Transitive $(q-1)$-fold packings of ${\mathrm{PG}}_{\mathrm{n}}(\mathrm{q})$

Abstract

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

arXiv comment5 pages