Transitive (q − 1)-fold packings of PGn(q)

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