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We prove the covering radius of the third-order Reed-Muller code RM(3,7) is20, which was previously known to be between 20 and 23 (inclusive). Thecovering radius of RM(3, 7) is the maximum third-order nonlinearity among all7-variable Boolean functions. It was known that there exist 7-variable Booleanfunctions with third-order nonlinearity 20. We prove the third-ordernonlinearity cannot achieve 21. According to the classification of the quotientspace of RM(6,6)/RM(3,6), we classify all 7-variable Boolean functions into 66types. Firstly, we prove 62 types (among 66) cannot have third-ordernonlinearity 21; Secondly, we prove function of the remaining 4 types can betransformed into a type (6, 10) function, if its third-order nonlinearity is21; Finally, we transform type (6, 10) functions into a specific form, andprove the functions in that form cannot achieve third-order nonlinearity 21(with the assistance of computers). By the way, we prove that the affinetransformation group over any finite field can be generated by two elements.

The Covering Radius of the Third-Order Reed-Muller Code RM(3,7) is 20