  In 1981, Schatz proved that the covering radius of the binary Reed-Muller
code $RM(2,6)$ is 18. For $RM(2,7)$, we only know that its covering radius is
between 40 and 44. In this paper, we prove that the covering radius of the
binary Reed-Muller code $RM(2,7)$ is at most 42. Moreover, we give a sufficient
and necessary condition for Boolean functions of 7-variable to achieve the
second-order nonlinearity 42.
