Erdös--Falconer Distance Problem under Hamming Metric in Vector Spaces over Finite Fields

Abstract

For a subset $I\subseteq \mathbb{F}_{q}^{n}$, let $\Delta(I)$ be the set of distances determined by the elements of $I.$ The Erdös--Falconer distance problem in $\mathbb{F}_{q}^{n}$ asks for a threshold on the cardinality $|I|$ so that $\Delta(I)$ contains a positive proportion of the whole distance set. In this paper, we consider the analogous question under Hamming distance, which is the most important metric in coding theory. When $q\geqslant 4$ is a fixed prime power and $n$ goes to infinity, our main result shows that, for arbitrary positive proportion $\alpha,$ we can find $\alpha n$ distinct Hamming distances in $\Delta(I)$ if $|I|>q^{(1-\beta)\cdot n},$ where $\beta$ is a positive number depending on $\alpha.$ Unlike using Fourier analytical method as usual, our main tools include the celebrated dependent random choice and some results from additive number theory and coding theory. Hence our bound is much smaller than the previously known bound which was obtained by Fourier analytic machinery.