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On regular sets of affine type in finite Desarguesian planes and related codes

Angela Aguglia, Bence Csajbók, Luca Giuzzi

TL;DR

This work studies regular affine-type point sets in finite Desarguesian planes, introducing broad construction methods that lift small examples to larger fields and combine algebraic curves with incidence geometry. A central result identifies a family of curves $\Gamma_a$ defined by $\mathrm{Tr}(y+ a x^{\sqrt{q}}) = \mathrm{N}(x)$, for square $q$, as yielding regular pointed-type sets in $\text{PG}(2,q^2)$ with four affine types $[q; q-2\sqrt{q}+1, q-\sqrt{q}+1, q+1, q+\sqrt{q}+1]$, and demonstrates that vertical lines meet in $q$ points while non-vertical lines meet in one of these four counts. Theorem main-1 follows by linking these intersection patterns to those with the Hermitian curve, via a system of equations that preserves the same affine-count structure. As a byproduct, the authors construct $\sqrt{q}$-divisible projective codes with five nonzero weights from $\Gamma_a$, and provide weight enumerator behavior modulo powers of $q$, highlighting a strong geometry–coding theory correspondence and yielding new infinite families of regular sets. These results advance understanding of incidence configurations in Desarguesian planes and their coding-theoretic implications.

Abstract

In this paper, we consider point sets of finite Desarguesian planes whose multisets of intersection numbers with lines is the same for all but one exceptional parallel class of lines. We call such sets regular of affine type. When the lines of the exceptional parallel class have the same intersection numbers, then we call these sets regular of pointed type. Classical examples are e.g. unitals; a detailed study and constructions of such sets with few intersection numbers is due to Hirschfeld and Szőnyi from 1991. We here provide some general construction methods for regular sets and describe a few infinite families. The members of one of these families have the size of a unital and meet affine lines of $\mathrm{PG}(2, q^2)$ in one of $4$ possible intersection numbers, each of them congruent to $1$ modulo $\sqrt{q}$. As a byproduct, we determine the intersection sizes of the Hermitian curve defined over $\mathrm{GF}(q^2)$ with suitable rational curves of degree $\sqrt{q}$ and we obtain $\sqrt{q}$-divisible codes with $5$ non-zero weights. We also determine the weight enumerator of the codes arising from the general constructions modulus some $q$-powers.

On regular sets of affine type in finite Desarguesian planes and related codes

TL;DR

This work studies regular affine-type point sets in finite Desarguesian planes, introducing broad construction methods that lift small examples to larger fields and combine algebraic curves with incidence geometry. A central result identifies a family of curves defined by , for square , as yielding regular pointed-type sets in with four affine types , and demonstrates that vertical lines meet in points while non-vertical lines meet in one of these four counts. Theorem main-1 follows by linking these intersection patterns to those with the Hermitian curve, via a system of equations that preserves the same affine-count structure. As a byproduct, the authors construct -divisible projective codes with five nonzero weights from , and provide weight enumerator behavior modulo powers of , highlighting a strong geometry–coding theory correspondence and yielding new infinite families of regular sets. These results advance understanding of incidence configurations in Desarguesian planes and their coding-theoretic implications.

Abstract

In this paper, we consider point sets of finite Desarguesian planes whose multisets of intersection numbers with lines is the same for all but one exceptional parallel class of lines. We call such sets regular of affine type. When the lines of the exceptional parallel class have the same intersection numbers, then we call these sets regular of pointed type. Classical examples are e.g. unitals; a detailed study and constructions of such sets with few intersection numbers is due to Hirschfeld and Szőnyi from 1991. We here provide some general construction methods for regular sets and describe a few infinite families. The members of one of these families have the size of a unital and meet affine lines of in one of possible intersection numbers, each of them congruent to modulo . As a byproduct, we determine the intersection sizes of the Hermitian curve defined over with suitable rational curves of degree and we obtain -divisible codes with non-zero weights. We also determine the weight enumerator of the codes arising from the general constructions modulus some -powers.
Paper Structure (8 sections, 9 theorems, 67 equations)

This paper contains 8 sections, 9 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. General constructions of regular sets of affine type
  3. Proofs of Theorems \ref{['t:32']} and \ref{['main-1']}
  4. Proof of Theorem \ref{['t:32']}
  5. The odd $q$ case.
  6. The even $q$ case.
  7. Proof of Theorem \ref{['main-1']}
  8. A class of $\sqrt{q}$-divisible codes over $\mathrm{GF}(q^2)$ and codes with a simple weight enumerator modulus a $q$-power

Key Result

Theorem 1.2

Let $q$ be a square prime power and $a\in \mathrm{GF}(q^2)^*$. Let $\Gamma_{a}$ denote the algebraic plane curve of affine equation Then the set of projective points of $\Gamma_a$ in $\mathrm{PG}(2,q^2)$ is a regular $(q^3+1)$-set of pointed type

Theorems & Definitions (18)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Example 2.4
  • Theorem 2.5
  • proof
  • ...and 8 more