Asymptotically optimal Boolean functions

The largest Hamming distance between a Boolean function in $n$ variables and the set of all affine Boolean functions in $n$ variables is known as the covering radius $\rho_n$ of the $[2^n,n+1]$ Reed-Muller code. This number determines how well Boolean functions can be approximated by linear Boolean functions. We prove that $$\lim_{n\to\infty}2^{n/2}-\rho_n/2^{n/2-1}=1,$$ which resolves a conjecture due to Patterson and Wiedemann from 1983.