Asymptotically optimal Boolean functions

TL;DR

This paper proves a long-standing conjecture by determining the asymptotic behavior of the covering radius of Reed-Muller codes, revealing how Boolean functions can be approximated by affine functions as the number of variables grows.

Contribution

It establishes the exact asymptotic limit of the covering radius for Reed-Muller codes, resolving a conjecture from 1983.

Findings

The limit of the normalized covering radius is 1 as n approaches infinity.

The result characterizes the asymptotic optimality of Boolean functions in relation to affine functions.

It provides a precise asymptotic measure of how well Boolean functions can be approximated by affine functions.

Abstract

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.