  We present methods for computing the distance from a Boolean polynomial on
$m$ variables of degree $m-3$ (i.e., a member of the Reed-Muller code
$RM(m-3,m)$) to the space of lower-degree polynomials ($RM(m-4,m)$). 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 $RM(4,8)$ in $RM(5,8)$ is 26
and the covering radius of $RM(5,9)$ in $RM(6,9)$ is between 28 and 32
inclusive, and we get improved lower bounds for higher~$m$. We also apply our
methods to various polynomials in the literature, thereby improving the known
bounds on the distance from 2-resilient polynomials to $RM(m-4,m)$.
