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

On the Covering Radius of the Second Order Reed-Muller Code of Length 128