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

We prove the covering radius of the third-order Reed-Muller code RM(3,7) is 20, which was known to be between 20 and 23 previously. The covering radius of RM(3, 7) is the maximum third-order nonlinearity among all 7-variable Boolean functions. It was known that there exist 7-variable Boolean functions with third-order nonlinearity 20. We prove the third-order nonlinearity cannot achieve 21. Because a 7-variable Boolean function is the concatenation of two 6-variable Boolean functions, our proof relies on the classification of RM(6,6)/RM(3,6); we classify all 7-variable Boolean functions into 66 types accordingly. Firstly, we prove 62 types (among 66) cannot have third-order nonlinearity 21; Secondly, we prove function of the remaining 4 types can be transformed into a type (6, 10) function, if its third-order nonlinearity is 21; Finally, we transform type (6, 10) functions into a specific form, and prove the functions in that form cannot achieve third-order nonlinearity 21 (with the assistance of computers).