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Small 3-fold blocking sets in $\mathrm{PG}(2,p^n)$

Bence Csajbók, Máté Róbert Kepes, Eszter Robin, Bence Sógor, Sherry Wang, Elias Williams

TL;DR

This work advances the understanding of minimal $t$-fold blocking sets in $ ext{PG}(2,p^n)$ by constructing explicit $3$-fold blocking sets of the conjectured size $3(p^n+p^{n(s-1)/s}+1)$. The approach combines three disjoint Rédei-type linear blocking sets, obtained as trace-type constructions and organized via the order-3 projectivity $(x:y:z) o(z:x:y)$, to realize a 3-fold blocking configuration on a common orbit. The authors first establish 2-fold blocking sets through disjoint trace-type copies, then extend the method to three copies, handling cases $h$ and $h=3$ with targeted algebraic arguments and existence proofs. The results provide concrete constructions that align with conjectured lower bounds, offering new tools for applications in coding theory and finite geometry, and illustrating the tightness of lower-bound heuristics under subfield size considerations. All mathematical notation is presented with explicit $ $ delimiters for clarity and searchability.

Abstract

A $t$-fold blocking set of the finite Desarguesian plane $\mathrm{PG}(2,p^n)$, $p$ prime, is a set of points meeting each line of the plane in at least $t$ points. The minimum size of such sets is of interest for numerous reasons; however, even the minimum size of nontrivial blocking sets (i.e. $1$-fold blocking sets not containing a line) in \(\mathrm{PG}(2,p^n)\) is an open question when $n\geq 5$ is odd. For $n>1$ the conjectured lower bound for this size is $(p^n+p^{n(s-1)/s}+1)$, where $p^{n/s}$ is the size of the largest proper subfield of $\mathbb{F}_{p^n}$. Since the union of $t$ pairwise disjoint nontrivial blocking sets is a $t$-fold blocking set, it is conjectured that when $p^{n/s}$ is large enough w.r.t. $t$, then the minimum size of a $t$-fold blocking set in $\mathrm{PG}(2,p^n)$ is $t(p^n+p^{n(s-1)/s}+1)$. If $n$ is even, then the decomposition of the plane into disjoint Baer subplanes gives a $t$-fold blocking set of this size. However, for odd $n$, the existence of such sets is an unsolved problem in most cases. In this paper, we construct $3$-fold blocking sets of conjectured size. These blocking sets are obtained as the disjoint union of three linear blocking sets of Rédei type, and they lie on the same orbit of the projectivity $(x:y:z)\mapsto (z:x:y)$.

Small 3-fold blocking sets in $\mathrm{PG}(2,p^n)$

TL;DR

This work advances the understanding of minimal -fold blocking sets in by constructing explicit -fold blocking sets of the conjectured size . The approach combines three disjoint Rédei-type linear blocking sets, obtained as trace-type constructions and organized via the order-3 projectivity , to realize a 3-fold blocking configuration on a common orbit. The authors first establish 2-fold blocking sets through disjoint trace-type copies, then extend the method to three copies, handling cases and with targeted algebraic arguments and existence proofs. The results provide concrete constructions that align with conjectured lower bounds, offering new tools for applications in coding theory and finite geometry, and illustrating the tightness of lower-bound heuristics under subfield size considerations. All mathematical notation is presented with explicit delimiters for clarity and searchability.

Abstract

A -fold blocking set of the finite Desarguesian plane , prime, is a set of points meeting each line of the plane in at least points. The minimum size of such sets is of interest for numerous reasons; however, even the minimum size of nontrivial blocking sets (i.e. -fold blocking sets not containing a line) in \(\mathrm{PG}(2,p^n)\) is an open question when is odd. For the conjectured lower bound for this size is , where is the size of the largest proper subfield of . Since the union of pairwise disjoint nontrivial blocking sets is a -fold blocking set, it is conjectured that when is large enough w.r.t. , then the minimum size of a -fold blocking set in is . If is even, then the decomposition of the plane into disjoint Baer subplanes gives a -fold blocking set of this size. However, for odd , the existence of such sets is an unsolved problem in most cases. In this paper, we construct -fold blocking sets of conjectured size. These blocking sets are obtained as the disjoint union of three linear blocking sets of Rédei type, and they lie on the same orbit of the projectivity .
Paper Structure (10 sections, 19 theorems, 45 equations, 1 figure)

This paper contains 10 sections, 19 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Blocking sets and multiple blocking sets containing lines
  4. Linear blocking sets
  5. The upper chromatic number of finite projective planes
  6. Construction of two disjoint blocking sets
  7. 3-fold blocking sets
  8. Preliminaries to the main construction
  9. The case of h>=4.
  10. The case of h=3

Key Result

Theorem 1.2

For each prime $p$ and integer $n>1$, if $s$ denotes the smallest prime divisor of $n$, then there exist $t$-fold blocking sets of the conjectured size $t(p^n+p^{n(s-1)/s}+1)$ in $\mathop{\mathrm{PG}}\nolimits(2,p^n)$ for $t=2,3$.

Figures (1)

  • Figure 1: $3$-fold blocking set of size $4q$

Theorems & Definitions (42)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4: The trace construction
  • Proposition 2.5
  • proof
  • Definition 2.6
  • ...and 32 more