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We present methods for computing the distance from a Boolean polynomial on <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> variables of degree <inline-formula> <tex-math notation="LaTeX">$m-3$ </tex-math></inline-formula> (i.e., a member of the Reed&#x2013;Muller code <inline-formula> <tex-math notation="LaTeX">$RM(m-3, m)$ </tex-math></inline-formula>) to the space of lower-degree polynomials (<inline-formula> <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> <tex-math notation="LaTeX">$RM(4,8)$ </tex-math></inline-formula> in <inline-formula> <tex-math notation="LaTeX">$RM(5,8)$ </tex-math></inline-formula> is 26 and the covering radius of <inline-formula> <tex-math notation="LaTeX">$RM(5,9)$ </tex-math></inline-formula> in <inline-formula> <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> <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> <tex-math notation="LaTeX">$RM(m-4, m)$ </tex-math></inline-formula>.

ieee transactions on information theory

The Covering Radius of the Reed&#x2013;Muller Code <italic>RM</italic>(<italic>m</italic> &#x2013; 4, <italic>m</italic>) in <italic>RM</italic>(<italic>m</italic> &#x2013; 3, <italic>m</italic>)

The Covering Radius of the Reed–Muller Code RM(m – 4, m) in RM(m – 3, m)