing extremal binary self-dual codes of different
lengths have the form G = (In | A); where In is the n x n identity matrix and A is
the n x n matrix fully determined by the first row. In this work, we define a generator
matrix in which A is a block matrix, where the blocks come from group rings and also,
A is not fully determined by the elements appearing in the first row. By applying our
construction over F2 +uF2 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 con-
struct many new binary self-dual [68,34,12]-codes with the rare parameters
$\gamma = 7$; $8$ and $9$ in $W_{68,2}$: In particular, we find 92 new binary
self-dual [68,34,12]-codes.
Citation
Bortos, M., Gildea, J., Kaya, A., Korban, A. & Tylyshchak,A. (2022). New self-dual codes of length 68 from a 2 × 2 block matrix construction and group rings. Advances in Mathematics of Communications, 16(2), 269-284. https://doi.org/10.3934/amc.2020111
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American Institute of Mathematical Sciences
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Advances in Mathematics of Communications
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Article
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This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Advances in Mathematics of Communications following peer review. The definitive publisher-authenticated version Bortos, M., Gildea, J., Kaya, A., Korban, A. & Tylyshchak,A. (2020). New self-dual codes of length 68 from a 2 × 2 block matrix construction and group rings. Advances in Mathematics of Communications. is available online at: https://www.aimsciences.org/article/doi/10.3934/amc.2020111
Series/Report no.
ISSN
1930-5346
EISSN
1930-5338