A note on the distinct distances problem over finite fields

TL;DR

This work studies a finite-field analogue of Erd\H{o}s's distinct distances problem under the Hamming metric, deriving a universal lower bound $|\Delta(S)| \ge \frac{\log |S|}{2\log(2nq)}$ and establishing tightness for sets of size polynomial in $n$. It leverages coding-theoretic tools, notably $q$-ary simplex codes and distance-preserving embeddings, to prove the bound and to construct sets with only a few distinct distances. The rainbow-sets analysis reveals that large sets with many distances do not necessarily yield large rainbow subsets, but positive results show the existence of sizeable rainbow subsets in $\mathbb{F}_2^n$ and that every large $S$ contains a nontrivial rainbow subset; these results are obtained through a blend of extremal combinatorics (Frankl–Wilson, Ray-Chaudhuri–Wilson) and classical coding theory, plus sum-distinctness arguments. Overall, the paper advances understanding of distance-structure phenomena over finite fields and links them to foundational coding-theoretic constructions and combinatorial results with potential implications for finite-field geometry and coding theory.

Abstract

We study a finite-field analogue of the Erdős distinct distances problem under the Hamming metric. For a set \(S\subseteq \mathbb{F}_q^n\) let $Δ(S)$ denote the set of Hamming distances determined by \(S\). We prove the lower bound \[ |Δ(S)| \;\ge\; \frac{\log |S|}{2\log(2nq)}, \] 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.

Theorems & Definitions (17)

• Theorem 1: xu2020erdos
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• proof : Proof of Theorem \ref{['thm:exist_few_distances']}
• Lemma 7
• Theorem 8: ray1975t
• proof : Proof of Lemma \ref{['thm:binary:large_set_many_distances']}