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Two-Distance Sets over Finite Fields

Jozsef Solymosi

TL;DR

This work investigates two-distance and equilateral point sets in finite-field geometries endowed with the standard quadratic distance dist$^2$ over $\mathbb{F}_p^d$. It shows that Blokhuis' quadratic upper bound $|X|\le\binom{d+2}{2}$ is sharp in the modular regime $p\mid(d+2)$ via a rank-collapse phenomenon of the Gram matrix and explicit constructions. The paper provides a two-pronged construction: (i) explicit $d+2$ equilateral sets when $p\mid(d+2)$, and (ii) a midpoint transformation that converts these seeds into two-distance sets of size $\binom{d+2}{2}$, achieving the bound. This sharpness contrasts with the Euclidean case and mirrors similar phenomena in Lorentz spaces, illustrating how finite-field geometry permits extremal configurations unavailable in positive definite settings.

Abstract

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

Two-Distance Sets over Finite Fields

TL;DR

This work investigates two-distance and equilateral point sets in finite-field geometries endowed with the standard quadratic distance dist over . It shows that Blokhuis' quadratic upper bound is sharp in the modular regime via a rank-collapse phenomenon of the Gram matrix and explicit constructions. The paper provides a two-pronged construction: (i) explicit equilateral sets when , and (ii) a midpoint transformation that converts these seeds into two-distance sets of size , achieving the bound. This sharpness contrasts with the Euclidean case and mirrors similar phenomena in Lorentz spaces, illustrating how finite-field geometry permits extremal configurations unavailable in positive definite settings.

Abstract

We show that Blokhuis' quadratic upper bound for two-distance sets is sharp over finite fields in almost all dimensions. Our construction complements Lisoněk's higher-dimensional maximal constructions that were carried out in Lorentz spaces.
Paper Structure (18 sections, 13 theorems, 94 equations)

This paper contains 18 sections, 13 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Distances over finite fields
  3. Equilateral sets and the $d+2$ phenomenon
  4. Midpoints and two distances
  5. Sharpness of Blokhuis' bound
  6. Graph-theoretic description.
  7. Acknowledgements
  8. Quadratic distance over Fp
  9. The Gram-matrix bound and the modular exception
  10. An explicit $d{+}2$ equilateral construction when p|(d+2)
  11. Midpoints (halving points) give a two-distance set
  12. Explicit equilateral and midpoint constructions for $p=3,5,7$
  13. Setup.
  14. The Gram-matrix bound: equilateral sets have size $\le d+1$ except in the modular case
  15. An explicit equilateral $(d+2)$-set for $(p,d)=(7,5)$
  16. ...and 3 more sections

Key Result

Lemma 1

For two distinct midpoints $M_{ij}\neq M_{k\ell}$ one has More precisely, if the edges $\{i,j\}$ and $\{k,\ell\}$ share a vertex then $\operatorname{dist}^2(M_{ij},M_{k\ell})=\Delta/4$, while if they are disjoint then $\operatorname{dist}^2(M_{ij},M_{k\ell})=\Delta/2$.

Theorems & Definitions (30)

  • Lemma 1
  • proof
  • Lemma 2: Gram matrix of an equilateral set
  • proof
  • Lemma 3: Rank of $I+J$ over $\mathbb{F}_p$
  • proof
  • Theorem 1: Equilateral size bound; exceptional case
  • proof
  • Proposition 1: $d{+}2$ equilateral points for $p\mid(d+2)$
  • proof
  • ...and 20 more