A note on the distinct distances problem over finite fields

2025-10-12

AI Summary

This paper investigates a variant of the Erdős distinct distances problem under the Hamming metric over the finite vector space $mathbb{F}_q^n$: given a set $S subseteq mathbb{F}_q^n$, determine a lower bound on the number $|Delta(S)|$ of distinct Hamming distances determined by $S$, and examine the existence of large rainbow subsets—i.e., subsets in which all pairwise Hamming distances are distinct. Employing combinatorial analysis, structural properties of finite fields, and explicit constructions, the authors establish a tight asymptotic lower bound $|Delta(S)| ge frac{log |S|}{2log(2nq)}$. They further demonstrate that high distance diversity does not guarantee the existence of large rainbow subsets. Moreover, they constructively prove that every sufficiently large $S$ contains a nontrivial rainbow subset. The main contributions are: (i) the first precise asymptotic lower bound on the number of distinct Hamming distances in this setting; and (ii) the revelation of a fundamental non-implication between distance richness and rainbow structure.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Other Foundations of Machine Learning
• Data Mining & Knowledge Management: Other Foundations of Data Mining & Knowledge Management

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study a finite-field analogue of the Erdős distinct distances problem under the Hamming metric. For a set (Ssubseteq 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|}{2log(2nq)}, ] and show this bound is tight when (|S|=O( ext{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 (Ssubseteq mathbb{F}_q^n) for which all (inom{|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.

Problem

• Studies distinct Hamming distances in finite fields using lower bounds.
• Investigates rainbow sets with pairwise distinct Hamming distances.
• Constructs counterexamples showing distinct distances don't guarantee rainbow sets.

Innovation

• Lower bound on Hamming distances in finite fields
• Tight bound for polynomial-sized subsets
• Existence of large rainbow sets in finite fields