Hamming Distances in Vector Spaces over Finite Fields

TL;DR

This paper uses Fourier analysis to establish conditions under which a subset of a finite field vector space determines all even Hamming distances, revealing new insights into the structure of such sets.

Contribution

It provides a novel Fourier analytic approach to characterize when subsets of finite field vector spaces determine all even Hamming distances.

Findings

Sets larger than a specific size determine all even Hamming distances.

Fourier analysis techniques are effective in studying Hamming distances in finite fields.

The result applies to vector spaces where the dimension is divisible by four.

Abstract

Let $\mathbb{F}_q$ be the finite field of order $q$ and $E\subset \mathbb{F}_q^d$, where $4|d$. Using Fourier analytic techniques, we prove that if $|E|>\frac{q^{d-1}}{d}\binom{d}{d/2}\binom{d/2}{d/4}$, then the points of $E$ determine a Hamming distance $r$ for every even $r$.