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- W3183388321 abstract "We introduce a consistent and efficient method to construct self-dual codes over <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$GF(q)$ </tex-math></inline-formula> using symmetric matrices and eigenvectors from a self-dual code over <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$GF(q)$ </tex-math></inline-formula> of smaller length where <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$q equiv 1 pmod 4$ </tex-math></inline-formula> . Using this method, which is called a ‘symmetric building-up’ construction, we improve the bounds of the best-known minimum weights of self-dual codes with lengths up to 40, which have not significantly improved for almost two decades. We focus on a class of self-dual codes, which includes double circulant codes. We obtain 2967 new self-dual codes over <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$GF(13)$ </tex-math></inline-formula> and <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$GF(17)$ </tex-math></inline-formula> up to equivalence. We also compute the minimum weights of quadratic residue(QR) codes that were previously unknown. These are a [20, 10, 10] QR self-dual code over <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$GF(23)$ </tex-math></inline-formula> , [24, 12, 12] QR self-dual codes over <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$GF(29)$ </tex-math></inline-formula> and <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$GF(41)$ </tex-math></inline-formula> , and a [32, 16, 14] QR self-dual code over <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$GF(19)$ </tex-math></inline-formula> . They have the highest minimum weights so far." @default.
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- W3183388321 date "2021-01-01" @default.
- W3183388321 modified "2023-09-25" @default.
- W3183388321 title "Self-Dual Codes, Symmetric Matrices, and Eigenvectors" @default.
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- W3183388321 doi "https://doi.org/10.1109/access.2021.3099434" @default.
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