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.

翻译：我们研究了Erdős不同距离问题在有限域上的Hamming度量类比。对于集合\(S\subseteq \mathbb{F}_q^n\)，令$\Delta(S)$表示由\(S\)确定的Hamming距离集合。我们证明了以下下界：\[ |\Delta(S)| \;\ge\; \frac{\log |S|}{2\log(2nq)}, \] 并说明当\(|S|=O(\text{poly}(n))\)时该界是紧的，其中比例常数仅依赖于$q$。随后我们还研究了寻找大\emph{彩虹集}的问题，即寻找子集\(S\subseteq \mathbb{F}_q^n\)使得$S$张成的所有\(\binom{|S|}{2}\)个两两Hamming距离互不相同。与欧氏空间情形相反，我们通过显式构造证明：具有大量不同距离的集合并不蕴含大彩虹集的存在性。尽管如此，我们确立了大彩虹集的存在性，并证明了\(\mathbb{F}_q^n\)中的每个大集合必然包含一个非平凡的彩虹子集。