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