Two-Distance Sets over Finite Fields

TL;DR

This paper demonstrates that Blokhuis' quadratic upper bound for two-distance sets over finite fields is nearly optimal in most dimensions, providing new constructions that complement existing higher-dimensional maximal sets.

Contribution

It proves the sharpness of Blokhuis' bound over finite fields and introduces new constructions that extend previous Lorentz space results.

Findings

Blokhuis' quadratic upper bound is sharp in almost all dimensions over finite fields.

New constructions complement existing maximal sets in Lorentz spaces.

The results advance understanding of two-distance sets in finite field geometries.

Abstract

We show that Blokhuis' quadratic upper bound for two-distance sets is sharp over finite fields in almost all dimensions. Our construction complements Lison\v{e}k's higher-dimensional maximal constructions that were carried out in Lorentz spaces.