Hamming Distances in Vector Spaces over Finite Fields

Abstract

Let ${\mathbb{F}}_q$ be the finite field of order $q$ and $E\subset {\mathbb{F}}_q^d$, where $4|d$. Using Fourier analytic techniques, we prove that if $|E|>\frac{q^{d-1}}{d}\left(\genfrac{}{}{0ex}{}{d}{d/2}\right)\left(\genfrac{}{}{0ex}{}{d/2}{d/4}\right)$, then the points of $E$ determine a Hamming distance $r$ for every even $r$.