  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 $|E|>\frac{q^{d-1}}{d}\binom{d}{d/2}\binom{d/2}{d/4}$, then the points of
$E$ determine a Hamming distance $r$ for every even $r$.
