The Maximum Nonlinearity of 8-variable Balanced Boolean Functions is 116

Does there exist an 8-variable balanced Boolean function with nonlinearity 118? This is a famous and long-standing open problem in Boolean functions. In this paper, we give a negative answer to this open problem, and prove that the maximum nonlinearity of balanced Boolean functions in 8 variables is 116, establishing the 8-variable instance of Dobbertin's conjectured recursion for the minimum Walsh peak of balanced functions. Equivalently, the covering radius of the Reed--Muller code RM(1,8) in 0-resilient Boolean functions is 116. We deduce the result by obtaining a quadratic form of polar rank 6, and reducing the problem to the existence of a 6-variable Boolean function with nonlinearity greater than 26 whose Hamming distance to a bent function is between 4 and 6.