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• W3211598911 abstract "We present methods for computing the distance from a Boolean polynomial on <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$m$ </tex-math></inline-formula> variables of degree <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$m-3$ </tex-math></inline-formula> (i.e., a member of the 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>$RM(m-3, m)$ </tex-math></inline-formula> ) to the space of lower-degree polynomials ( <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$RM(m-4, m)$ </tex-math></inline-formula> ). The methods give verifiable certificates for both the lower and upper bounds on this distance. By applying these methods to representative lists of polynomials, we show that the covering radius of <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$RM(4,8)$ </tex-math></inline-formula> in <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$RM(5,8)$ </tex-math></inline-formula> is 26 and the covering radius of <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$RM(5,9)$ </tex-math></inline-formula> in <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$RM(6,9)$ </tex-math></inline-formula> is between 28 and 32 inclusive, and we get improved lower bounds for higher <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$m$ </tex-math></inline-formula> . We also apply our methods to various polynomials in the literature, thereby improving the known bounds on the distance from 2-resilient polynomials to <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$RM(m-4, m)$ </tex-math></inline-formula> ." @default.
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• W3211598911 date "2022-01-01" @default.
• W3211598911 modified "2023-10-16" @default.
• W3211598911 title "The Covering Radius of the Reed–Muller Code <i>RM</i>(<i>m</i> – 4, <i>m</i>) in <i>RM</i>(<i>m</i> – 3, <i>m</i>)" @default.
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• W3211598911 doi "https://doi.org/10.1109/tit.2021.3120754" @default.
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