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A note on strong blocking sets and higgledy-piggledy sets of lines

Stefano Lia, Geertrui Van de Voorde

Abstract

This paper studies {\em strong blocking sets} in the $N$-dimensional finite projective space $\mathrm{PG}(N,q)$. We first show that certain unions of blocking sets cannot form strong blocking sets, which leads to a new lower bound on the size of a strong blocking set in $\mathrm{PG}(N,q)$. Our second main result shows that, for $q>\frac{2}{ln(2)}(N+1)$, there exists a subset of $2N-2$ lines of a Desarguesian line spread in $\mathrm{PG}(N,q)$, $N$ odd, in {\em higgledy-piggledy arrangement}; thus giving rise to a strong blocking set of size $(2N-2)(q+1)$.

A note on strong blocking sets and higgledy-piggledy sets of lines

Abstract

This paper studies {\em strong blocking sets} in the -dimensional finite projective space . We first show that certain unions of blocking sets cannot form strong blocking sets, which leads to a new lower bound on the size of a strong blocking set in . Our second main result shows that, for , there exists a subset of lines of a Desarguesian line spread in , odd, in {\em higgledy-piggledy arrangement}; thus giving rise to a strong blocking set of size .
Paper Structure (5 sections, 7 theorems, 20 equations)

This paper contains 5 sections, 7 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Strong blocking sets and linear codes
  3. Strong blocking sets and higgledy-piggledy sets
  4. Strong blocking sets arising from the union of blocking sets
  5. Construction of a higgledy-piggledy set of lines of size arising from a spread

Key Result

Lemma 2.2

Let $q\geq N^2$, $N\geq 3$. Let $\mathcal{S}=\bigcup_{i=1}^k B_i$ be a strong blocking set of $\mathrm{PG}(N,q)$, given by the union of $k$ disjoint minimal blocking sets $B_i$ that are either lines are Baer subplanes. Then no hyperplane contains all but at most $N-1$ of the sets $B_i$.

Theorems & Definitions (14)

  • Remark 1.3
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • Theorem 2.6
  • proof
  • Corollary 2.7
  • Remark 2.8
  • ...and 4 more