A note on the distinct distances problem over finite fields

We study a finite-field analogue of the Erd\H{o}s distinct distances problem under the Hamming metric. For a set $S\subseteq \mathbb{F}_q^n$ let $\Delta(S)$ denote the set of Hamming distances determined by $S$. We prove the lower bound $$|\Delta(S)| \;\ge\; \frac{\log |S|}{2\log(2nq)},$$ and show this bound is tight when $|S|=O(\text{poly}(n))$, where the constant of proportionality depends only on $q$. We then also study the problem of finding a large \emph{rainbow set}, that is, a subset $S\subseteq \mathbb{F}_q^n$ for which all $\binom{|S|}{2}$ pairwise Hamming distances spanned by $S$ are distinct. In contrast to the Euclidean setting, we show that a set with many distinct distances does not imply the existence of a large rainbow set, by giving an explicit construction. Nevertheless, we establish the existence of large rainbow sets, and prove that every large set in $\mathbb{F}_q^n$ necessarily contains a non-trivial rainbow subset.