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- W4285202422 abstract "Kim et al. (2021) gave a method to embed a given binary <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$[n,k]$ </tex-math></inline-formula> code <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$mathcal {C},,(k = 3, 4)$ </tex-math></inline-formula> into a self-orthogonal code of the shortest length which has the same dimension <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$k$ </tex-math></inline-formula> and minimum distance <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$d' ge d(mathcal {C})$ </tex-math></inline-formula> . We extend this result by proposing a new method related to a special matrix, called the self-orthogonality matrix <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$SO_{k}$ </tex-math></inline-formula> , obtained by shortening a Reed-Muller code <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>${mathcal R}(2,k)$ </tex-math></inline-formula> . Using this approach, we can extend binary linear codes to many optimal self-orthogonal codes of dimensions 5 and 6. Furthermore, we partially disprove the conjecture (Kim et al. (2021)) by showing that if <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$31 le n le 256$ </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>$nequiv 14,22,29 pmod {31}$ </tex-math></inline-formula> , then there exist optimal <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$[n], [5]$ </tex-math></inline-formula> codes which are self-orthogonal. We also construct optimal self-orthogonal <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$[n], [6]$ </tex-math></inline-formula> codes when <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$41 le n le 256$ </tex-math></inline-formula> satisfies <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$n ne 46, 54, 61$ </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>$n equiv !!!!!/~7, 14, 22, 29, 38, 45, 53, 60 pmod {63}$ </tex-math></inline-formula> ." @default.
- W4285202422 created "2022-07-14" @default.
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- W4285202422 date "2022-11-01" @default.
- W4285202422 modified "2023-09-27" @default.
- W4285202422 title "Self-Orthogonality Matrix and Reed-Muller Codes" @default.
- W4285202422 doi "https://doi.org/10.1109/tit.2022.3186316" @default.
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