Hamming Distances in Vector Spaces over Finite Fields

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 $\mathrm{\mid }E\mathrm{\mid }>\frac{q^{d-1}}{d}\left(\genfrac{}{}{0px}{}{d}{d\mathrm{/}2}\right)\left(\genfrac{}{}{0px}{}{d\mathrm{/}2}{d\mathrm{/}4}\right)$, then the points of $E$ determine a Hamming distance $r$ for every even $r$.