# Highly nonlinear functions over finite fields

## TL;DR

This paper proves a long-standing conjecture about the maximum Hamming distance of functions from finite fields to finite fields, extending previous results and determining the asymptotic behavior of Reed-Muller codes.

## Contribution

It generalizes the Patterson-Wiedemann conjecture to all finite fields, using advanced number theory and probabilistic methods, and determines the asymptotic covering radius.

## Key findings

- Proves the conjecture for most finite fields unconditionally.
- Establishes the asymptotic maximum distance for functions as dimension grows.
- Determines the asymptotic covering radius of Reed-Muller codes.

## Abstract

We consider a generalisation of a conjecture by Patterson and Wiedemann from 1983 on the Hamming distance of a function from $\mathbb{F}_q^n$ to $\mathbb{F}_q$ to the set of affine functions from $\mathbb{F}_q^n$ to $\mathbb{F}_q$. We prove the conjecture for each $q$ such that the characteristic of $\mathbb{F}_q$ lies in a subset of the primes with density $1$ and we prove the conjecture for all $q$ by assuming the generalised Riemann hypothesis. Roughly speaking, we show the existence of functions for which the distance to the affine functions is maximised when $n$ tends to infinity. This also determines the asymptotic behaviour of the covering radius of the $[q^n,n+1]$ Reed-Muller code over $\mathbb{F}_q$ and so answers a question raised by Leducq in 2013. Our results extend the case $q=2$, which was recently proved by the author and which corresponds to the original conjecture by Patterson and Wiedemann. Our proof combines evaluations of Gauss sums in the semiprimitive case, probabilistic arguments, and methods from discrepancy theory.