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Transitive $(q-1)$-fold packings of $\rm{PG}_n(q)$

Daniel R. Hawtin

Abstract

A $t$-fold packing of a projective space $\rm{PG}_n(q)$ is a collection $\mathcal{P}$ of line-spreads such that each line of $\rm{PG}_n(q)$ occurs in precisely $t$ spreads in $\mathcal{P}$. A $t$-fold packing $\mathcal{P}$ is transitive if a subgroup of $\rm{PΓL}_{n+1}(q)$ preserves and acts transitively on $\mathcal{P}$. We give a construction for a transitive $(q-1)$-fold packing of $\rm{PG}_n(q)$, where $q=2^k$, for any odd positive integers $n$ and $k$, such that $n\geq 3$. This generalises a construction of Baker from 1976 for the case $q=2$.

Transitive $(q-1)$-fold packings of $\rm{PG}_n(q)$

Abstract

A -fold packing of a projective space is a collection of line-spreads such that each line of occurs in precisely spreads in . A -fold packing is transitive if a subgroup of preserves and acts transitively on . We give a construction for a transitive -fold packing of , where , for any odd positive integers and , such that . This generalises a construction of Baker from 1976 for the case .
Paper Structure (2 sections, 8 theorems, 13 equations)

This paper contains 2 sections, 8 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. The construction

Key Result

Theorem 1.1

Let $n$ and $k$ be odd positive integers, with $n\geqslant 3$, let $q=2^k$, and let $\mathcal{P}$ be as in (eq:partition). Then $\mathcal{P}$ is a transitive $(q-1)$-fold packing of $\mathop{\mathrm{PG}}\nolimits_n(q)$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 6 more