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 $$\underset{n\to \mathrm{\infty }}{\lim }{2}^{n\mathrm{/}2}-{\rho }_n\mathrm{/}{2}^{n\mathrm{/}2-1}=1,$$ which resolves a conjecture due to Patterson and Wiedemann from 1983.