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.
Boolean functions are widely used in cryptography and coding theory. A key challenge lies in finding the weight spectrum for all $n$-variable Boolean functions of degree at most $r$, also known as the Reed--Muller code $\RM(r,n)$. In 2023, Carlet and Solé determined the weight spectra of $\RM(n-c,n)$ for $c=3,4$, and conjectured the weight spectrum of $\RM(n-c, n$) for $n\ge2c$. The weight spectra of $\RM(n-5, n)$ and $\RM(n-6,n)$ were determined by Carlet in 2024 and by Lou and Wang in 2025, respectively, confirming the Carlet-Solé conjecture for the cases $c=5,6$. Very recently, Leuenberger et al. and Wei et al. investigated the weight spectrum of $\RM(7,14)$ independently, and there are eight potential weights which have not been determined. Up until now, the weight spectra of the Reed--Muller codes $RM(r,n)$ are still unknown for $r=3,...,n-7$. In this paper, we determine the exact weight spectrum of $\RM(m-7, m)$ for every $m\ge 14$. By combining affine geometry and Fourier analysis, we exclude the conjectured eight weights for $\RM(7,14)$. We then extend the exclusions to all higher dimensions using a geometric transfer. Consequently, we provide a counterexample to the Carlet-Solé conjecture.