  It was proved by J. Schatz that the covering radius of the second order
Reed--Muller code $RM(2, 6)$ is 18 (IEEE Trans Inf Theory 27: 529--530, 1985).
However, the covering radius of $RM(2,7)$ has been an open problem for many
years. In this paper, we prove that the covering radius of $RM(2,7)$ is 40,
which is the same as the covering radius of $RM(2,7)$ in $RM(3,7)$. As a
corollary, we also find new upper bounds for $RM(2,n)$, $n=8,9,10$.
