  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.
