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• W4287888918 abstract "The binary quadratic-residue codes and the punctured Reed-Muller codes <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>${mathcal {R}}_{2}((m-1)/2, m))$ </tex-math></inline-formula> are two families of binary cyclic codes with parameters <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$[n, (n+1)/2, d geq sqrt {n}]$ </tex-math></inline-formula> . These two families of binary cyclic codes are interesting partly due to the fact that their minimum distances have a square-root bound. The objective of this paper is to construct two families of binary cyclic codes of length <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$2^{m}-1$ </tex-math></inline-formula> and dimension near <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$2^{m-1}$ </tex-math></inline-formula> with good minimum distances. When <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$m geq 3$ </tex-math></inline-formula> is odd, the codes become a family of duadic codes with parameters <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$[2^{m}-1, 2^{m-1}, d]$ </tex-math></inline-formula> , where <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$d geq 2^{(m-1)/2}+1$ </tex-math></inline-formula> if <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$m equiv 3 pmod {4}$ </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>$d geq 2^{(m-1)/2}+3$ </tex-math></inline-formula> if <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$m equiv 1 pmod {4}$ </tex-math></inline-formula> . The two families of binary cyclic codes contain some optimal binary cyclic codes." @default.
• W4287888918 created "2022-07-26" @default.
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• W4287888918 date "2022-12-01" @default.
• W4287888918 modified "2023-10-15" @default.
• W4287888918 title "Binary [<i>n</i>, (<i>n</i> + 1)/2] Cyclic Codes With Good Minimum Distances" @default.
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• W4287888918 doi "https://doi.org/10.1109/tit.2022.3193715" @default.
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